[seqfan] Rainbow definitions, with a/b = c/d

Tue Oct 8 23:39:32 CEST 2019

Hello SeqFans,

I was looking for a seq S (of distinct integers) where 
one could pick any 4 successive terms a, b, c, d and 
have a/b = c/d.
I quickly found https://oeis.org/A038754 
    [Original definition of the seq is between brackets]
    [a(2n) = 3^n, a(2n+1) = 2*3^n].
    S = 1, 2, 3, 6, 9, 18, 27, 54, 81, 162, 243, 486,... 
    The above seq starts with 1,2,3.

The start 1,2,4 produces the powers of 2.
The start 1,2,5 produces https://oeis.org/A026383
    [a(n) = 5a(n-2), starting 1,2].
The start 1,2,6, produces https://oeis.org/A026549
    [Ratios of successive terms are 2,3,2,3,2,3,2,3...]
The start 1,2,7 produces https://oeis.org/A123752
    [a(n) = 7*a(n-2), a(0) = 1, a(1) = 2].
The start 1,2,8 produces https://oeis.org/A098232
    [Largest power of 2 <= 3^n].
The start 1,2,9 produces https://oeis.org/A083423
    [a(n) = (5*3^n + (-3)^n)/6].
The start 1,2,10 produces https://oeis.org/A004643
    [Powers of 2 written in base 4].
etc.
(I love the rainbow of definitions!)

The start 1,3,2 produces https://oeis.org/A164073 
    [a(n) = 2*a(n-2) for n > 2; a(1) = 1, a(2) = 3].
The start 1,3,4 produces https://oeis.org/A084221
    [a(n+2) = 4*a(n), with a(0)=1, a(1)=3].
The start 1,3,5 produces https://oeis.org/A056487
    [a(n) = 5^(n/2) for n even, a(n) = 3*5^((n-1)/2)
    for n odd].
....(!?) and produces also https://oeis.org/A111386
    [a(1) = 1, a(2) = 3; for n >= 3, take a(n) to be 
    the smallest odd number not occurring earlier such
    that a(n-1) divides the concatenation a(n-2)a(n)].
The start 1,3,6 produces https://oeis.org/A026532
    [Ratios of successive terms are 3,2,3,2,3,2,3,2...]
The start 1,3,7 produces 1,3,7,21,49,147,343,1029,...
    _not in the OEIS_.
The start 1,3,8 produces https://oeis.org/A096886
    [Expansion of (1+3*x)/(1-8*x^2)].
The start 1,3,9 produces the powers of 3
The start 1,3,10 produces https://oeis.org/A004663
    [Powers of 3 written in base 9].
etc.

The start 1,4,2 produces https://oeis.org/A143095 
    [(1, 2, 4, 8,...) interleaved with (4, 8, 16, 32,...)].
    But we have repeated terms there.
The start 1,4,3 produces https://oeis.org/A166552
    [a(n) = 3*a(n-2) for n > 2; a(1) = 1; a(2) = 4]
The start 1,4,5 produces https://oeis.org/A133632
    [a(1)=1, a(n)=(p-1)*a(n-1), if n is even, else 
    a(n)=p*a(n-2), where p=5].
The start 1,4,6 produces https://oeis.org/A164532
    [a(n) = 6*a(n-2) for n > 2; a(1) = 1, a(2) = 4].
The start 1,4,7 produces 1,4,7,28,49,196,343,1372,...
    _nt in the OEIS_
The start 1,4,8 produces https://oeis.org/A094015
    [Expansion of (1+4x)/(1-8x^2)].
The start 1,4,9 produces https://oeis.org/A133125
    [A133080 * A000244].
The start 1,4,10 produces https://oeis.org/A136859
    [Numbers n such that n and the square of n use only
    the digits 0, 1, 4 and 6].
etc. (another rainbow!)

The start 1,5,2 produces 1,5,2,10,4,20,8,40,16,80,32,160,...
    _not in the OEIS_.
The start 1,5,3 produces https://oeis.org/A166465
    [a(n) = 3*a(n-2) for n > 2; a(1) = 1, a(2) = 5].
The start 1,5,4 produces 1,5,4,20,16,80,64,320,256,...
    _not in the OEIS_.
The start 1,5,6 produces https://oeis.org/A166023
    [a(n) = 6*a(n-2) for n > 2; a(1) = 1, a(2) = 5]
The start 1,5,7 produces 1,5,7,35,49,245,343,...
   _not in the OEIS_.
The start 1,5,8 produces 1,5,8,40,64,320,512,...
   _not in the OEIS_.
The start 1,5,9 produces 1,5,9,45,81,405,729,...
   _not in the OEIS_.
The start 1,5,10 produces https://oeis.org/A268100
   [a(n) = 2^((n-1) mod 2)*5*10^floor((n-1)/2)]
etc.

