A105754 Lucas 8-step numbers.
1, 3, 7, 15, 31, 63, 127, 255, 502, 1003, 2003, 3999, 7983, 15935, 31807, 63487, 126719, 252936, 504869, 1007735, 2011471, 4014959, 8013983, 15996159, 31928831, 63730943, 127208950, 253913031, 506818327, 1011625183, 2019235407
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.
- Spiros D. Dafnis, Andreas N. Philippou, Ioannis E. Livieris, An Alternating Sum of Fibonacci and Lucas Numbers of Order k, Mathematics (2020) Vol. 9, 1487.
- 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).
Programs
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Mathematica
a={-1, -1, -1, -1, -1, -1, -1, 8}; Table[s=Plus@@a; a=RotateLeft[a]; a[[ -1]]=s, {n, 50}] 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 *) -
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
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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
Formula
a(n) = Sum_{k=1..8} a(n-k) for n > 0, a(0)=8, a(n)=-1 for n=-7..-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)/( -1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 ). - R. J. Mathar, Jun 20 2011