OFFSET
0,3
COMMENTS
Analogous bit string description and o.g.f. (1-x)/(1-2x+x^{k+1}) works for nonzero k-nacci numbers.
Compositions of n into parts <= 7. - Joerg Arndt, Aug 06 2012
LINKS
T. D. Noe, Table of n, a(n) for n = 0..200
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.
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.
Zhao Hui Du, Link giving derivation and proof of the formula
Omar Khadir, László Németh, and László Szalay, Tiling of dominoes with ranked colors, Results in Math. (2024) Vol. 79, Art. No. 253. See p. 2.
László Németh and László Szalay, Explicit solution of system of two higher-order recurrences, arXiv:2408.12196 [math.NT], 2024. See p. 10.
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, Fibonacci n-Step Number
Eric Weisstein's World of Mathematics, Heptanacci Number
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1).
FORMULA
O.g.f.: 1/(1 - x - x^2 - x^3 - x^4 - x^5 - x^6 - x^7).
a(n) = Sum_{i=n-7..n-1} a(i).
a(n) = round((r-1)/((t+1)*r - 2*t) * r^(n-1)), where r is the heptanacci constant, the real root of the equation x^(t+1) - 2*x^t + 1 = 0 which is greater than 1. The formula could also be used for a k-step Fibonacci sequence if r is replaced by the k-bonacci constant, as in A000045, A000073, A000078, A001591, A001592. - Zhao Hui Du, Aug 24 2008
a(n) = 2*a(n-1) - a(n-8). - Vincenzo Librandi, Dec 20 2010
MATHEMATICA
a[0] = a[1] = 1; a[2] = 2; a[3] = 4; a[4] = 8; a[5] = 16; a[6] = 32; a[7] = 64; a[n_] := 2*a[n - 1] - a[n - 8]; Array[a, 31, 0]
CoefficientList[ Series[(1 - x)/(1 - 2 x + x^8), {x, 0, 30}], x]
LinearRecurrence[{1, 1, 1, 1, 1, 1, 1}, {1, 1, 2, 4, 8, 16, 32}, 40] (* Harvey P. Dale, Nov 16 2014 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Len Smiley, Dec 14 2001
EXTENSIONS
Definition corrected by Vincenzo Librandi, Dec 20 2010
STATUS
approved