A121625 Real part of (n + n*i)^n.

1, 1, 0, -54, -1024, -12500, 0, 6588344, 268435456, 6198727824, 0, -9129973459552, -570630428688384, -19384006821904192, 0, 56050417968750000000, 4722366482869645213696, 211773507042902211629312, 0, -1012950863698080557631477248, -107374182400000000000000000000

Offset

0,4

Links

Table of n, a(n) for n=0..20.

Formula

a(n) = Re(n + n*i)^n.

From Chai Wah Wu, Feb 15 2024: (Start)

a(n) = n^n*Re((1+i)^n) = n^n*A146559(n) = n^n*Sum_{n=0..floor(n/2)} binomial(n,2j)*(-1)^j.

a(n) = 0 if and only if n==2 mod 4, as (1+i)^2=2i is purely imaginary, (1+i)^4=-4 is a nonzero real and (1+i) and (1+i)^3=-2+2i both have nonzero real parts.

(End)

Example

a(7) = 6588344 since (7 + 7i)^7 = (6588344 - 6588344i).

Mathematica

a[n_] := Re[(n + n*I)^n]; Array[a, 19] (* Robert G. Wilson v, Aug 17 2006 *)

Prog

(PARI) a(n) = real((n + n*I)^n); \\ Michel Marcus, Dec 19 2020
(Python)
def A121625(n): return n**n*((1, 1, 0, -2)[n&3]<<((n>>1)&-2))*(-1 if n&4 else 1) # Chai Wah Wu, Feb 16 2024

Crossrefs

Cf. A115415, A121626, A146559.

Sequence in context: A324952 A298718 A245832 * A341940 A190399 A071803

Adjacent sequences: A121622 A121623 A121624 * A121626 A121627 A121628

Keyword

sign

Author

Gary W. Adamson, Aug 12 2006

Extensions

More terms from Robert G. Wilson v, Aug 17 2006

a(0)=1 prepended by Alois P. Heinz, Dec 19 2020

Status

approved