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A074048 Pentanacci numbers with initial conditions a(0)=5, a(1)=1, a(2)=3, a(3)=7, a(4)=15. 32
5, 1, 3, 7, 15, 31, 57, 113, 223, 439, 863, 1695, 3333, 6553, 12883, 25327, 49791, 97887, 192441, 378329, 743775, 1462223, 2874655, 5651423, 11110405, 21842481, 42941187, 84420151, 165965647, 326279871, 641449337, 1261056193, 2479171199, 4873922247 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

These pentanacci numbers follow the same pattern as Lucas, generalized tribonacci(A001644) and generalized tetranacci (A073817) numbers: Binet's formula is a(n)=r1^n+r^2^n+r3^n+r4^n+r5^n, with r1, r2, r3, r4, r5 roots of the characteristic polynomial. a(n) is also the trace of A^n, where A is the pentamatrix ((1,1,0,0,0),(1,0,1,0,0),(1,0,0,1,0),(1,0,0,0,1),(1,0,0,0,0)).

For n >= 5, a(n) is the number of cyclic sequences consisting of n zeros and ones that do not contain five consecutive ones provided the positions of the zeros and ones are fixed on a circle. This is proved in Charalambides (1991) and Zhang and Hadjicostas (2015). (For n=1,2,3,4 the statement is still true provided we allow the sequence to wrap around itself on a circle). - Petros Hadjicostas, Dec 18 2016

a(3407) has 1001 decimal digits. - Michael De Vlieger, Dec 28 2016

LINKS

Michael De Vlieger, Table of n, a(n) for n = 0..3406

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, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.4.

S. Saito, T. Tanaka, N. Wakabayashi, Combinatorial Remarks on the Cyclic Sum Formula for Multiple Zeta Values, Journal of Integer Sequences, 14 (2011), # 11.2.4, Table 3.

Yüksel Soykan, On A Generalized Pentanacci Sequence, Asian Research Journal of Mathematics (2019) Vol. 14, No. 3, 1-9.

Yüksel Soykan, Sum Formulas for Generalized Fifth-Order Linear Recurrence Sequences, Journal of Advances in Mathematics and Computer Science (2019) Vol. 34, No. 5, 1-14.

E. Weisstein, Fibonacci n-Step

L. Zhang and P. Hadjicostas, On sequences of independent Bernoulli trials avoiding the pattern '11..1', Math. Scientist, 40 (2015), 89-96.

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

FORMULA

a(n) = a(n-1) +a(n-2) +a(n-3) +a(n-4) +a(n-5).

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

a(n) = 2*a(n-1) -a(n-6), n>5. [Vincenzo Librandi, Dec 20 2010]

For k>0 and n>=0, a(n+5*k) = a(k)*a(n+4*k) - A123127(k-1)*a(n+3*k) + A123126(k-1)*a(n+2*k) - A074062(k)*a(n+k) + a(n). For example, if k=4, n=3, we have a(n+5*k) = a(23) = 5651423, a(4)*a(19) - A123127(3)*a(15) + A123126(3)*a(1695) - A074062(4)*a(7) + a(3) = (15)*(378329) - (1)*(25327) + (1)*(1695) - (-1)*(113) + (7) = 5651423. - Kai Wang, Sep 13 2020

From Kai Wang, Dec 16 2020: (Start)

For k >= 0,

    | a(k+4) a(k+5) a(k+6) a(k+7) a(k+8) |

    | a(k+3) a(k+4) a(k+5) a(k+6) a(k+7) |

det | a(k+2) a(k+3) a(k+4) a(k+5) a(k+6) | = 9584 = A106273(5).

    | a(k+1) a(k+2) a(k+3) a(k+4) a(k+5) |

    | a(k)   a(k+1) a(k+2) a(k+3) a(k+4) |

(End)

MATHEMATICA

CoefficientList[Series[(5-4*x-3*x^2-2*x^3-x^4)/(1-x-x^2-x^3-x^4-x^5), {x, 0, 30}], x]

LinearRecurrence[{1, 1, 1, 1, 1}, {5, 1, 3, 7, 15}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2012 *)

PROG

(PARI) polsym(polrecip(1-x-x^2-x^3-x^4-x^5), 33) \\ Joerg Arndt, Jan 28 2019

CROSSREFS

Cf. A000078, A001630, A001644, A000032, A073817, A106297 (Pisano Periods).

Essentially the same as A023424.

Cf. A123126, A123127, A074062.

Cf. A106273.

Sequence in context: A242303 A214803 A225984 * A244350 A176321 A248130

Adjacent sequences:  A074045 A074046 A074047 * A074049 A074050 A074051

KEYWORD

easy,nonn,changed

AUTHOR

Mario Catalani (mario.catalani(AT)unito.it), Aug 14 2002

STATUS

approved

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Last modified January 22 02:33 EST 2021. Contains 340359 sequences. (Running on oeis4.)