A016161 Expansion of g.f. 1/((1-5*x)*(1-7*x)).

1, 12, 109, 888, 6841, 51012, 372709, 2687088, 19200241, 136354812, 964249309, 6798573288, 47834153641, 336059778612, 2358521965909, 16540171339488, 115933787267041, 812299450322412, 5689910849522509

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Also, this is the number of incongruent integer-edged Heron triangles whose circumdiameter is the product of n distinct primes each of shape 4k + 1. Cf. A003462, A109021. - R. K. Guy, Jan 31 2007

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (12,-35).

Crossrefs

Cf. A003462, A080962, A081200, A109021.

Cf. A016162, A016163, A016164, A016165, A016166.

Programs

Magma

[n le 2 select 12^(n-1) else 12*Self(n-1) -35*Self(n-2): n in [1..30]]; // G. C. Greubel, Nov 09 2024

Mathematica

CoefficientList[Series[1/((1-5x)(1-7x)),{x,0,30}],x] (* or *) LinearRecurrence[ {12,-35},{1,12},30] (* Harvey P. Dale, Nov 16 2021 *)

PARI

Vec(1/((1-5*x)*(1-7*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 24 2012

SageMath

A016161=BinaryRecurrenceSequence(12,-35,1,12)
[A016161(n) for n in range(31)] # G. C. Greubel, Nov 09 2024

Formula

From R. J. Mathar, Sep 18 2008: (Start)

a(n) = (7^(n+1) - 5^(n+1))/2 = A081200(n+1).

Binomial transform of A080962. (End)

a(n) = 7*a(n-1) + 5^n. - Vincenzo Librandi, Feb 09 2011

E.g.f.: exp(5*x)*(7*exp(2*x) - 5)/2. - Stefano Spezia, Oct 25 2023