A293296 a(n) = 2*n^2 - floor(n/4).

0, 2, 8, 18, 31, 49, 71, 97, 126, 160, 198, 240, 285, 335, 389, 447, 508, 574, 644, 718, 795, 877, 963, 1053, 1146, 1244, 1346, 1452, 1561, 1675, 1793, 1915, 2040, 2170, 2304, 2442, 2583, 2729, 2879, 3033, 3190, 3352, 3518, 3688, 3861, 4039, 4221, 4407, 4596

Offset: 0

Author

Peter Luschny, Oct 08 2017

Keywords

• nonn
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Crossrefs

Cf. A001105, A007877.

Programs

Maple

a := n -> 2*n^2 - floor(n/4): seq(a(n), n=0..48);

Mathematica

LinearRecurrence[{2, -1, 0, 1, -2, 1}, {0, 2, 8, 18, 31, 49}, 49]
Table[2n^2-Floor[n/4],{n,0,60}] (* Harvey P. Dale, Jan 08 2022 *)

PARI

a(n) = 2*n^2-n\4; \\ Altug Alkan, Oct 08 2017

Python

def A293296(n): return (n**2<<1)-(n>>2) # Chai Wah Wu, Jan 26 2023

Formula

a(n) = [x^n] (-x*(2+4*x+4*x^2+3*x^3+3*x^4)/((x+1)*(x^2+1)*(x-1)^3)).

a(n) = n! [x^n] (3*exp(x)-exp(-x)+14*exp(x)*x+16*exp(x)*x^2-2*cos(x)-2*sin(x))/8.

a(n) = a(n-6) - 2*a(n-5) + a(n-4) - a(n-2) + 2*a(n-1) for n >= 6.

(-1)^n*(a(n+3) - 3*a(n+2) + 3*a(n+1) - a(n)) = sqrt(n^2 mod 8) = A007877(n).