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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Polytope Wiki, Triangular-antitegmatic Icosachoron
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

First differences of A000584.

Column k=4 of array A047969.

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

Programs

Magma

[(n+1)^5-n^5: n in [0..30]]; // Vincenzo Librandi, Nov 23 2011

Mathematica

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

PARI

a(n)=(n+1)^5-n^5 \\ Charles R Greathouse IV, Sep 24 2015

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