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A105754 Lucas 8-step numbers.

%I A105754 #37 Feb 16 2025 08:32:57
%S A105754 1,3,7,15,31,63,127,255,502,1003,2003,3999,7983,15935,31807,63487,
%T A105754 126719,252936,504869,1007735,2011471,4014959,8013983,15996159,
%U A105754 31928831,63730943,127208950,253913031,506818327,1011625183,2019235407
%N A105754 Lucas 8-step numbers.
%H A105754 G. C. Greubel, <a href="/A105754/b105754.txt">Table of n, a(n) for n = 1..2500</a> (terms 1..200 from T. D. Noe)
%H A105754 Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, <a href="https://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 A105754 C. A. Charalambides, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/29-4/charalambides.pdf">Lucas numbers and polynomials of order k and the length of the longest circular success run</a>, The Fibonacci Quarterly, 29 (1991), 290-297.
%H A105754 Spiros D. Dafnis, Andreas N. Philippou, Ioannis E. Livieris, <a href="https://doi.org/10.3390/math8091487">An Alternating Sum of Fibonacci and Lucas Numbers of Order k</a>, Mathematics (2020) Vol. 9, 1487.
%H A105754 Tony D. Noe and Jonathan Vos Post, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Noe/noe5.html">Primes in Fibonacci n-step and Lucas n-step Sequences,</a> J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4
%H A105754 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Lucasn-StepNumber.html">Lucas n-Step Number</a>
%H A105754 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,1,1,1,1,1).
%F A105754 a(n) = Sum_{k=1..8} a(n-k) for n > 0, a(0)=8, a(n)=-1 for n=-7..-1.
%F A105754 G.f.: -x*(1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 7*x^6 + 8*x^7)/( -1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 ). - _R. J. Mathar_, Jun 20 2011
%t A105754 a={-1, -1, -1, -1, -1, -1, -1, 8}; Table[s=Plus@@a; a=RotateLeft[a]; a[[ -1]]=s, {n, 50}]
%t A105754 CoefficientList[Series[-x*(1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 7*x^6 + 8*x^7 + 9*x^8)/(-1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9), {x, 0, 50}], x] (* _G. C. Greubel_, Dec 18 2017 *)
%o A105754 (PARI) a(n)=([0,1,0,0,0,0,0,0; 0,0,1,0,0,0,0,0; 0,0,0,1,0,0,0,0; 0,0,0,0,1,0,0,0; 0,0,0,0,0,1,0,0; 0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,1; 1,1,1,1,1,1,1,1]^(n-1)*[1;3;7;15;31;63;127;255])[1,1] \\ _Charles R Greathouse IV_, Jun 14 2015
%o A105754 (PARI) x='x+O('x^30); Vec(-x*(1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 7*x^6 + 8*x^7 + 9*x^8)/(-1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9)) \\ _G. C. Greubel_, Dec 18 2017
%Y A105754 Cf. A000032, A001644, A073817, A074048, A074584, A104621, A105755 (Lucas n-step numbers).
%K A105754 nonn,easy
%O A105754 1,2
%A A105754 _T. D. Noe_, Apr 22 2005