one version of the "top 300" sequences (very long message)

[Joseph Biberstine]

Here's a cleaner version (fixed exponent parsing and trimmed start of line)

A002720 Number of partial permutations of an n-set; number of n X n
binary matrices with at most one 1 in each row and column.
A005802 Number of permutations in S_n with longest increasing
subsequence of length <= 3 (i.e. 1234-avoiding permutations); vexillary
permutations (i.e. 2143-avoiding).
A008620 Positive integers repeated three times.
A004253 Pythagoras' theorem generalized. Also domino tilings in K_3 X
P_2n (or in S_4 X P_2n).
A005928 G.f.: s(1)^3/s(3), where s(k) := subs(q=q^k, eta(q)) and eta(q)
is Dedekind's function, cf. A010815. [Fine]
A000119 Number of representations of n as a sum of distinct Fibonacci
numbers.
A069283 a(n) = -1 + number of odd divisors of n.
A007425 d_3(n), or tau_3(n), the number of ordered factorizations of n
as n = rst.
A039966 a(0) = 1, a(3n+2) = 0, a(3n) = a(3n+1) = a(n).
A002878 Bisection of Lucas sequence.
A007325 G.f.: Prod_{k>0}
(1-x^{5k-1})*(1-x^{5k-4})/((1-x^{5k-2})*(1-x^{5k-3})).
A000258 E.g.f.: exp(exp(exp(x)-1)-1).
A003239 Number of rooted planar trees with n non-root nodes: circularly
cycling the subtrees at the root gives equivalent trees.
A034444 ud(n) = number of unitary divisors of n (d such that d divides
n, GCD(d,n/d)=1).
A029838 Expansion of square root of q times normalized Hauptmodul for
Gamma(4) in powers of q^8.
A001788 n*(n+1)*2^(n-2).
A005717 Construct triangle in which n-th row is obtained by expanding
(1+x+x^2)^n and take the next-to-central column.
A005185 Hofstadter Q-sequence: a(1) = a(2) = 1;
a(n)=a(n-a(n-1))+a(n-a(n-2)) for n > 2.
A000579 Figurate numbers or binomial coefficients C(n,6).
A033716 Number of integer solutions to the equation x^2+3y^2=n.
A003149 Sum_{k=0..n} k!(n-k)!.
A001791 Binomial coefficients C(2n,n-1).
A003484 Radon function, also called Hurwitz-Radon numbers.
A007153 Dedekind numbers: monotone Boolean functions or antichains of
subsets of an n-set containing at least one nonempty set.
A001047 3^n - 2^n.
A036987 Fredholm-Rueppel sequence.
A001055 Number of ways of factoring n with all factors >1.
A003500 a(n) = 4a(n-1) - a(n-2).
A000712 Number of partitions of n into parts of 2 kinds.
A008615 [n/2] - [n/3].
A000127 Maximal number of regions obtained by joining n points around a
circle by straight lines. Also number of regions in 4-space formed by
n-1 hyperplanes.
A052849 a(0) = 0; a(n+1) = 2*n! (n >= 0).
A001599 Harmonic or Ore numbers: numbers n such that harmonic mean of
divisors of n is an integer.
A101330 Array read by antidiagonals: T(n,k) = Knuth's Fibonacci (or
circle) product of n and k ("n o k").
A007240 McKay-Thompson series of class 1A for Monster; another version
of j-function.
A003605 Unique monotonic sequence of nonnegative integers satisfying
a(a(n)) = 3n.
A003849 The infinite Fibonacci word (start with 0, apply 0->01, 1->0,
take limit).
A010683 Let S(x,y) = number of lattice paths from (0,0) to (x,y) that
use the step set { (0,1), (1,0), (2,0), (3,0), ....} and never pass
below y = x. Sequence gives S(n-1,n) = number of `Schroeder' trees with
n+1 leaves and root of deg. 2.
A002530 Denominators of continued fraction convergents to sqrt(3).
A001678 Number of series-reduced planted trees with n nodes.
A002605 a(n+2) = 2*a(n+1) + 2*a(n).
A005259 Apery (Ap\'{e}ry) numbers: Sum_{k=0..n}
(binomial(n,k)*binomial(n+k,k))^2.
