A245489 a(n) = (1^n + (-2)^n + 4^n)/3.

1, 1, 7, 19, 91, 331, 1387, 5419, 21931, 87211, 349867, 1397419, 5593771, 22366891, 89483947, 357903019, 1431677611, 5726579371, 22906579627, 91625794219, 366504225451, 1466014804651, 5864063412907, 23456245263019, 93824997829291, 375299957762731

Offset: 0

Author

Michael Somos, Jul 23 2014

Keywords

• nonn
• easy

Examples

			G.f. = 1 + x + 7*x^2 + 19*x^3 + 91*x^4 + 331*x^5 + 1387*x^6 + 5419*x^7 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,6,-8).

Crossrefs

Cf. A001045, A018240, A129362.

Programs

GAP

List([0..30], n-> (1 +(-2)^n +4^n)/3); # G. C. Greubel, Sep 21 2019

Magma

[(1^n + (-2)^n + 4^n) / 3 : n in [0..30]]; // Vincenzo Librandi, Jul 25 2014

Maple

seq((1 +(-2)^n +4^n)/3, n=0..30); # G. C. Greubel, Sep 21 2019

Mathematica

CoefficientList[Series[(1-2x-2x^2)/((1-x)(1+2x)(1-4x)), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 25 2014 *)
LinearRecurrence[{3,6,-8},{1,1,7},30] (* Harvey P. Dale, Dec 04 2018 *)

PARI

{a(n) = (1^n + (-2)^n + 4^n) / 3};

PARI

{a(n) = if( n<0, 4^n, 1) * polcoeff( (1 - 2*x - 2*x^2) / ((1 - x) * (1 + 2*x) * (1 - 4*x)) + x * O(x^abs(n)), abs(n))};

Sage

[(1 +(-2)^n +4^n)/3 for n in (0..30)] # G. C. Greubel, Sep 21 2019

Formula

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

0 = 8*a(n) - 6*a(n+1) - 3*a(n+2) + a(n+3) for all n in Z.

a(2*n) = A018240(4*n + 3). a(2*n + 1) = A129362(4*n).

a(n) = A001045(3*n)/(3*A001045(n)) for n >= 1. - Peter Bala, Apr 06 2015

E.g.f.: (exp(x) + exp(4*x) + exp(-2*x))/3. - G. C. Greubel, Sep 21 2019