A195163 - OEIS

%I #56 Sep 15 2022 13:07:45

%S 0,1,1000,2997,5992,9985,14976,20965,27952,35937,44920,54901,65880,

%T 77857,90832,104805,119776,135745,152712,170677,189640,209601,230560,

%U 252517,275472,299425,324376,350325,377272,405217,434160,464101,495040,526977,559912,593845,628776

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

%C 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

%H Vincenzo Librandi, <a href="/A195163/b195163.txt">Table of n, a(n) for n = 0..10000</a>

%H M. A. Asiru, <a href="http://dx.doi.org/10.1080/0020739X.2016.1164346">All square chiliagonal numbers</a>, Int J Math Educ Sci Technol, 47:7(2016), 1123-1134.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F 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

%F G.f.: x*(1+997*x)/(1-x)^3. - _R. J. Mathar_, Sep 12 2011

%F E.g.f.: exp(x)*x*(1 + 499*x). - _Ilya Gutkovskiy_, Apr 10 2016

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 4. - _Muniru A Asiru_, Sep 12 2017

%p A195163:=n->n*(499*n - 498): seq(A195163(n), n=0..50); # _Wesley Ivan Hurt_, Sep 16 2017

%t LinearRecurrence[{3,-3,1},{0,1,1000},50] (* _Vincenzo Librandi_, Nov 25 2011 *)

%t PolygonalNumber[1000,Range[0,40]] (* _Harvey P. Dale_, Sep 15 2022 *)

%o (PARI) a(n)=n*(499*n-498) \\ _Charles R Greathouse IV_, Sep 11 2011

%o (PARI) x='x+O('x^99); concat(0, Vec(x*(1+997*x)/(1-x)^3)) \\ _Altug Alkan_, Apr 10 2016

%o (Magma) [n*(499*n-498): n in [0..45]]; // _Vincenzo Librandi_, Nov 25 2011

%o (JavaScript) function a(n){return 998*n*(n-1)/2+n} // _Sergey Pavlov_, Aug 14 2015

%o (GAP)

%o 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

%Y Cf. A000578, A092182.

%K nonn,easy

%O 0,3

%A _Kausthub Gudipati_, Sep 10 2011