A105755 Lucas 9-step numbers.
1, 3, 7, 15, 31, 63, 127, 255, 511, 1013, 2025, 4047, 8087, 16159, 32287, 64511, 128895, 257535, 514559, 1028105, 2054185, 4104323, 8200559, 16384959, 32737631, 65410751, 130692607, 261127679, 521740799, 1042453493, 2082852801
Offset: 1
Links
- G. C. Greubel, Table of n, a(n) for n = 1..2500 (terms 1..200 from T. D. Noe)
- Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
- C. A. Charalambides, Lucas numbers and polynomials of order k and the length of the longest circular success run, The Fibonacci Quarterly, 29 (1991), 290-297.
- Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4.
- Eric Weisstein's World of Mathematics, Lucas n-Step Number
- Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1,1,1).
Programs
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Mathematica
a={-1, -1, -1, -1, -1, -1, -1, -1, 9}; Table[s=Plus@@a; a=RotateLeft[a]; a[[ -1]]=s, {n, 50}] -
Maxima
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); makelist(a(n),n,1,17); /* Vladimir Kruchinin, Aug 10 2011 */
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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
Formula
a(n) = Sum_{k=1..9} a(n-k) for n > 0, a(0)=9, a(n)=-1 for n=-8..-1
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
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