A317298 a(n) = (1/2)*(1 + (-1)^n + 2*n + 4*n^2).

1, 3, 11, 21, 37, 55, 79, 105, 137, 171, 211, 253, 301, 351, 407, 465, 529, 595, 667, 741, 821, 903, 991, 1081, 1177, 1275, 1379, 1485, 1597, 1711, 1831, 1953, 2081, 2211, 2347, 2485, 2629, 2775, 2927, 3081, 3241, 3403, 3571, 3741, 3917, 4095, 4279, 4465, 4657

Offset: 0

Author

Stefano Spezia, Jan 22 2019

Keywords

• nonn
• easy

Comments

For n > 0, first differences of A304487.

All the terms of this sequence are odd numbers.

Links

• Stefano Spezia, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

Cf. A000290, A004526, A005408, A033567, A033999, A188135, A304487.

Cf. A306362 (prime numbers subsequence).

Programs

GAP

Flat(List([0..50], n->(1/2)*(1 + (-1)^n + 2*n + 4*n^2)));

Magma

[(1/2)*(1+(-1)^n+2*n+4*n^2): n in [0..50]];

Maple

a:=n->(1/2)*(1 + (-1)^n + 2*n + 4*n^2): seq(a(n), n=0..50);

Mathematica

a[n_]:=(1/2)*(1 + (-1)^n + 2*n + 4*n^2); Array[a, 50, 0]

Maxima

makelist((1/2)*(1+(-1)^n+2*n+4*n^2), n, 0, 50);

PARI

a(n) = (1/2)*(1+(-1)^n+2*n+4*n^2);

Python

[(1+(-1)**n+2*n+4*n**2)/2 for n in range(0,50)]

Formula

a(n) = (1/2)*(A033999(n) + A005408(n) + 4*A000290(n)).

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

a(2*n) = A188135(n).

a(2*n-1) = A033567(n), for n > 0.

O.g.f.: -(1 + x + 5*x^2 + x^3)/(-1 + x)^3*(1 + x).

E.g.f.: (1/2)*exp(-x)*(1 + exp(2*x)*(1 + 6*x + 4*x^2)).

Sum_{n>0} 1/a(n) = (1/4)*(Pi - log(4)) + i*(polygamma(0, 1/8 - i*sqrt(7)/8) - polygamma(0, 1/8 + i*sqrt(7)/8))/(2*sqrt(7)) = 1.603596691017309384564895..., where i is the imaginary unit. - Stefano Spezia, Feb 10 2019

a(n) = 1 + 2*(n^2 + floor(n/2)). - Stefano Spezia, Dec 08 2021