Quite confused about two sequences
creigh at o2online.de
creigh at o2online.de
Mon Aug 23 16:16:43 CEST 2004
Ahh.. thanks to Ralf Stefan and Neil Sloane (whose message
"The Gessel link at A057681 works for me!" I just recieved
a few minutes ago ). Superseeker appears to work admirably.
(I wasn't aware that it could handle negative numbers- should
have read the help file first.)
Report on [ -1,-1,6,-15,27,-36,27,27,-162,405,-729,972,-729,-729,4374]:
Many tests are carried out, but only potentially useful information
(if any) is reported here.
Even though there are a large number of sequences in the table, at least
one of yours is not there! Please send it to me using
the submission form on the sequence web page
http://www.research.att.com/~njas/sequences/Submit.html
and I will (probably) add it! Include a brief description. Thanks!
SUGGESTION: GUESSGF FOUND ONE OR MORE GENERATING FUNCTIONS
WARNING: THESE MAY BE ONLY APPROXIMATIONS!
Generating function(s) and type(s) are:
1 + 4 x
[- --------------, ogf]
2
3 x + 1 + 3 x
1/2 1/2 1/2
[- 5/3 3 exp(- 3/2 x) sin(1/2 3 x) - exp(- 3/2 x) cos(1/2 3 x), egf]
1/2 1/2 1/2
[- 5/3 3 exp(- 3/2 x) sin(1/2 3 x) - exp(- 3/2 x) cos(1/2 3 x), egf]
2
1 - 6 x - 12 x
[- ------------------------, lgdogf]
2 3
-15 x - 1 - 7 x - 12 x
SUGGESTION: LISTTOREC FOUND ONE OR MORE RECURRENCES
WARNING: THESE MAY BE ONLY APPROXIMATIONS!
Recurrence(s) and type(s) are:
2
[{a(0) = -1, a(1) = -1, 3 a(n) + (3 n + 3) a(n + 1) + (2 + 3 n + n ) a(n
+ 2)},
egf]
SUGGESTION: LISTTOALGEQ FOUND ONE OR MORE ALGEBRAIC
EQUATIONS SATISFIED BY THE GEN. FN.
WARNING: THESE MAY BE ONLY APPROXIMATIONS!
Equation(s) and type(s) are:
2
[1 + 4 n + (1 + 3 n + 3 n ) a(n), ogf]
2
[n + (1 + 3 n) a(n) + (4 + 3 n) a(n) , revogf]
2 2 3
[-6780 + 40680 n + 81360 n + (6089 + 51606 n + 109992 n + 81360 n ) a(n)
2 3 2
+ (10365 n + 691 + 4837 n + 8292 n ) a(n) , lgdogf]
Types of generating functions that may have been mentioned above:
ogf = ordinary generating function
egf = exponential generating function
revogf = reversion of ordinary generating function
revegf = reversion of exponential generating function
lgdogf = logarithmic derivative of ordinary generating function
lgdegf = logarithmic derivative of exponential generating function
TRY "GUESSS", HARM DERKSEN'S PROGRAM FOR GUESSING A GENERATING FUNCTION FOR A
SEQUENCE.
Guesss - guess a sequence, by Harm Derksen (hderksen at math.mit.edu)
Guesss suggests that the generating function F(x)
may satisfy the following algebraic or differential equation:
4/3*x+1/3+(x^2+x+1/3)*F(x) = 0
If this is correct the next 6 numbers in the sequence are:
[-10935, 19683, -26244, 19683, 19683, -118098]
TEST: APPLY VARIOUS TRANSFORMATIONS TO SEQUENCE AND LOOK IT
UP IN THE ENCYCLOPEDIA AGAIN
SUCCESS
(limited to 40 matches):
Transformation T100 gave a match with:
%I A023126
%S A023126 1,1,1,1,1,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,3,1,2,3,1,2,3,1,2,3,
%T A023126 1,2,3,1,2,3,1,2,3,1,2,3,1,4,2,3,1,4,2,3,1,4,2,3,1,4,2,3,1,4,2,3,1,
%U A023126 4,2,3,1,4,2,3,1,4,2,5,3,1,4,2,5,3,1
%N A023126 Signature sequence of 1/e^2 (arrange the numbers i+j*x (i,j
>= 1) in increasing order; the sequence of i's is the signature of x).
%D A023126 C. Kimberling, "Fractal Sequences and Interspersions", Ars Combinatoria,
vol 45 pp. 157-168, 1997.
