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A264850 a(n) = n*(n + 1)*(n + 2)*(7*n - 5)/12.

%I A264850 #18 Feb 16 2025 08:33:27
%S A264850 0,1,18,80,230,525,1036,1848,3060,4785,7150,10296,14378,19565,26040,
%T A264850 34000,43656,55233,68970,85120,103950,125741,150788,179400,211900,
%U A264850 248625,289926,336168,387730,445005,508400,578336,655248,739585,831810,932400,1041846
%N A264850 a(n) = n*(n + 1)*(n + 2)*(7*n - 5)/12.
%C A264850 Partial sums of 16-gonal (or hexadecagonal) pyramidal numbers. Therefore, this is the case k=7 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 A264850 OEIS Wiki, <a href="https://oeis.org/wiki/Figurate_numbers">Figurate numbers</a>
%H A264850 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PyramidalNumber.html">Pyramidal Number</a>
%H A264850 <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>
%H A264850 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A264850 G.f.: x*(1 + 13*x)/(1 - x)^5.
%F A264850 a(n) = Sum_{k = 0..n} A172076(k).
%F A264850 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 A264850 Table[n (n + 1) (n + 2) (7 n - 5)/12, {n, 0, 50}]
%t A264850 LinearRecurrence[{5,-10,10,-5,1},{0,1,18,80,230},40] (* _Harvey P. Dale_, Sep 27 2018 *)
%o A264850 (Magma) [n*(n+1)*(n+2)*(7*n-5)/12: n in [0..50]]; // _Vincenzo Librandi_, Nov 27 2015
%o A264850 (PARI) a(n)=n*(n+1)*(n+2)*(7*n-5)/12 \\ _Charles R Greathouse IV_, Jul 26 2016
%Y A264850 Cf. A172076.
%Y A264850 Cf. similar sequences with formula n*(n+1)*(n+2)*(k*n-k+2)/12: A000292 (k=0), A002415 (which arises from k=1), A002417 (k=2), A002419 (k=3), A051797 (k=4), A051799 (k=5), A220212 (k=6), this sequence (k=7), A264851 (k=8), A264852 (k=9).
%K A264850 nonn,easy
%O A264850 0,3
%A A264850 _Ilya Gutkovskiy_, Nov 26 2015