A034851 Rows of Losanitsch's triangle (n >= 0, k >= 0).
A000172 Franel number a(n) = Sum C(n,k)^3, k=0..n.
A000325 2^n - n.
A001317 Pascal's triangle mod 2 converted to decimal.
A005117 Square-free numbers.
A008441 Number of ways of writing n as the sum of 2 triangular numbers.
A000625 Number of n-node steric rooted ternary trees; number of n carbon
alkyl radicals C(n)H(2n+1) taking stereoisomers into account
A084938 Triangle of numbers T(n,k), 0<=n, 0<=k: T(n,k)= sum(j>=0)
j!*T(n-j-1, k-1)).
A005251 a(n)=a(n-1)+a(n-2)+a(n-4).
A108299 Triangle read by rows, 0 <= k <= n:
T(n,k)=binomial(n-[(k+1)/2],[k/2])*(-1)^[(k+1)/2].
A001339 a(n) = Sum (k+1)! C(n,k), k = 0..n.
A001608 Perrin (or Ondrej Such) sequence: a(n) = a(n-2) + a(n-3).
A000016 a(n) = number of distinct (infinite) output sequences from
binary n-stage shift register which feeds back the complement of the
last stage. E.g. for n=6 there are 6 such sequences.
A075886 Number of cubes at generation n when building fractal cube with
edge ratio of 1/2.
A001500 Number of stochastic matrices of integers: n X n arrays of
nonnegative integers with all row and column sums equal to 3.
A002321 Mertens's function: Sum_{1<=k<=n} mu(k), where mu = Moebius
function (A008683).
A006190 a(n) = 3*a(n-1) + a(n-2).
A000793 Landau's function g(n): largest order of permutation of n
elements. Equivalently, largest lcm of partitions of n.
A011371 n minus (number of 1's in binary expansion of n). Also highest
power of 2 dividing n!.
A006721 Somos-5 sequence: a(n) = (a(n-1)a(n-4)+a(n-2)a(n-3))/a(n-5).
A004001 Hofstadter-Conway $10000 sequence: a(n) = a(a(n-1))+a(n-a(n-1))
with a(1) = a(2) = 1.
A006516 2^(n-1)*(2^n - 1).
A034807 Triangle T(n,k) of coefficients of Lucas (or Cardan) polynomials.
A005811 Number of runs in binary expansion of n (n>0); number of 1's in
Gray code for n.
A003714 Fibbinary numbers: if n = F_i1+F_i2+...+F_ik is the Zeckendorf
representation of n (i.e. write n in Fibonacci number system) then a(n)
= 2^{i1-2}+2^{i2-2}+...+2^{ik-2}.
A002457 (2n+1)!/n!^2.
A001563 n*n! = (n+1)!-n!.
A007583 (2^(2n+1) + 1)/3.
A007070 a(n)=4a(n-1)-2a(n-2).
A008302 Triangle of Mahonian numbers T(n,k): coefficients in expansion
of Product (1+x+...+x^k); k=0..n.
A037027 Skew Fibonacci-Pascal triangle read by rows.
A004016 Theta series of planar hexagonal lattice A_2.
A001950 Upper Wythoff sequence (a Beatty sequence): a(n) =
floor(n*phi^2), where phi = (1+sqrt(5))/2.
A004011 Theta series of D_4 lattice; Fourier coefficients of Eisenstein
series E_{gamma,2}.
A000088 Number of graphs on n unlabeled nodes.
A000928 Irregular primes: p is regular if and only if the numerators of
the Bernoulli numbers B_2, B_4, ..., B_{p-3} (A000367) are not divisible
by p.
A000118 Number of ways of writing n as a sum of 4 squares; theta series
of lattice Z^4.
A003422 Left factorials: !n = Sum k!, k=0..n-1.
A002866 a(0) = 1; for n>0, a(n) = 2^(n-1)*n!.
A008297 Triangle of Lah numbers.
A000244 Powers of 3.
A026741 a(n) = n if n odd, n/2 if n even.
A008438 Sum of divisors of 2n+1.
A053175 Catalan-Larcombe-French sequence.
A000169 Number of labeled rooted trees with n nodes: n^(n-1).
