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%I #43 Aug 25 2023 08:28:12
%S 0,0,1,0,1,2,4,8,15,30,59,116,228,448,881,1732,3405,6694,13160,25872,
%T 50863,99994,196583,386472,759784,1493696,2936529,5773064,11349545,
%U 22312618,43865452,86237208,169537887,333302710,655255875,1288199132
%N Expansion of g.f. x^3*(1 - x)/(1 - x - x^2 - x^3 - x^4 - x^5).
%C Second column of the n-th power of pentanacci matrix {{1,1,1,1,1},{1,0,0,0,0}, {0,1,0,0,0}, {0,0,1,0,0}, {0,0,0,1,0}} read from bottom to top gives 5 numbers starting from position n.
%C a(n+5) equals the number of n-length binary words avoiding runs of zeros of lengths 5i+4, (i=0,1,2,...). - _Milan Janjic_, Feb 26 2015
%H Robert Israel, <a href="/A124312/b124312.txt">Table of n, a(n) for n = 1..3407</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 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,1,1).
%p f:= gfun:-rectoproc({a(n)+a(n+1)+a(n+2)+a(n+3)+a(n+4)-a(n+5), a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 0}, a(n), remember):
%p seq(f(n),n=1..30); # _Robert Israel_, Apr 13 2017
%t CoefficientList[Series[(x^3-x^4)/(1-x-x^2-x^3-x^4-x^5), {x,0,50}], x]
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 50); [0,0] cat Coefficients(R!( x^3*(1-x)^2/(1-2*x+x^6) )); // _G. C. Greubel_, Aug 25 2023
%o (SageMath)
%o def A124312_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( x^2*(1-x)^2/(1-2*x+x^6) ).list()
%o A124312_list(50) # _G. C. Greubel_, Aug 25 2023
%Y Cf. A001591, A124311, A124313, A124314.
%K nonn,easy
%O 1,6
%A _Artur Jasinski_, Oct 25 2006
%E Edited by _N. J. A. Sloane_, Oct 29 2006, Jul 14 2007
%E Name corrected by _Robert Israel_, Apr 13 2017
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