A219721 Expansion of (1+7*x+5*x^2+7*x^3+x^4)/(1-x-x^4+x^5).

1, 8, 13, 20, 22, 29, 34, 41, 43, 50, 55, 62, 64, 71, 76, 83, 85, 92, 97, 104, 106, 113, 118, 125, 127, 134, 139, 146, 148, 155, 160, 167, 169, 176, 181, 188, 190, 197, 202, 209, 211, 218, 223, 230, 232, 239, 244, 251, 253, 260, 265, 272, 274, 281, 286, 293

Offset: 0

Author

Bruno Berselli, Nov 26 2012

Keywords

• nonn
• easy

Comments

Positive values of y in the Diophantine equation 21*x+1 = y^2; the corresponding values of x are given in A219391.

Equivalently, numbers that are congruent to {1,8,13,20} mod 21.

The product of any two terms belongs to the sequence and therefore also a(n)^2, a(n)^3, a(n)^4 etc.

Links

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

Crossrefs

Cf. A219391.

Programs

Magma

I:=[1,8,13,20,22]; [n le 5 select I[n] else Self(n-1) +Self(n-4)-Self(n-5): n in [1..60]]; // Vincenzo Librandi, Aug 18 2013

Mathematica

CoefficientList[Series[(1 + 7 x + 5 x^2 + 7 x^3 + x^4)/(1 - x - x^4 + x^5), {x, 0, 60}], x]
LinearRecurrence[{1, 0, 0, 1, -1}, {1, 8, 13, 20, 22}, 60] (* Vincenzo Librandi, Aug 18 2013 *)

Maxima

makelist(coeff(taylor((1+7*x+5*x^2+7*x^3+x^4)/(1-x-x^4+x^5), x, 0, n), x, n), n, 0, 60);

PARI

Vec((1+7*x+5*x^2+7*x^3+x^4)/(1-x-x^4+x^5)+O(x^60))

Formula

G.f.: (1+7*x+5*x^2+7*x^3+x^4)/((1+x)*(1-x)^2*(1+x^2)).

a(n) = -a(-n-1) = (42*n-6*i^(n*(n-1))-7*(-1)^n+5)/8 +2, where i=sqrt(-1).