A127843 a(1) = 1, a(2) = ... = a(9) = 0, a(n) = a(n-9)+a(n-8) for n>9.

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 0, 1, 4, 6, 4, 1, 0, 0, 0, 1, 5, 10, 10, 5, 1, 0, 0, 1, 6, 15, 20, 15, 6, 1, 0, 1, 7, 21, 35, 35, 21, 7, 1, 1, 8, 28, 56, 70, 56, 28, 8, 2, 9

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.

Apart from offset same as A017867. - Georg Fischer, Oct 07 2018

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 [Apparently unpublished as of May 2016]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
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,1,1).

Programs

GAP

a:=[1,0,0,0,0,0,0,0,0];;  for n in [10..90] do a[n]:=a[n-8]+a[n-9]; od; a; # Muniru A Asiru, Oct 07 2018

Mathematica

LinearRecurrence[{0,0,0,0,0,0,0,1,1},{1,0,0,0,0,0,0,0,0},120] (* Harvey P. Dale, Jun 15 2017 *)
CoefficientList[Series[(1-x)*(1+x)*(1+x^2)*(1+x^4) / (1-x^8-x^9), {x, 0, 50}], x] (* Stefano Spezia, Oct 08 2018 *)

PARI

Vec(x*(1-x)*(1+x)*(1+x^2)*(1+x^4)/(1-x^8-x^9) + O(x^100)) \\ Colin Barker, May 30 2016

Formula

Binet-like formula: a(n) = Sum_{i=1..9} (r_i^n)/(8(r_i)^2+9(r_i)) where r_i is a root of x^9=x+1.

G.f.: x*(1-x)*(1+x)*(1+x^2)*(1+x^4) / (1-x^8-x^9). - Colin Barker, May 30 2016