A002387 Least k such that H(k) > n, where H(k) is the harmonic number
sum_{i=1..k} 1/i.
A011973 Triangle of numbers {C(n-k,k), n >= 0, 0<=k<=[ n/2 ]}; or,
triangle of coefficients of Fibonacci polynomials.
A003114 Number of partitions of n into parts 5k+1 or 5k-1. Coefficients
in expansion of one of the Rogers-Ramanujan identities.
A002054 Binomial coefficient binomial(2n+1,n-1).
A001285 Thue-Morse sequence: let A_k denote the first 2^k terms; then
A_0 = 1, and for k >= 0, A_{k+1} = A_k B_k, where B_k is obtained from
A_k by interchanging 1's and 2's.
A003406 Expansion of Ramanujan's function R(x) = 1 + Sum_{n >= 1} {
x^(n*(n+1)/2) / ((1+x)(1+x^2)(1+x^3)...(1+x^n)) }.
A002024 n appears n times.
A001067 Numerator of Bernoulli(2n)/(2n).
A000680 (2n)!/2^n.
A015128 G.f.: Product_{m=1..inf} (1+q^m)/(1-q^m); also (Sum (-q)^(m^2),
m = -inf .. inf )^(-1).
A005248 Bisection of Lucas numbers: A000032(2n).
A033282 Triangle read by rows: T(n,k) is the number of diagonal
dissections of a convex n-gon into k+1 regions.
A000001 Number of groups of order n.
A005493 a(n) = number of partitions of [n+1] with a distinguished block.
For example, a(1) counts (12), (1)-2, 1-(2) where dashes separate blocks
and the distinguished block is parenthesized.
A060843 Busy Beaver problem: maximal number of steps that an n-state
Turing machine can make on an initially blank tape before eventually
halting.
A011782 Expansion of (1-x)/(1-2x) in powers of x.
A007526 a(n) = n(a(n-1) + 1).
A000337 (n-1)*2^n + 1.
A063007 Triangle: T(n,k) = C(n,k)*C(n+k,k) read by rows.
A000531 From area of cyclic polygon of 2n +1 sides.
A014466 Dedekind numbers: monotone Boolean functions, or nonempty
antichains of subsets of an n-set
A004526 Integers repeated.
A001498 Triangle of coefficients of Bessel polynomials (exponents in
increasing order).
A018819 Binary partition function: number of partitions of n into powers
of 2.
A000290 The squares: a(n) = n^2.
A003159 Numbers n such that binary representation ends in even number of
zeros.
A000312 Number of labeled mappings from n points to themselves
(endofunctions): n^n.
A001481 Numbers that are the sum of 2 squares.
A000326 Pentagonal numbers: n(3n-1)/2.
A000578 The cubes: a(n) = n^3.
A002088 Sum of totient function: a(n) = Sum_{k=1..n} phi(k) (cf. A000010).
A003273 Congruent numbers: positive integers n for which there exists a
right triangle having area n and rational sides.
A001400 G.f.: 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)).
A000170 Number of ways of placing n nonattacking queens on n X n board.
A046092 2n(n+1).
A003319 Number of connected permutations of [1..n] (those not fixing
[1..j] for 0<j<n). Also called indecomposable permutations.
A000296 Number of partitions of an n-set into blocks of size >1. Also
number of cyclically spaced (or feasible) partitions.
A002522 n^2 + 1.
A001834 a(0) = 1, a(1) = 5, a(n) = 4a(n-1) - a(n-2).
A000629 Number of necklaces of sets of labeled beads.
A005875 Theta series of cubic lattice; also number of ways of writing a
nonnegative integer n as a sum of 3 squares (zero being allowed).
A000295 Eulerian numbers 2^n - n - 1. (Column 2 of Euler's triangle
A008292.)
A000112 Number of partially ordered sets ("posets") with n unlabeled
elements.
A000389 Binomial coefficients C(n,5).
A000055 Number of trees with n unlabeled nodes.
A000254 Stirling numbers of first kind s(n,2): a(n+1)=(n+1)*a(n)+n!.
A000084 Number of series-parallel networks with n unlabeled edges. Also
called yoke-chains by Cayley and MacMahon.
A005836 Numbers n such that base 3 representation contains no 2.
