A323225 a(n) = ((2^n*n + i*(1 - i)^n - i*(1 + i)^n))/4, where i is the imaginary unit.

0, 1, 3, 7, 16, 38, 92, 220, 512, 1160, 2576, 5648, 12288, 26592, 57280, 122816, 262144, 557184, 1179904, 2490624, 5242880, 11009536, 23067648, 48233472, 100663296, 209717248, 436211712, 905973760, 1879048192, 3892305920, 8053047296, 16642981888, 34359738368

Offset: 0

Author

Peter Luschny, Mar 18 2019

Keywords

• nonn
• easy

Comments

Related to Clifford algebras (see A323100 and A323346).

Links

• Robert Israel, Table of n, a(n) for n = 0..3308
• Index entries for linear recurrences with constant coefficients, signature (6,-14,16,-8).

Crossrefs

Antidiagonal sums of A323346.

Cf. A001787, A009545, A321959, A323100.

Programs

Maple

a := n -> ((2^n*n + I*(1 - I)^n - I*(1 + I)^n))/4:
seq(a(n), n=0..32);

Mathematica

LinearRecurrence[{6, -14, 16, -8}, {0, 1, 3, 7}, 40] (* Jean-François Alcover, Mar 20 2019 *)
Table[((2^n n + I (1 - I)^n - I (1 + I)^n))/4, {n, 0, 29}] (* Alonso del Arte, Mar 27 2020 *)

Formula

a(n) = Sum_{k = 0..n} A323346(n - k, k - 1).

a(n) = (A001787(n) + A009545(n))/2.

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

a(n) = n! [x^n] (exp(2*x)*x + exp(x)*sin(x))/2.

a(n) = (4*n*a(n-3) + (2 - 6*n)*a(n-2) + (4*n - 2)*a(n-1))/(n - 1) for n >= 3.