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%I #40 Feb 11 2023 16:21:01
%S 0,1,50,147,292,485,726,1015,1352,1737,2170,2651,3180,3757,4382,5055,
%T 5776,6545,7362,8227,9140,10101,11110,12167,13272,14425,15626,16875,
%U 18172,19517,20910,22351,23840,25377,26962,28595,30276,32005,33782,35607,37480
%N 50-gonal numbers: a(n) = 48*n*(n-1)/2 + n.
%C According to the common formula for the polygonal numbers: (s-2)*n*(n-1)/2 + n (here s = 50).
%C 96*a(n) + 23^2 is a square.
%H <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = n*(24*n - 23).
%F G.f.: x*(1+47*x)/(1-x)^3. - _Vincenzo Librandi_, Aug 17 2015
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Vincenzo Librandi_, Aug 17 2015
%F E.g.f.: exp(x)*(x + 24*x^2). - _Nikolaos Pantelidis_, Feb 10 2023
%p A261343:=n->n*(24*n-23): seq(A261343(n), n=0..40); # _Wesley Ivan Hurt_, Aug 20 2015
%t PolygonalNumber[50,Range[0,40]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Oct 11 2019 *)
%o (JavaScript) function a(n){return 48*n*(n-1)/2+n}
%o (PARI) first(m)=vector(m,n,n--;n*(24*n-23)) \\ _Anders Hellström_, Aug 15 2015
%o (Magma) [n*(24*n-23): n in [0..40]]; // _Vincenzo Librandi_, Aug 17 2015
%Y Cf. A001107, A051872, A254474.
%K nonn,easy
%O 0,3
%A _Sergey Pavlov_, Aug 15 2015
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