A127840 a(1)=1, a(2)=...=a(6)=0, a(n) = a(n-6)+a(n-5) for n>6.

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 2, 1, 0, 0, 1, 3, 3, 1, 0, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 2, 6, 15, 20, 15, 7, 8, 21, 35, 35, 22, 15, 29, 56, 70, 57, 37, 44, 85, 126, 127, 94, 81, 129, 211, 253, 221, 175, 210, 340, 464, 474, 396, 385, 550

Offset: 1

Author

Stephen Suter (sms5064(AT)psu.edu), Apr 02 2007

Keywords

• nonn
• easy

Comments

Part of the phi_k family of sequences defined by a(1)=1, a(2)=...=a(k)=0, a(n)=a(n-k)+a(n-k+1) for n>k. phi_2 is a shift of the Fibonacci sequence and phi_3 is a shift of the Padovan sequence.

References

• S. Suter, Binet-like formulas for recurrent sequences with characteristic equation x^k=x+1, preprint, 2007. [Apparently unpublished as of May 2016]

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Sadjia Abbad and Hacène Belbachir, The r-Fibonacci polynomial and its companion sequences linked with some classical sequences, Integers (2025), Vol. 25, Art. No. A38. See p. 17.
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1,1).

Programs

PARI

Vec(x*(1-x)*(1+x+x^2+x^3+x^4)/(1-x^5-x^6) + O(x^100)) \\ Colin Barker, May 30 2016

Formula

Binet-like formula: a(n) = Sum_{i=1..6} (r_i^n)/(5(r_i)^2+6(r_i)) where r_i is a root of x^6=x+1.

a(n) = A017837(n-6). - R. J. Mathar, Sep 20 2012

G.f.: x*(1-x)*(1+x+x^2+x^3+x^4) / (1-x^5-x^6). - Colin Barker, May 30 2016