A238738 Expansion of (1 + 2*x + 2*x^2)/(1 - x - 2*x^3 + 2*x^4 + x^6 - x^7).

1, 3, 5, 7, 11, 15, 18, 24, 30, 34, 42, 50, 55, 65, 75, 81, 93, 105, 112, 126, 140, 148, 164, 180, 189, 207, 225, 235, 255, 275, 286, 308, 330, 342, 366, 390, 403, 429, 455, 469, 497, 525, 540, 570, 600, 616, 648, 680, 697, 731, 765, 783, 819, 855, 874

Offset: 0

Author

Bruno Berselli, Mar 04 2014

Keywords

• nonn
• easy

Comments

Subsequence of A008732: a(n) = A008732(A047212(n+1)).

See also Deléham's example in A008732: these numbers are in the first (A000566), third (A005475) and fifth (A028895) column.

Examples

			G.f.: 1 + 3*x + 5*x^2 + 7*x^3 + 11*x^4 + 15*x^5 + 18*x^6 + 24*x^7 + ...
		

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• Bruno Berselli, Illustration of the initial terms.
• Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Crossrefs

Cf. A000212 (see illustration above), A000217, A008732, A211538.

Programs

Magma

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+2*x+2*x^2)/(1-x-2*x^3+2*x^4+x^6-x^7)));

Mathematica

CoefficientList[Series[(1 + 2 x + 2 x^2)/(1 - x - 2 x^3 + 2 x^4 + x^6 - x^7), {x, 0, 60}], x]

Maxima

makelist(coeff(taylor((1+2*x+2*x^2)/(1-x-2*x^3+2*x^4+x^6-x^7), x, 0, n), x, n), n, 0, 60);

PARI

Vec((1+2*x+2*x^2)/(1-x-2*x^3+2*x^4+x^6-x^7)+O(x^60))

Sage

m = 60; L. = PowerSeriesRing(ZZ, m); f = (1+2*x+2*x^2)/(1-x-2*x^3+2*x^4+x^6-x^7); print(f.coefficients())

Formula

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

a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7), with n>6.

a(3k) = k*(5*k + 7)/2 + 1 (A000566);

a(3k+1) = k*(5*k + 11)/2 + 3 (A005475);

a(3k+2) = k*(5*k + 15)/2 + 5 (A028895).

a(n) = (floor(n/3)+1)*(4*n-7*floor(n/3)+2)/2. [Luce ETIENNE, Jun 14 2014]