A000048 Number of n-bead necklaces with beads of 2 colors and primitive
period n, when turning over is not allowed but the two colors can be
interchanged.
A000695 Moser-de Bruijn sequence: sums of distinct powers of 4.
A000031 Number of n-bead necklaces with 2 colors when turning over is
not allowed; number of output sequences from a simple n-stage cycling
shift register; number of binary irreducible polynomials whose degree
divides n.
A008288 Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by
antidiagonals.
A040051 Parity of partition function A000041.
A001813 Quadruple factorial numbers: (2n)!/n!.
A006003 n(n^2+1)/2.
A008683 Moebius (or Mobius) function mu(n).
A002654 Number of ways of writing n as a sum of at most two nonzero
squares, where order matters; also (number of divisors of n of form
4m+1) - (number of divisors of form 4m+3).
A015518 a(n) = 2 a(n-1) + 3 a(n-2), a(0)=0, a(1)=1.
A005425 a(n) = 2*a(n-1)+(n-1)*a(n-2).
A028246 Triangular array of numbers a(n,k) = Sum_{i=0..k}
(-1)^(k-i)*C(k,i)*i^n; n >= 1, 1<=k<=n.
A001075 a(0) = 1, a(1) = 2, a(n) = 4a(n-1) - a(n-2).
A001076 Denominators of continued fraction convergents to sqrt(5).
A000311 Schroeder's fourth problem; also phylogenetic trees with n
nodes; also total partitions of n.
A000265 Remove 2's from n; or largest odd divisor of n; or odd part of n.
A002415 4-dimensional pyramidal numbers: n^2*(n^2-1)/12.
A000598 Number of rooted ternary trees with n nodes; number of n-carbon
alkyl radicals C(n)H(2n+1) ignoring stereoisomers.
A027907 Triangle of trinomial coefficients T(n,k) (n >= 0, 0<=k<=2n),
read by rows (n-th row is obtained by expanding (1+x+x^2)^n).
A002182 Highly composite numbers, definition (1): where d(n), the number
of divisors of n (A000005), increases to a record.
A001970 Functional determinants; partitions of partitions; Euler
transform applied twice to all 1's sequence.
A000537 Sum of first n cubes; or n-th triangular number squared.
A007317 Binomial transform of Catalan numbers.
A000069 Odious numbers: odd number of 1's in binary expansion.
A000215 Fermat numbers: 2^(2^n) + 1.
A001469 Genocchi numbers (of first kind); unsigned coefficients give
expansion of tan(x/2).
A007051 (3^n + 1)/2.
A000521 Coefficients of modular function j as power series in q=e^(2Pi i t).
A000367 Numerators of Bernoulli numbers B_2n.
A001035 Number of partially ordered sets ("posets") with n labeled
elements (or labeled acyclic transitive digraphs).
A004018 Theta series of square lattice (or number of ways of writing n
as a sum of 2 squares).
A000931 Padovan sequence: a(n) = a(n-2) + a(n-3).
A019538 Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n
>= 1, 1 <= k <= n).
A001110 Numbers that are both triangular and square: a(n) = 34a(n-1) -
a(n-2) + 2.
A003215 Hex (or centered hexagonal) numbers: 3n(n+1)+1 (crystal ball
sequence for hexagonal lattice).
A003415 a(n) = n' = derivative of n: a(0) = a(1) = 0, a(prime) = 1,
a(mn) = m*a(n) + n*a(m).
A000372 Dedekind numbers: number of monotone Boolean functions of n
variables or number of antichains of subsets of an n-set.
A007814 Exponent of highest power of 2 dividing n (the binary carry
sequence).
A001835 a(n) = 4a(n-1) - a(n-2); a(0)=a(1)=1.
A000245 3(2n)!/((n+2)!(n-1)!).
A001227 Number of odd divisors of n.
A005408 The odd numbers: a(n) = 2n+1.
A053121 Catalan triangle (with 0's). Inverse lower triangular matrix of
A049310(n,m) (coefficients of Chebyshev's S polynomials).
A001615 Dedekind psi function: n * Product_{p|n, p prime} (1 + 1/p).
A000930 a(n) = a(n-1) + a(n-3).
A001710 Order of alternating group A_n, or number of even permutations
of n letters.
