A168084 - OEIS

%I #23 Apr 05 2018 23:23:23

%S 0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,4,8,16,32,64,128,256,512,1024,2048,

%T 4096,8191,16381,32760,65516,131024,262032,524032,1048000,2095872,

%U 4191488,8382464,16763904,33525760,67047424,134086657,268156933,536281106,1072496696

%N Fibonacci 13-step numbers.

%H G. C. Greubel, <a href="/A168084/b168084.txt">Table of n, a(n) for n = 1..1000</a>

%H Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, <a href="http://www.emis.de/journals/JIS/VOL18/Szczyrba/sz3.html">Analytic Representations of the n-anacci Constants and Generalizations Thereof</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.

%H M. Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Janjic/janjic63.html">On Linear Recurrence Equations Arising from Compositions of Positive Integers</a>, J. Int. Seq. 18 (2015) # 15.4.7.

%H <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1).

%F Another form of the g.f. f: f(z)=(z^(k-1)-z^(k))/(1-2*z+z^(k+1)) with k=13. then a(n)=sum((-1)^i*binomial(n-k+1-k*i,i)*2^(n-k+1-(k+1)*i),i=0..floor((n-k+1)/(k+1)))-sum((-1)^i*binomial(n-k-k*i,i)*2^(n-k-(k+1)*i),i=0..floor((n-k)/(k+1))) with k=13 and convention sum(alpha(i),i=m..n)=0 for m>n. - _Richard Choulet_, Feb 22 2010

%p k:=13:a:=taylor((z^(k-1)-z^(k))/(1-2*z+z^(k+1)),z=0,51);for p from 0 to 50 do j(p):=coeff(a,z,p):od :seq(j(p),p=0..50); k:=13:for n from 0 to 50 do l(n):=sum((-1)^i*binomial(n-k+1-k*i,i)*2^(n-k+1-(k+1)*i),i=0..floor((n-k+1)/(k+1)))-sum((-1)^i*binomial(n-k-k*i,i)*2^(n-k-(k+1)*i),i=0..floor((n-k)/(k+1))):od:seq(l(n),n=0..50); # _Richard Choulet_, Feb 22 2010

%t a={1,0,0,0,0,0,0,0,0,0,0,0,0};Flatten[Prepend[Table[s=Plus@@a;a=RotateLeft[a];a[[ -1]]=s,{n,60}],Table[0,{m,Length[a]-1}]]]

%t LinearRecurrence[{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, 50]

%t With[{nn=13},LinearRecurrence[Table[1,{nn}],Join[Table[0,{nn-1}],{1}],50]] (* _Harvey P. Dale_, Aug 17 2013 *)

%Y Cf. A000078, A001591, A001592, A122189, A079262, A168083.

%K nonn,easy

%O 1,15

%A _Vladimir Joseph Stephan Orlovsky_, Nov 18 2009