A017847 Expansion of 1/(1 - x^6 - x^7).

1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 3, 3, 1, 0, 0, 1, 4, 6, 4, 1, 0, 1, 5, 10, 10, 5, 1, 1, 6, 15, 20, 15, 6, 2, 7, 21, 35, 35, 21, 8, 9, 28, 56, 70, 56, 29, 17, 37, 84, 126, 126, 85, 46, 54, 121, 210, 252, 211, 131, 100, 175, 331, 462, 463, 342, 231, 275, 506, 793, 925

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of compositions of n into parts 6 and 7. - Joerg Arndt, Jun 27 2013

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Johann Cigler, Recurrences for certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials, arXiv:2212.02118 [math.NT], 2022.
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1,1).

Crossrefs

Column k=6 of A306713.

Programs

Magma

m:=70; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-x^6-x^7))); // Vincenzo Librandi, Jun 27 2013

Magma

I:=[1,0,0,0,0,0,1]; [n le 7 select I[n] else Self(n-6)+Self(n-7): n in [1..70]]; // Vincenzo Librandi, Jun 27 2013

Mathematica

CoefficientList[Series[1/(1-x^6-x^7), {x, 0, 70}], x] (* or *)  LinearRecurrence[{0, 0, 0, 0, 0, 1, 1}, {1, 0, 0, 0, 0, 0, 1}, 70] (* Harvey P. Dale, Dec 15 2012 *)
CoefficientList[Series[1 / (1 - Total[x^Range[6, 7]]),{x, 0, 70}], x] (* Vincenzo Librandi, Jun 27 2013 *)

Formula

a(0)=1, a(1)=0,a(2)=0, a(3)=0, a(4)=0, a(5)=0, a(6)=1; for n>6, a(n) = a(n-6)+a(n-7). - Harvey P. Dale, Dec 15 2012

a(n) = Sum_{k=0..floor(n/6)} binomial(k,n-6*k). - Seiichi Manyama, Oct 01 2024