A276283 Expansion of (1 + x + 3*x^2 + x^3)/((1 - x)^2*(1 + x^2)).

1, 3, 7, 11, 13, 15, 19, 23, 25, 27, 31, 35, 37, 39, 43, 47, 49, 51, 55, 59, 61, 63, 67, 71, 73, 75, 79, 83, 85, 87, 91, 95, 97, 99, 103, 107, 109, 111, 115, 119, 121, 123, 127, 131, 133, 135, 139, 143, 145, 147, 151, 155, 157, 159, 163, 167, 169, 171, 175, 179, 181, 183, 187, 191, 193, 195, 199, 203, 205, 207, 211

Offset: 0

Author

Ilya Gutkovskiy, Aug 27 2016

Keywords

• nonn
• easy

Comments

Primes in this sequence: 3, 7, 11, 13, 19, 23, 31, 37, 43, 47, 59, 61, 67, 71, 73, 79, 83, 97, 103, 107, 109, 127, 131, 139, 151, 157, 163, 167, 179, 181, 191, 193, 199, ... (A040116, offset 2).

Links

• Carauleanu Marc, Table of n, a(n) for n = 0..4444
• Ilya Gutkovskiy, Illustration
• Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1)

Crossrefs

Cf. A005408, A040116.

Programs

Maple

a:=series((1+x+3*x^2+x^3)/((1-x)^2*(1+x^2)),x=0,71): seq(coeff(a,x,n),n=0..70); # Paolo P. Lava, Mar 27 2019

Mathematica

LinearRecurrence[{2, -2, 2, -1}, {1, 3, 7, 11}, 71]
Table[3 n - Sin[Pi (n/2)] + 1, {n, 0, 70}]
Table[(6 n - I ((-I)^n - I^n + 2 I))/2, {n, 0, 70}]

PARI

Vec((1+x+3*x^2+x^3)/((1-x)^2*(1+x^2)) + O(x^99)) \\ Altug Alkan, Aug 27 2016

Formula

O.g.f.: (1 + x + 3*x^2 + x^3)/((1 - x)^2*(1 + x^2)).

E.g.f.: (1 + 3*x)*exp(x) - sin(x).

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

a(n) = 3*n - sin(Pi*n/2) + 1.

a(n) = (6*n - i*((-i)^n - i^n + 2*i))/2, where i is the imaginary unit.