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A105755 Lucas 9-step numbers. 14
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 (list; graph; refs; listen; history; text; internal format)
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, 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
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
MATHEMATICA
a={-1, -1, -1, -1, -1, -1, -1, -1, 9}; Table[s=Plus@@a; a=RotateLeft[a]; a[[ -1]]=s, {n, 50}]
PROG
(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 */
(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
CROSSREFS
Cf. A000032, A001644, A073817, A074048, A074584, A104621, A105754 (Lucas n-step numbers).
Sequence in context: A305493 A213247 A043755 * A043764 A213248 A336700
KEYWORD
nonn,easy
AUTHOR
T. D. Noe, Apr 22 2005
STATUS
approved

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Last modified March 29 11:14 EDT 2024. Contains 371278 sequences. (Running on oeis4.)