A074584 Esanacci (hexanacci or "6-anacci") numbers.

6, 1, 3, 7, 15, 31, 63, 120, 239, 475, 943, 1871, 3711, 7359, 14598, 28957, 57439, 113935, 225999, 448287, 889215, 1763832, 3498707, 6939975, 13766015, 27306031, 54163775, 107438335, 213112838, 422726969, 838513963, 1663261911, 3299217791, 6544271807

Offset

0,1

Comments

These esanacci numbers follow the same pattern as Lucas, generalized tribonacci (A001644), generalized tetranacci (A073817), and generalized pentanacci (A074048) numbers.

The closed form is a(n) = r1^n + r^2^n + r3^n + r4^n + r5^n + r6^n, with r1, r2, r3, r4, r5, r6 roots of the characteristic polynomial.

a(n) is also the trace of A^n, where A is the matrix ((1, 1, 0, 0, 0, 0), (1, 0, 1, 0, 0, 0), (1, 0, 0, 1, 0, 0), (1, 0, 0, 0, 1, 0), (1, 0, 0, 0, 0, 1), (1, 0, 0, 0, 0, 0).

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.

Mario Catalani, Polymatrix and Generalized Polynacci Numbers, arXiv:math/0210201 [math.CO], 2002.

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.

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.

Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1).

Formula

a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5) + a(n-6), a(0)=6, a(1)=1, a(2)=3, a(3)=7, a(4)=15, a(5)=31.

G.f.: (6-5*x-4*x^2-3*x^3-2*x^4-x^5)/(1-x-x^2-x^3-x^4-x^5-x^6).

a(n) = 2*a(n-1) - a(n-7) for n >= 7. - Vincenzo Librandi, Dec 20 2010

Mathematica

CoefficientList[Series[(6-5*x-4*x^2-3*x^3-2*x^4-x^5)/(1-x-x^2-x^3-x^4-x^5-x^6), {x, 0, 40}], x]
LinearRecurrence[{1, 1, 1, 1, 1, 1}, {6, 1, 3, 7, 15, 31}, 40] (* Harvey P. Dale, Nov 08 2011 *)

Prog

(PARI) polsym(polrecip(1-x-x^2-x^3-x^4-x^5-x^6), 40) \\ G. C. Greubel, Apr 22 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (6-5*x-4*x^2-3*x^3-2*x^4-x^5)/(1-x-x^2-x^3-x^4-x^5-x^6) )); // G. C. Greubel, Apr 22 2019
(Sage) ((6-5*x-4*x^2-3*x^3-2*x^4-x^5)/(1-x-x^2-x^3-x^4-x^5-x^6)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 22 2019
(Python)
def aupton(nn):
  alst = [6, 1, 3, 7, 15, 31]
  for n in range(6, nn+1): alst.append(sum(alst[n-6:n]))
  return alst[:nn+1]
print(aupton(33)) # Michael S. Branicky, Jun 01 2021

Crossrefs

Cf. A000078, A001630, A001644, A000032, A073817, A074048.

Sequence in context: A296504 A360779 A188859 * A195478 A259731 A176399

Adjacent sequences: A074581 A074582 A074583 * A074585 A074586 A074587

Keyword

easy,nonn

Author

Mario Catalani (mario.catalani(AT)unito.it), Aug 26 2002

Status

approved