%I #15 Feb 16 2025 08:33:27

%S 0,1,20,90,260,595,1176,2100,3480,5445,8140,11726,16380,22295,29680,

%T 38760,49776,62985,78660,97090,118580,143451,172040,204700,241800,

%U 283725,330876,383670,442540,507935,580320,660176,748000,844305,949620,1064490,1189476

%N a(n) = n*(n + 1)*(n + 2)*(4*n - 3)/6.

%C Partial sums of 18-gonal (or octadecagonal) pyramidal numbers. Therefore, this is the case k=8 of the general formula n*(n + 1)*(n + 2)*(k*n - k + 2)/12, which is related to 2*(k+1)-gonal pyramidal numbers.

%H OEIS Wiki, <a href="https://oeis.org/wiki/Figurate_numbers">Figurate numbers</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PyramidalNumber.html">Pyramidal Number</a>

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

%F G.f.: x*(1 + 15*x)/(1 - x)^5.

%F a(n) = Sum_{k = 0..n} A172078(k).

%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - _Vincenzo Librandi_, Nov 27 2015

%t Table[n (n + 1) (n + 2) (4 n - 3)/6, {n, 0, 50}]

%o (Magma) [n*(n + 1)*(n + 2)*(4*n - 3)/6: n in [0..50]]; // _Vincenzo Librandi_, Nov 27 2015

%o (PARI) a(n)=n*(n+1)*(n+2)*(4*n-3)/6 \\ _Charles R Greathouse IV_, Jul 26 2016

%Y Cf. A172078.

%Y Cf. similar sequences with formula n*(n+1)*(n+2)*(k*n-k+2)/12 listed in A264850.

%K nonn,easy,changed

%O 0,3

%A _Ilya Gutkovskiy_, Nov 26 2015