A317793 a(n) = (4^n + (-3)^n + 2^n + (-1)^n)/2.

1, 15, 22, 177, 406, 2445, 7162, 36177, 121486, 554325, 2009602, 8656377, 32761366, 136617405, 529712842, 2169039777, 8525430046, 34553579685, 136858084882, 551499730377, 2193794127526, 8811785649165, 35137304693722, 140878711512177, 562526325893806

Offset: 1

Author

Jinyuan Wang, Aug 07 2018

Keywords

• nonn
• easy

Comments

This sequence is an extension of A014551; the sequences A014551(n) = 2^n + (-1)^n, a(n) = 4^n + (-3)^n + 2^n + (-1)^n and b(n) = 6^n + (-5)^n + 4^n + (-3)^n + 2^n + (-1)^n, ... can be considered to be of the same type.

For k>0, a(4k-2)/5, a(2k)/3 and a(2k+1)/2 are integers.

Links

• Index entries for linear recurrences with constant coefficients, signature (2,13,-14,-24).

Crossrefs

Cf. A014551, A087451.

Programs

Magma

[(4^n+(-3)^n+2^n+(-1)^n)/2: n in [1..30]]; // Vincenzo Librandi, Aug 08 2018

Mathematica

CoefficientList[ Series[(-48x^3 - 21x^2 + 13x + 1)/(24x^4 + 14x^3 - 13x^2 - 2x + 1), {x, 0, 25}], x]  (* or *)LinearRecurrence[{2, 13, -14, -24}, {1, 15, 22, 177}, 26] (* Robert G. Wilson v, Aug 07 2018 *)

PARI

Vec(x*(1 + 13*x - 21*x^2 - 48*x^3) / ((1 + x)*(1 - 2*x)*(1 + 3*x)*(1 - 4*x)) + O(x^40)) \\ Colin Barker, Aug 07 2018

Formula

a(n) = (4^n + (-3)^n + 2^n + (-1)^n)/2 for n > 0.

From Colin Barker, Aug 07 2018: (Start)

G.f.: x*(1 + 13*x - 21*x^2 - 48*x^3) / ((1 + x)*(1 - 2*x)*(1 + 3*x)*(1 - 4*x)).

a(n) = 2*a(n-1) + 13*a(n-2) - 14*a(n-3) - 24*a(n-4) for n>4.

(End)

E.g.f.: (cosh(3*x/2) + cosh(7*x/2))*(cosh(x/2) + sinh(x/2)) - 2. - Stefano Spezia, Mar 20 2022

Extensions

More terms from Colin Barker, Aug 07 2018