%H A023126 C. Kimberling, <a href="http://faculty.evansville.edu/ck6/integer/intersp.
html">Interspersions</a>
%Y A023126 Sequence in context: A000034 A040001 A066788 this_sequence A082898
A058656 A033111
%Y A023126 Adjacent sequences: A023123 A023124 A023125 this_sequence A023127
A023128 A023129
%K A023126 nonn
%O A023126 0,9
%A A023126 Clark Kimberling (ck6(AT)evansville.edu)
%I A010882
%S A010882 1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,
%T A010882 1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,
%U A010882 1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,1,2,3
%N A010882 Simple periodic sequence.
%F A010882 G.f.:(1+2x+3x^2)/(1-x^3) - Paul Barry (pbarry(AT)wit.ie), May
25 2003
%Y A010882 Sequence in context: A047896 A073645 A082846 this_sequence A054073
A059832 A065365
%Y A010882 Adjacent sequences: A010879 A010880 A010881 this_sequence A010883
A010884 A010885
%K A010882 nonn
%O A010882 0,2
%A A010882 njas
Transformation T109 gave a match with:
%I A078343
%S A078343 1,2,3,8,19,46,111,268,647,1562,3771,9104,21979,53062,128103,309268,746639,
%T A078343 1802546,4351731,10506008,25363747,61233502,147830751,356895004,861620759,
%U A078343 2080136522,5021893803,12123924128,29269742059,70663408246,170596558551
%V A078343 -1,2,3,8,19,46,111,268,647,1562,3771,9104,21979,53062,128103,309268,746639,
%W A078343 1802546,4351731,10506008,25363747,61233502,147830751,356895004,861620759,
%X A078343 2080136522,5021893803,12123924128,29269742059,70663408246,170596558551
%N A078343 a(0) = -1, a(1) = 2; a(n) = 2*a(n-1) + a(n-2).
%D A078343 H. S. M. Coxeter, 1998, Numerical distances among the circles
in a loxodromic sequence, Nieuw Arch. Wisk, 16, pp. 1-9.
%F A078343 For the unsigned version: a(1)=1; a(2)=2; a(n) = sum(k=2,n-1,
a(k) + a(k-1) ).
%F A078343 a(n) is asymptotic to (1/4)*(8-5*sqrt(2))*(1+sqrt(2))^n.
%F A078343 a(n) = A048746(n-3) + 2, for n>2. - Ralf Stephan (ralf(AT)ark.
in-berlin.de), Oct 17 2003
%p A078343 f:=proc(n) option remember; if n=0 then RETURN(-1); fi; if n=1
then RETURN(2); fi; 2*f(n-1)+f(n-2); end;
%Y A078343 Sequence in context: A041205 A002356 A041281 this_sequence A077269
A007999 A006609
%Y A078343 Adjacent sequences: A078340 A078341 A078342 this_sequence A078344
A078345 A078346
%K A078343 sign
%O A078343 0,2
%A A078343 Benoit Cloitre (abcloitre(AT)wanadoo.fr), Nov 22 2002
%E A078343 Entry revised by njas, Apr 29 2004
List of transformations used:
T100 binomial transform: b(n)=SUM C(n,k)a(k), k=0..n
T109 invert: define b by 1+SUM a(n)x^n = 1/(1 - SUM b(n)x^n)
Abbreviations used in the above list of transformations:
u[j] = j-th term of the sequence
v[j] = u[j]/(j-1)!
Sn(z) = ordinary generating function
En(z) = exponential generating function
TEST: APPLY VARIOUS TRANSFORMATIONS TO SEQUENCE AND
CHECK AGAINST FIRST DIFFERENCES OF SEQUENCES IN OEIS
SUCCESS
(limited to 40 matches):
Transformation T100 gave a match with first differences of:
%I A047254
%S A047254 2,3,5,8,9,11,14,15,17,20,21,23,26,27,29,32,33,35,38,39,41,44,45,47,50,
%T A047254 51,53,56,57,59,62,63,65,68,69,71,74,75,77,80,81,83,86,87,89,92,93,95,
%U A047254 98,99,101,104,105,107,110
%N A047254 Numbers that are congruent to {2, 3, 5} mod 6.
%Y A047254 Sequence in context: A047607 A047372 A027756 this_sequence A051214
A013634 A025033
%Y A047254 Adjacent sequences: A047251 A047252 A047253 this_sequence A047255
A047256 A047257
%K A047254 nonn
%O A047254 0,1
%A A047254 njas
%I A069705
%S A069705 1,2,4,1,2,4,1,2,4,1,2,4,1,2,4,1,2,4,1,2,4,1,2,4,1,2,4,1,2,4,1,2,4,1,2,
%T A069705 4,1,2,4,1,2,4,1,2,4,1,2,4,1,2,4,1,2,4,1,2,4,1,2,4,1,2,4,1,2,4,1,2,4,1,
%U A069705 2,4,1,2,4,1,2,4,1,2,4,1,2,4,1,2,4,1,2,4,1,2,4,1,2,4,1,2,4,1,2,4,1,2,4
%N A069705 2^n mod 7.