A001840 Expansion of x/((1 - x)^2*(1 - x^3)).
A008619 Positive integers repeated.
A006356 Number of distributive lattices; also number of paths with n
turns when light is reflected from 3 glass plates.
A001462 Golomb's sequence: a(n) is the number of times n occurs,
starting with a(1) = 1.
A000165 Double factorial numbers: (2n)!! = 2^n*n!.
A003313 Length of shortest addition chain for n.
A004009 Expansion of Eisenstein series E_4(q) (alternate convention
E_2(q)); theta series of E_8 lattice.
A000957 Fine's sequence (or Fine numbers): number of relations of
valence >= 1 on an n-set; also number of ordered rooted trees with n
edges having root of even degree.
A000207 Number of inequivalent ways of dissecting a regular (n+2)-gon
into n triangles by n-1 non-intersecting diagonals under rotations and
reflections; also the number of planar 2-trees.
A000002 Kolakoski sequence: a(n) is length of n-th run; a(1) = 1;
sequence consists just of 1's and 2's.
A001316 Gould's sequence: Sum_{k=0..n} (C(n,k) mod 2): number of odd
entries in row n of Pascal's triangle (A007318).
A001037 Number of degree-n irreducible polynomials over GF(2); number of
n-bead necklaces with beads of 2 colors when turning over is not allowed
and with primitive period n; number of binary Lyndon words of length n.
A000071 Fibonacci numbers - 1.
A006720 Somos-4 sequence: a(n)=(a(n-1)a(n-3)+a(n-2)^2)/a(n-4).
A033184 Catalan triangle A009766 transposed.
A000792 a(n) = max{ (n-i)a(i) : i<n}; a(0) = 1.
A009766 Catalan's triangle T(n,k) (read by rows): each term is the sum
of the entries above and to the left, i.e. T(n,k)=sum(T(n-1,j),j=0..k).
A001190 Wedderburn-Etherington numbers: binary rooted trees (every node
has out-degree 0 or 2) with n endpoints (and 2n-1 nodes in all).
A001629 Fibonacci numbers convolved with themselves.
A027641 Numerator of Bernoulli number B_n.
A001108 a(n)-th triangular number is a square: a(n+1) = 6*a(n)-a(n-1)+2.
A001541 a(0) = 1, a(1) = 3; for n > 1, a(n) = 6a(n-1) - a(n-2).
A002212 Number of restricted hexagonal polyominoes with n cells.
A000669 Number of series-reduced planted trees with n leaves. Also the
number of essentially series series-parallel networks with n edges; also
the number of essentially parallel series-parallel networks with n
edges; also the number of total orderings of n unlabeled points.
A000123 Number of binary partitions: number of partitions of 2n into
powers of 2.
A000073 Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3).
A000032 Lucas numbers (beginning at 2): L(n) = L(n-1) + L(n-2). (Cf.
A000204.)
A001844 Centered square numbers: 2n(n+1)+1. Sums of two consecutive
squares. Also, consider all Pythagorean triples (X,Y,Z=Y+1) ordered by
increasing Z; then sequence gives Z values.
A001318 Generalized pentagonal numbers: n(3n-1)/2, n=0, +- 1, +- 2,....
A000255 a(n) = n*a(n-1) + (n-1)*a(n-2), a(0) = 1, a(1) = 1.
A002034 Smarandache numbers: smallest number m such that n divides m!.
A000096 n(n+3)/2.
A000178 Superfactorials: product of first n factorials.
A002315 NSW numbers: a(n) = 6*a(n-1) - a(n-2); also a(n)^2 - 2*b(n)^2 =
-1 with b(n)=A001653(n).
A001652 a(n)=6a(n-1)-a(n-2)+2 with a(0) = 0, a(1) = 3.
A000332 Binomial coefficients binomial(n,4).
A000182 Tangent (or "Zag") numbers: expansion of tan x. Also expansion
of tanh(x).
A002378 Oblong (or pronic, or heteromecic) numbers: n(n+1).
A001792 (n+2)*2^(n-1).
A000262 Number of "sets of lists": number of partitions of {1,..,n} into
any number of lists, where a list means an ordered subset.
A005773 Number of directed animals of size n (or directed n-ominoes in
standard position).