The start 1,6,2 produces https://oeis.org/A163864 
    [a(n) = 2*a(n-2) for n > 2; a(1) = 1, a(2) = 6].
The start 1,6,3 produces https://oeis.org/A166450
    [a(n) = 3*a(n-2) for n > 2; a(1) = 1, a(2) = 6].
The start 1,6,4 produces https://oeis.org/A081631
   [2*2^n-(-2)^n].
The start 1,6,5 produces 1,6,5,30,25,150,125,750,...
   _not in the OEIS_.
The start 1,6,7 produces 1,6,7,42,49,...
   _not in the OEIS_.
The start 1,6,8 produces https://oeis.org/A164640
    [a(n) = 8*a(n-2) for n > 2; a(1) = 1, a(2) = 6].
The start 1,6,9 produces 1,6,9,54,81,...
   _not in the OEIS_.
The start 1,6,10 produces 1,6,10,60,100,600,1000,6000,10000
   _not in the OEIS_.
etc.

The start 1,7,2 produces 1,7,2,14,4,28,8,56,16,112,32,224,... 
    _not in the OEIS_.
The start 1,7,3 produces https://oeis.org/A166481
    [a(n) = 3*a(n-2) for n > 2; a(1) = 1; a(2) = 7].
The start 1,7,4 produces 1,7,4,28,16,96,64,...
   _not in the OEIS_.
The start 1,7,5 produces 1,7,5,35,25,175,...
   _not in the OEIS_.
The start 1,7,6 produces 1,7,6,42,36,252,...
   _not in the OEIS_.
The start 1,7,8 produces 1,7,8,56,64,...
   _not in the OEIS_.
The start 1,7,9 produces 1,7,9,63,81,...
   _not in the OEIS_.
The start 1,7,10 produces 1,7,10,70,100,700,1000,7000,...
   _not in the OEIS_.
etc. 

The start 1,8,2 produces https://oeis.org/A164587
    [a(n) = 2*a(n - 2) for n > 2; a(1) = 1, a(2) = 8].
The start 1,8,3 produces 1,8,3,24,9,72,27,216,81,648,243,...
   _not in the OEIS_.
The start 1,8,4 produces almost https://oeis.org/A135520
    [erase the 1st term, then a(n) = 4*a(n-2)].
The start 1,8,5 produces 1,8,5,40,25,200,125,...
   _not in the OEIS_.
The start 1,8,6 produces 1,8,6,48,36,...
   _not in the OEIS_.
The start 1,8,7 produces 1,8,7,56,49,...
   _not in the OEIS_.
The start 1,8,9 produces 1,8,9,72,81,...
   _not in the OEIS_.
The start 1,8,10 produces 1,8,10,80,100,800,1000,...
   _not in the OEIS_.
etc.

The start 1,9,2 produces 1,9,2,18,4,36,8,72,16,144,32,288,...
    _not in the OEIS_.
The start 1,9,3 produces almost https://oeis.org/A162852
    [a(n) = 3*a(n-2) for n > 2; a(1) = 3, a(2) = -1].
    But, again, a lot of not-distinct terms.
The start 1,9,4 produces 1,9,4,36,16,144,...
   _not in the OEIS_.
The start 1,9,5 produces 1,9,5,45,25,225,...
   _not in the OEIS_.
The start 1,9 6 produces 1,9,6,54,36,...
   _notin the OEIS_.
The start 1,9,7 produces 1,9,7,63,49,...
   _not in the OEIS_.
The start 1,9,8 produces 1,9,8,72,64,...
   _not in the OEIS_.
The start 1,9,10 produces 1,9,10,90,100,900,1000,...
   _not in the OEIS_.
etc.

The start 1,10,2 produces 1,10,2,20,4,40,8,80,16,160,32,320,...
    _not in the OEIS_.
The start 1,10,3 produces 1,10,3,30,9,90,27,270,...
   _not in the OEIS_.
etc.

The start 2,1,6 produces 2,1,6,3,18,9,54,27,...
   _not in the OEIS_.

As 19 is prime, the start 20,19 produces 20,19,40,38,80,72,... 
   _not in the OEIS_ (and never in the OEIS, I guess).
etc.

Note that the remark here https://oeis.org/A164073, by a friend
of mine ("Absolute second differences are the sequence itself"),
is always true for some seqs above that follow the pattern: 
1, a, 2, 2a, 4, 4a, 8, 8a, 16, 16a... for a = 0 to k.

Oh, and what about picking 5 (instead of 4) successive integers 
a, b, c, d, e in a seq of distinct terms such that a/b is always
equal to d/e?
Never mind, this has been already submitted by Paul Curtz:
https://oeis.org/A133464
Too good, the OEIS'authors!-))

My two,three, four cents.
É.
P.-S.#1
Many thanks to Klaus Brockhaus who computed 10 years ago a lot 
of the above, and forgive the errors I left in this mail, despite
my carefull rerereading).

P.-S.#2
Let me pick again (in a similar T) 5 successive integers
a, b, c, d, e -- but such that a/c is now equal to d/e:
T = 1, 2, 3, 4, 12, 24, 96, 576,...
Yessss, _T is not in the OEIS_!-))