A022521 - OEIS

%I #77 Apr 29 2026 15:16:48

%S 1,31,211,781,2101,4651,9031,15961,26281,40951,61051,87781,122461,

%T 166531,221551,289201,371281,469711,586531,723901,884101,1069531,

%U 1282711,1526281,1803001,2115751,2467531,2861461,3300781,3788851,4329151,4925281,5580961,6300031,7086451

%N a(n) = (n+1)^5 - n^5.

%C 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

%C In Conway and Guy, these numbers are called nexus numbers of order 5. - _M. F. Hasler_, Jan 27 2013

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

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

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

%H Vincenzo Librandi, <a href="/A022521/b022521.txt">Table of n, a(n) for n = 0..10000</a>

%H William Cheah and David Treeby, <a href="https://arxiv.org/abs/2604.20889">Structure and Growth of Galileo Sequences</a>, arXiv:2604.20889 [math.GM], 2026. See p. 7.

%H Polytope Wiki, <a href="https://polytope.miraheze.org/wiki/Triangular-antitegmatic_icosachoron">Triangular-antitegmatic Icosachoron</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

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

%F 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

%F 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

%F 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

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

%t Table[(n+1)^5-n^5, {n,0,30}] (* _Vincenzo Librandi_, Nov 23 2011 *)

%o (Magma) [(n+1)^5-n^5: n in [0..30]]; // _Vincenzo Librandi_, Nov 23 2011

%o (PARI) a(n)=(n+1)^5-n^5 \\ _Charles R Greathouse IV_, Sep 24 2015

%Y First differences of A000584.

%Y Column k=4 of array A047969.

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

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_