%F A069705 n=0 mod 3 -> a(n)=1 n=1 mod 3 -> a(n)=2 n=2 mod 3 -> a(n)=4
%F A069705 a(n)=2^mod(n,3) - Paul Barry (pbarry(AT)wit.ie), Oct 06 2003
%e A069705 a(4)=16 mod 7=2, a(5)=32 mod 7=4, a(6)=64 mod 7=1
%Y A069705 Sequence in context: A087419 A050979 A053450 this_sequence A062039
A035492 A057176
%Y A069705 Adjacent sequences: A069702 A069703 A069704 this_sequence A069706
A069707 A069708
%K A069705 nonn
%O A069705 0,2
%A A069705 Jon Perry (perry(AT)globalnet.co.uk), Jan 14 2003
%I A047236
%S A047236 1,2,4,7,8,10,13,14,16,19,20,22,25,26,28,31,32,34,37,38,40,43,44,46,49,
%T A047236 50,52,55,56,58,61,62,64,67,68,70,73,74,76,79,80,82,85,86,88,91,92,94,
%U A047236 97,98,100,103,104,106,109
%N A047236 Numbers that are congruent to {1, 2, 4} mod 6.
%Y A047236 Sequence in context: A075663 A056231 A047540 this_sequence A039581
A093701 A045601
%Y A047236 Adjacent sequences: A047233 A047234 A047235 this_sequence A047237
A047238 A047239
%K A047236 nonn
%O A047236 0,2
%A A047236 njas
%I A047267
%S A047267 0,2,5,6,8,11,12,14,17,18,20,23,24,26,29,30,32,35,36,38,41,42,44,47,48,
%T A047267 50,53,54,56,59,60,62,65,66,68,71,72,74,77,78,80,83,84,86,89,90,92,95,
%U A047267 96,98,101,102,104,107,108
%N A047267 Numbers that are congruent to {0, 2, 5} mod 6.
%Y A047267 Sequence in context: A081083 A024798 A045751 this_sequence A058591
A059009 A026179
%Y A047267 Adjacent sequences: A047264 A047265 A047266 this_sequence A047268
A047269 A047270
%K A047267 nonn
%O A047267 0,2
%A A047267 njas
%I A047231
%S A047231 0,3,4,6,9,10,12,15,16,18,21,22,24,27,28,30,33,34,36,39,40,42,45,46,48,
%T A047231 51,52,54,57,58,60,63,64,66,69,70,72,75,76,78,81,82,84,87,88,90,93,94,
%U A047231 96,99,100,102,105,106,108
%N A047231 Numbers that are congruent to {0, 3, 4} mod 6.
%Y A047231 Sequence in context: A080336 A082692 A032706 this_sequence A050131
A089986 A005122
%Y A047231 Adjacent sequences: A047228 A047229 A047230 this_sequence A047232
A047233 A047234
%K A047231 nonn
%O A047231 0,2
%A A047231 njas
%I A047242
%S A047242 0,1,3,6,7,9,12,13,15,18,19,21,24,25,27,30,31,33,36,37,39,42,43,45,48,
%T A047242 49,51,54,55,57,60,61,63,66,67,69,72,73,75,78,79,81,84,85,87,90,91,93,
%U A047242 96,97,99,102,103,105,108
%N A047242 Numbers that are congruent to {0, 1, 3} mod 6.
%Y A047242 Sequence in context: A026406 A047558 A082847 this_sequence A083951
A026227 A026232
%Y A047242 Adjacent sequences: A047239 A047240 A047241 this_sequence A047243
A047244 A047245
%K A047242 nonn
%O A047242 0,3
%A A047242 njas
%I A047259
%S A047259 1,4,5,7,10,11,13,16,17,19,22,23,25,28,29,31,34,35,37,40,41,43,46,47,
%T A047259 49,52,53,55,58,59,61,64,65,67,70,71,73,76,77,79,82,83,85,88,89,91,94,
%U A047259 95,97,100,101,103,106,107,109
%N A047259 Numbers that are congruent to {1, 4, 5} mod 6.
%Y A047259 Sequence in context: A052147 A035266 A035264 this_sequence A039577
A013951 A013947
%Y A047259 Adjacent sequences: A047256 A047257 A047258 this_sequence A047260
A047261 A047262
%K A047259 nonn
%O A047259 0,2
%A A047259 njas
List of transformations used:
T100 binomial transform: b(n)=SUM C(n,k)a(k), k=0..n
Abbreviations used in the above list of transformations:
u[j] = j-th term of the sequence
v[j] = u[j]/(j-1)!
Sn(z) = ordinary generating function
En(z) = exponential generating function
More information about the SeqFan
mailing list