A193771 Expansion of 1 / (1 - x - x^3 + x^6) in powers of x.

1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 17, 23, 31, 41, 54, 72, 96, 127, 168, 223, 296, 392, 519, 688, 912, 1208, 1600, 2120, 2809, 3721, 4929, 6530, 8651, 11460, 15181, 20111, 26642, 35293, 46753, 61935, 82047, 108689, 143982, 190736, 252672, 334719, 443408, 587391

Offset: 0

Author

Michael Somos, Jan 01 2013

Keywords

• nonn

Examples

			G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 5*x^6 + 7*x^7 + 10*x^8 + 13*x^9 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• Index entries for linear recurrences with constant coefficients, signature (1, 0, 1, 0, 0, -1).

Crossrefs

Cf. A000931, A008621, A017818.

Programs

Magma

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-x-x^3+x^6)));  // G. C. Greubel, Aug 10 2018

Mathematica

CoefficientList[Series[1/(1-x-x^3+x^6),{x,0,50}],x] (* or *) LinearRecurrence[ {1,0,1,0,0,-1},{1,1,1,2,3,4},50] (* Harvey P. Dale, Jul 25 2017 *)

PARI

{a(n) = if( n<0, n = -n; polcoeff( - x^6 / (1 - x^3 - x^5 + x^6) + x * O(x^n), n), polcoeff( 1 / (1 - x - x^3 + x^6) + x * O(x^n), n))};

Formula

G.f.: 1 / (1 - x - x^3 + x^6) = 1 / (1 - x / (1 - x^2 / (1 + x^2 / (1 - x / (1 + x / (1 + x^2 / (1 - x^2))))))).

a(n) = a(n-1) + a(n-3) - a(n-6) for all n in Z.

Convolution of A008621 and A000931. PSUM transform of A017818.