cp's OEIS Frontend

A105755 Lucas 9-step numbers.

%I A105755 #45 Feb 16 2025 08:32:57
%S A105755 1,3,7,15,31,63,127,255,511,1013,2025,4047,8087,16159,32287,64511,
%T A105755 128895,257535,514559,1028105,2054185,4104323,8200559,16384959,
%U A105755 32737631,65410751,130692607,261127679,521740799,1042453493,2082852801
%N A105755 Lucas 9-step numbers.
%H A105755 G. C. Greubel, <a href="/A105755/b105755.txt">Table of n, a(n) for n = 1..2500</a> (terms 1..200 from T. D. Noe)
%H A105755 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 A105755 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 A105755 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 A105755 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Lucasn-StepNumber.html">Lucas n-Step Number</a>
%H A105755 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,1,1,1,1,1,1).
%F A105755 a(n) = Sum_{k=1..9} a(n-k) for n > 0, a(0)=9, a(n)=-1 for n=-8..-1
%F A105755 G.f.: -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 ). - _R. J. Mathar_, Jun 20 2011
%F A105755 a(n) = n*Sum_{k=1..n} (Sum_{i=0..floor((n-k)/9)} (-1)^i*binomial(k, k-i)*binomial(n-9*i-1, k-1))/k. - _Vladimir Kruchinin_, Aug 10 2011
%t A105755 a={-1, -1, -1, -1, -1, -1, -1, -1, 9}; Table[s=Plus@@a; a=RotateLeft[a]; a[[ -1]]=s, {n, 50}]
%o A105755 (Maxima)
%o A105755 a(n):=n*sum(sum((-1)^i*binomial(k,k-i)*binomial(n-9*i-1,k-1),i,0,(n-k)/9)/k,k,1,n);
%o A105755 makelist(a(n),n,1,17); /* _Vladimir Kruchinin_, Aug 10 2011 */
%o A105755 (PARI) a(n)=([0,1,0,0,0,0,0,0,0; 0,0,1,0,0,0,0,0,0; 0,0,0,1,0,0,0,0,0; 0,0,0,0,1,0,0,0,0; 0,0,0,0,0,1,0,0,0; 0,0,0,0,0,0,1,0,0; 0,0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,0,1; 1,1,1,1,1,1,1,1,1]^(n-1)*[1;3;7;15;31;63;127;255;511])[1,1] \\ _Charles R Greathouse IV_, Jun 15 2015
%Y A105755 Cf. A000032, A001644, A073817, A074048, A074584, A104621, A105754 (Lucas n-step numbers).
%K A105755 nonn,easy
%O A105755 1,2
%A A105755 _T. D. Noe_, Apr 22 2005