A264850 - OEIS

%I #17 Sep 08 2022 08:46:14

%S 0,1,18,80,230,525,1036,1848,3060,4785,7150,10296,14378,19565,26040,

%T 34000,43656,55233,68970,85120,103950,125741,150788,179400,211900,

%U 248625,289926,336168,387730,445005,508400,578336,655248,739585,831810,932400,1041846

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

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

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

%H <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</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 + 13*x)/(1 - x)^5.

%F a(n) = Sum_{k = 0..n} A172076(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) (7 n - 5)/12, {n, 0, 50}]

%t LinearRecurrence[{5,-10,10,-5,1},{0,1,18,80,230},40] (* _Harvey P. Dale_, Sep 27 2018 *)

%o (Magma) [n*(n+1)*(n+2)*(7*n-5)/12: n in [0..50]]; // _Vincenzo Librandi_, Nov 27 2015

%o (PARI) a(n)=n*(n+1)*(n+2)*(7*n-5)/12 \\ _Charles R Greathouse IV_, Jul 26 2016

%Y Cf. A172076.

%Y 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 nonn,easy

%O 0,3

%A _Ilya Gutkovskiy_, Nov 26 2015