A002450 (4^n - 1)/3.
A003462 (3^n - 1)/2.
A000219 Number of planar partitions of n.
A000934 Chromatic number (or Heawood number) Chi(n) of surface of genus n.
A000070 Sum_{k=0..n} p(k) where p(k) = number of partitions of k (A000041).
A005316 Meandric numbers: number of ways a river can cross a road n times.
A001263 Triangle of Narayana numbers T(n,k) = C(n-1,k-1)C(n,k-1)/k with
1<=k<=n, read by rows. Also called the Catalan triangle.
A005044 Alcuin's sequence: expansion of x^3/((1-x^2)*(1-x^3)*(1-x^4)).
A002623 G.f.: 1/((1-x)^3*(1-x^2)).
A002061 Central polygonal numbers: n^2 - n + 1.
A005130 Robbins numbers: Product_{k=0..n-1} (3k+1)!/(n+k)!; descending
plane partitions whose parts do not exceed n; alternating sign n X n
matrices (ASM's).
A000330 Square pyramidal numbers: 0^2+1^2+2^2+...+n^2 = n(n+1)(2n+1)/6.
A000027 The natural numbers. Also called the whole numbers, the counting
numbers or the positive integers.
A000700 Expansion of product (1+x^(2k+1)), k=0..inf; number of
partitions of n into distinct odd parts; number of self-conjugate
partitions; number of symmetric Ferrers graphs with n nodes.
A000058 Sylvester's sequence: a(n+1) = a(n)^2 - a(n) + 1, with a(0) = 2.
A000124 Central polygonal numbers (the Lazy Caterer's sequence):
n(n+1)/2 + 1; or, maximal number of pieces formed when slicing a pancake
with n cuts.
A000081 Number of rooted trees with n nodes (or connected functions with
a fixed point).
A010815 From Euler's Pentagonal Theorem: coefficient of q^n in Product
(1-q^m), m=1.. infinity. Also the q-expansion of the Dedekind eta
function without the q^(1/24) factor.
A000043 Primes p such that 2^p - 1 is prime. 2^p - 1 is then called a
Mersenne prime.
A000364 Euler (or secant or "Zig") numbers: expansion of sec x.
A001850 Central Delannoy numbers: Sum_{k=0..n} C(n,k)*C(n+k,k).
A000272 Number of labeled trees on n nodes: n^(n-2).
A000120 1's-counting sequence: number of 1's in binary expansion of n.
A001405 Central binomial coefficients: C(n,floor(n/2)).
A005043 Motzkin sums: a(n) = (n-1)*(2*a(n-1)+3*a(n-2))/(n+1). Also
called Riordan numbers or ring numbers.
A001353 a(n) = 4a(n-1) - a(n-2).
A000292 Tetrahedral (or pyramidal) numbers: C(n+2,3) = n(n+1)(n+2)/6.
A006125 2^{n(n-1)/2}.
A000201 Lower Wythoff sequence (a Beatty sequence): a(n) = floor(n*phi),
where phi = (1+sqrt(5))/2.
A000720 pi(n), the number of primes <= n. Sometimes called PrimePi(n) to
distinguish it from the number 3.14159...
A008275 Triangle read by rows of Stirling numbers of first kind, s(n,k),
n >= 1, 1<=k<=n.
A001764 Binomial(3n,n)/(2n+1) (enumerates ternary trees and also
non-crossing trees).
A035263 Trajectory of 1 under the morphism 1 -> 10, 0 -> 11.
A014486 List of totally balanced sequences of 2n binary digits written
in base 10. Binary expansion of each term contains n 0's and n 1's, and
reading from left to right (the most significant to the least
significant bit), the number of 0's never exceeds the number of 1's.
A004148 Generalized Catalan numbers: a(n+1)=a(n)+ Sum a(k)a(n-1-k),
k=1..n-1.
A000005 d(n) (also called tau(n) or sigma_0(n)), the number of divisors
of n.
A000203 sigma(n) = sum of divisors of n. Also called sigma_1(n).
A008292 Triangle of Eulerian numbers T(n,k) read by rows.
A000009 Expansion of Product (1 + x^m), m=1..inf; number of partitions
of n into distinct parts; number of partitions of n into odd parts.
