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

1,18

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 [Apparently unpublished as of May 2016]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).

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

Prog

(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

Crossrefs

Sequence in context: A036866 A144225 A159854 * A017837 A145153 A255517

Adjacent sequences: A127837 A127838 A127839 * A127841 A127842 A127843

Keyword

nonn,easy

Author

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

Status

approved