A127839 a(1)=1, a(2)=...=a(5)=0, a(n) = a(n-5) + a(n-4) for n > 5.

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 1, 3, 3, 1, 1, 4, 6, 4, 2, 5, 10, 10, 6, 7, 15, 20, 16, 13, 22, 35, 36, 29, 35, 57, 71, 65, 64, 92, 128, 136, 129, 156, 220, 264, 265, 285, 376, 484, 529, 550, 661, 860, 1013, 1079, 1211

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

Links

• Harvey P. Dale, 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 (0,0,0,1,1)

Programs

Mathematica

LinearRecurrence[{0,0,0,1,1},{1,0,0,0,0},70] (* Harvey P. Dale, Mar 19 2012 *)

Formula

Binet-like formula: a(n) = Sum_{i=1...5} (r_i^n)/(4(r_i)^2+5(r_i)) where r_i is a root of x^5=x+1.

G.f.: x*(x^4-1)/(x^5+x^4-1). - Harvey P. Dale, Mar 19 2012

a(n) = A017827(n-6) for n >= 6. - R. J. Mathar, May 09 2013