A084169 A Pell Jacobsthal product.

0, 1, 2, 15, 60, 319, 1470, 7267, 34680, 168435, 810898, 3921103, 18918900, 91381991, 441150502, 2130258075, 10285325040, 49663079099, 239791814010, 1157823924167, 5590452446700, 26993130847215, 130334271942158

Offset: 0

Author

Paul Barry, May 18 2003

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,13,4,-4).

Crossrefs

Cf. A000129, A001045, A007985, A057087.

Programs

Magma

[0] cat [(2^n-(-1)^n)*Evaluate(DicksonSecond(n-1,-1), 2)/3: n in [1..40]]; // G. C. Greubel, Oct 11 2022

Mathematica

LinearRecurrence[{2,13,4,-4}, {0,1,2,15}, 41] (* G. C. Greubel, Oct 11 2022 *)

SageMath

def A084169(n): return (2^n-(-1)^n)*lucas_number1(n,2,-1)/3
[A084169(n) for n in range(41)] # G. C. Greubel, Oct 11 2022

Formula

a(n) = (2^n - (-1)^n)*( (1+sqrt(2))^n - (1-sqrt(2))^n )/(6*sqrt(2)).

a(n) = A001045(n)*A000129(n).

G.f.: x*(1-2*x^2)/((1+2*x-x^2)*(1-4*x-4*x^2)). - Colin Barker, May 01 2012

a(n) = (A007985 + 2*A057087(n))/3. - R. J. Mathar, Sep 29 2020