A022521 a(n) = (n+1)^5 - n^5.

1, 31, 211, 781, 2101, 4651, 9031, 15961, 26281, 40951, 61051, 87781, 122461, 166531, 221551, 289201, 371281, 469711, 586531, 723901, 884101, 1069531, 1282711, 1526281, 1803001, 2115751, 2467531, 2861461, 3300781, 3788851, 4329151, 4925281, 5580961, 6300031, 7086451

Offset

0,2

Comments

Last digit of a(n) is always 1. Last two digits of a(n) (i.e., a(n) mod 100) are repeated periodically with palindromic part of period 20 {1,31,11,81,1,51,31,61,81,51,51,81,61,31,51,1,81,11,31,1}. Last three digits of a(n) (i.e., a(n) mod 1000) are repeated periodically with palindromic part of period 200. - Alexander Adamchuk, Aug 11 2006

In Conway and Guy, these numbers are called nexus numbers of order 5. - M. F. Hasler, Jan 27 2013

Numbers that can be arranged in a triangular-antitegmatic icosachoron (the 4D version of "rhombic dodecahedal numbers" (A005917)). - Steven Lu, Mar 28 2023

References

John H. Conway and Richard K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 54.

Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), pages 205-206.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

William Cheah and David Treeby, Structure and Growth of Galileo Sequences, arXiv:2604.20889 [math.GM], 2026. See p. 7.

Polytope Wiki, Triangular-antitegmatic Icosachoron

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

Formula

a(n) = A003215(n) + 24 * A006322(n). - Xavier Acloque, Oct 11 2003

G.f.: (-1-x^4-26*x^3-66*x^2-26*x)/(x-1)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009

G.f.: polylog(-5, x)*(1-x)/x. See the g.f. of the rows of A008292 by Vladeta Jovovic, Sep 02 2002. - Wolfdieter Lang, May 10 2021

Sum_{n>=0} 1/a(n) = c1*tanh(c2/2) - c2*tanh(c1/2), where c1 = tan(3*Pi/10)*Pi and c2 = tan(Pi/10)*Pi. - Amiram Eldar, Jan 27 2022

E.g.f.: exp(x)*(1 + 30*x + 75*x^2 + 40*x^3 + 5*x^4). - Stefano Spezia, Oct 28 2025

Mathematica

Table[(n+1)^5-n^5, {n, 0, 30}] (* Vincenzo Librandi, Nov 23 2011 *)

Prog

(Magma) [(n+1)^5-n^5: n in [0..30]]; // Vincenzo Librandi, Nov 23 2011
(PARI) a(n)=(n+1)^5-n^5 \\ Charles R Greathouse IV, Sep 24 2015

Crossrefs

First differences of A000584.

Column k=4 of array A047969.

Cf. A003215, A005917, A006322, A008292, A019916, A019952, A022522.

Sequence in context: A096906 A181124 A142328 * A152730 A361700 A090027

Adjacent sequences: A022518 A022519 A022520 * A022522 A022523 A022524

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved