A121671 Real part of (1 + n*i)^5.

1, -4, 41, 316, 1121, 2876, 6121, 11516, 19841, 31996, 49001, 71996, 102241, 141116, 190121, 250876, 325121, 414716, 521641, 647996, 796001, 967996, 1166441, 1393916, 1653121, 1946876, 2278121, 2649916, 3065441, 3527996, 4041001, 4607996, 5232641, 5918716

Offset

0,2

Comments

The imaginary term considered as an unsigned real integer = A121672(n). The companion sequence A121672 uses the operation (n + i)^5.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

From Bruno Berselli, Mar 01 2012: (Start)

G.f.: (1-9*x+71*x^2+61*x^3-4*x^4)/(1-x)^5.

a(n) = 5*n^4 - 10*n^2 + 1. (End)

a(n) = (1+n^2)^(5/2)*cos(5*arctan(n)). - Gerry Martens, Apr 06 2024

From Elmo R. Oliveira, Jun 05 2026: (Start)

E.g.f.: exp(x)*(1 - 5*x + 25*x^2 + 30*x^3 + 5*x^4).

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).

a(n) = 5*A178392(n) + 1. (End)

Example

a(4) = 1121 since (1 + 4i)^5 = (1121 + 404i) where 404 = A121672(4).

Mathematica

Table[Re[(1 + n*I)^5], {n, 0, 35}] (* T. D. Noe, Mar 01 2012 *)
(* Alternative: *)
LinearRecurrence[{5, -10, 10, -5, 1}, {1, -4, 41, 316, 1121}, 40] (* Harvey P. Dale, Apr 21 2019 *)

Prog

(PARI) a(n) = real((1 + n*I)^5); \\ Michel Marcus, Dec 19 2020
(Python)
def A121671(n): return 5*n**2*(n**2-2)+1 # Chai Wah Wu, Jun 05 2026

Crossrefs

Cf. A121672, A178392.

Sequence in context: A057419 A389193 A089664 * A089454 A193368 A109109

Adjacent sequences: A121668 A121669 A121670 * A121672 A121673 A121674

Keyword

sign,easy

Author

Gary W. Adamson, Aug 14 2006

Extensions

Corrected and extended by T. D. Noe, Mar 01 2012

Status

approved