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A168083 Fibonacci 12-step numbers.

%I A168083 #30 Dec 23 2020 20:35:55
%S A168083 0,0,0,0,0,0,0,0,0,0,0,1,1,2,4,8,16,32,64,128,256,512,1024,2048,4095,
%T A168083 8189,16376,32748,65488,130960,261888,523712,1047296,2094336,4188160,
%U A168083 8375296,16748544,33492993,66977797,133939218,267845688,535625888,1071120816
%N A168083 Fibonacci 12-step numbers.
%C A168083 From _Ruediger Jehn_, Nov 30 2020: (Start)
%C A168083 a(n+12) is the number of compositions of n with no part greater than 12.
%C A168083 a(n+12) is the number of ways of throwing n with an unstated number of dodecahedra (here dice with numbers from 1 to 12).
%C A168083 (End)
%H A168083 G. C. Greubel, <a href="/A168083/b168083.txt">Table of n, a(n) for n = 1..1000</a>
%H A168083 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 A168083 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,1,1,1,1,1,1,1,1,1).
%F A168083 Another form of the g.f. f: f(z)= (z^(k-1)-z^(k))/(1-2*z+z^(k+1)) with k=12. 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=12 and convention sum(alpha(i),i=m..n)=0 for m>n. - _Richard Choulet_, Feb 22 2010
%p A168083 k:=12: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); 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); # _Richard Choulet_, Feb 22 2010
%t A168083 a={1,0,0,0,0,0,0,0,0,0,0,0};Flatten[Prepend[Table[s=Plus@@a;a=RotateLeft[a];a[[ -1]]=s,{n,60}],Table[0,{m,Length[a]-1}]]]
%t A168083 LinearRecurrence[{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, 50]
%t A168083 With[{nn=12},LinearRecurrence[Table[1,{nn}],Join[Table[0,{nn-1}],{1}], 50]] (* _Harvey P. Dale_, Aug 17 2013 *)
%K A168083 nonn
%O A168083 1,14
%A A168083 _Vladimir Joseph Stephan Orlovsky_, Nov 18 2009