A002426 Central trinomial coefficient: largest coefficient of (1+x+x^2)^n.
A000010 Euler totient function phi(n): count numbers <= n and prime to n.
A000594 Ramanujan's tau function (or tau numbers).
A000111 Euler or up/down numbers: expansion of sec x + tan x . Also
number of alternating permutations on n letters.
A001399 Number of partitions of n into at most 3 parts; also partitions
of n+3 in which the greatest part is 3; also multigraphs with 3 nodes
and n edges.
A001906 F(2n) = bisection of Fibonacci sequence: a(n)=3a(n-1)-a(n-2).
A000225 2^n - 1.
A010060 Thue-Morse sequence: let A_k denote the first 2^k terms; then
A_0 = 0, and for k >= 0, A_{k+1} = A_k B_k, where B_k is obtained from
A_k by interchanging 0's and 1's.
A001511 2^a(n) divides 2n. Or, a(n) = 2-adic valuation of 2n.
A001147 Double factorial numbers: (2n-1)!! = 1.3.5....(2n-1).
A000085 Number of self-inverse permutations on n letters, also known as
involutions; number of Young tableaux with n cells.
A001333 Numerators of continued fraction convergents to sqrt(2).
A008277 Triangle of Stirling numbers of 2nd kind, S2(n,k), n >= 1, 1<=k<=n.
A001700 C(2n+1, n+1): number of ways to put n+1 indistinguishable balls
into n+1 distinguishable boxes = number of (n+1)-st degree monomials in
n+1 variables = number of monotone maps from 1..n+1 to 1..n+1.
A000522 Total number of arrangements of a set with n elements: a(n) =
Sum_{k=0..n} n!/k!.
A001787 n*2^(n-1).
A000079 Powers of 2: a(n) = 2^n.
A006318 Large Schroeder numbers.
A002487 Stern's diatomic series: a(0) = 0, a(1) = 1; for n >= 1, a(2n) =
a(n), a(2n+1) = a(n) + a(n+1).
A000166 Subfactorial or rencontres numbers, or derangements: number of
permutations of n elements with no fixed points.
A000129 Pell numbers: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) +
a(n-2).
A007318 Pascal's triangle read by rows: C(n,k) = binomial(n,k) =
n!/(k!*(n-k)!), 0<=k<=n.
A001653 Numbers n such that 2*n^2 - 1 is a square.
A001519 a(n) = F(2n-1) = bisection of Fibonacci sequence A000045:
a(n)=3a(n-1)-a(n-2).
A001109 a(n)^2 is a triangular number: a(n) = 6*a(n-1) - a(n-2).
A000670 Number of preferential arrangements of n labeled elements; or
number of weak orders on n labeled elements.
A002620 Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently,
floor(n^2/4).
A000142 Factorial numbers: n! = 1*2*3*4*...*n (order of symmetric group
S_n, number of permutations of n letters).
A001006 Motzkin numbers: number of ways of drawing any number of
nonintersecting chords among n points on a circle.
A000984 Central binomial coefficients: C(2n,n) = (2n)!/(n!)^2.
A001003 Schroeder's second problem (generalized parentheses); also
called super-Catalan numbers or little Schroeder numbers.
A001045 Jacobsthal sequence: a(n) = a(n-1) + 2a(n-2).
A000217 Triangular numbers: a(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+...+n.
A000040 The prime numbers.
A000041 a(n) = number of partitions of n (the partition numbers).
A000110 Bell or exponential numbers: ways of placing n labeled balls
into n indistinguishable boxes.
A000045 Fibonacci numbers: F(n) = F(n-1) + F(n-2), F(0) = 0, F(1) = 1,
F(2) = 1, ...
A000108 Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!).
Also called Segner numbers.

Ed Pegg Jr wrote:
> Is there a version with just the %N lines?
> 
> Ed Pegg Jr
> 
> N. J. A. Sloane wrote:
>> David W. asked for the full text of
>> the top 300 seqs according to that list.  Here it is (there were two
>> error messages when I did this because two lines were too long for
>> the local version of awk).
>> Neil
>>
>> %N A002720 Number of partial permutations of an n-set; number of n X n
>> binary matrices with at most one 1 in each row and column.
>>   
>