This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A124312 #49 Jun 22 2025 16:28:09
%S A124312 0,0,1,0,1,2,4,8,15,30,59,116,228,448,881,1732,3405,6694,13160,25872,
%T A124312 50863,99994,196583,386472,759784,1493696,2936529,5773064,11349545,
%U A124312 22312618,43865452,86237208,169537887,333302710,655255875,1288199132
%N A124312 Expansion of g.f. x^3*(1 - x)/(1 - x - x^2 - x^3 - x^4 - x^5).
%C A124312 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 A124312 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 A124312 Robert Israel, <a href="/A124312/b124312.txt">Table of n, a(n) for n = 1..3407</a>
%H A124312 Martin Burtscher, Igor Szczyrba, and 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 A124312 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,1,1).
%F A124312 a(n) = A001591(n+1) - A001591(n). - _Greg Dresden_ and Leo Zhang, Jun 20 2025
%p A124312 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 A124312 seq(f(n),n=1..30); # _Robert Israel_, Apr 13 2017
%t A124312 CoefficientList[Series[(x^3-x^4)/(1-x-x^2-x^3-x^4-x^5), {x,0,50}], x]
%o A124312 (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 A124312 (SageMath)
%o A124312 def A124312_list(prec):
%o A124312 P.<x> = PowerSeriesRing(ZZ, prec)
%o A124312 return P( x^2*(1-x)^2/(1-2*x+x^6) ).list()
%o A124312 A124312_list(50) # _G. C. Greubel_, Aug 25 2023
%Y A124312 Cf. A001591, A124311, A124313, A124314.
%K A124312 nonn,easy
%O A124312 1,6
%A A124312 _Artur Jasinski_, Oct 25 2006
%E A124312 Edited by _N. J. A. Sloane_, Oct 29 2006, Jul 14 2007
%E A124312 Name corrected by _Robert Israel_, Apr 13 2017