A195163 1000-gonal numbers: a(n) = n*(499*n - 498).

0, 1, 1000, 2997, 5992, 9985, 14976, 20965, 27952, 35937, 44920, 54901, 65880, 77857, 90832, 104805, 119776, 135745, 152712, 170677, 189640, 209601, 230560, 252517, 275472, 299425, 324376, 350325, 377272, 405217, 434160, 464101, 495040, 526977, 559912, 593845, 628776

Offset: 0

Author

Kausthub Gudipati, Sep 10 2011

Keywords

• nonn
• easy

Comments

a(A271470(n)) is a perfect square. In fact, a(A271470(n)) = A271105(n) if the first term of a(n) is 1. - Muniru A Asiru, Apr 10 2016

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• M. A. Asiru, All square chiliagonal numbers, Int J Math Educ Sci Technol, 47:7(2016), 1123-1134.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000578, A092182.

Programs

GAP

a:=[0,1,1000];; for n in [4..10^2] do a[n]:=3*a[n-1]-3*a[n-2]+*a[n-3]; od; a;  # Muniru A Asiru, Sep 12 2017

JavaScript

function a(n){return 998*n*(n-1)/2+n} // Sergey Pavlov, Aug 14 2015

Magma

[n*(499*n-498): n in [0..45]]; // Vincenzo Librandi, Nov 25 2011

Maple

A195163:=n->n*(499*n - 498): seq(A195163(n), n=0..50); # Wesley Ivan Hurt, Sep 16 2017

Mathematica

LinearRecurrence[{3,-3,1},{0,1,1000},50] (* Vincenzo Librandi, Nov 25 2011 *)
PolygonalNumber[1000,Range[0,40]] (* Harvey P. Dale, Sep 15 2022 *)

PARI

a(n)=n*(499*n-498) \\ Charles R Greathouse IV, Sep 11 2011

PARI

x='x+O('x^99); concat(0, Vec(x*(1+997*x)/(1-x)^3)) \\ Altug Alkan, Apr 10 2016

Formula

a(n) = 998*n*(n-1)/2 + n, according to the common formula for s-gonal numbers, a(n) = (s-2)*n*(n-1)/2 + n. - Sergey Pavlov, Aug 14 2015

G.f.: x*(1+997*x)/(1-x)^3. - R. J. Mathar, Sep 12 2011

E.g.f.: exp(x)*x*(1 + 499*x). - Ilya Gutkovskiy, Apr 10 2016

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 4. - Muniru A Asiru, Sep 12 2017