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

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..2500 (terms 1..200 from T. D. Noe)

Martin Burtscher, Igor Szczyrba, and 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, and Ioannis E. Livieris, An Alternating Sum of Fibonacci and Lucas Numbers of Order k, Mathematics (2020) Vol. 9, 1487.

Bowie Liu, Dennis Wong, Chan-Tong Lam, and Sio-Kei Im, Generating a Cyclic 2-Gray Code for Lucas Words in Constant Amortized Time, 36th Ann. Symp. Combin. Pattern Match. (CPM 2025), Art. 22. See references.

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).

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

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 *)

Prog

(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
(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

Crossrefs

Cf. A000032, A001644, A073817, A074048, A074584, A104621, A105755 (Lucas n-step numbers).

Sequence in context: A116082 A245269 A043747 * A116690 A267258 A305493

Adjacent sequences: A105751 A105752 A105753 * A105755 A105756 A105757

Keyword

nonn,easy

Author

T. D. Noe, Apr 22 2005

Status

approved