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

0, 1, 18, 80, 230, 525, 1036, 1848, 3060, 4785, 7150, 10296, 14378, 19565, 26040, 34000, 43656, 55233, 68970, 85120, 103950, 125741, 150788, 179400, 211900, 248625, 289926, 336168, 387730, 445005, 508400, 578336, 655248, 739585, 831810, 932400, 1041846

Offset: 0

Author

Ilya Gutkovskiy, Nov 26 2015

Keywords

• nonn
• easy

Comments

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.

Links

• OEIS Wiki, Figurate numbers
• Eric Weisstein's World of Mathematics, Pyramidal Number
• Index to sequences related to pyramidal numbers
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A172076.

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).

Programs

Magma

[n*(n+1)*(n+2)*(7*n-5)/12: n in [0..50]]; // Vincenzo Librandi, Nov 27 2015

Mathematica

Table[n (n + 1) (n + 2) (7 n - 5)/12, {n, 0, 50}]
LinearRecurrence[{5,-10,10,-5,1},{0,1,18,80,230},40] (* Harvey P. Dale, Sep 27 2018 *)

PARI

a(n)=n*(n+1)*(n+2)*(7*n-5)/12 \\ Charles R Greathouse IV, Jul 26 2016

Formula

G.f.: x*(1 + 13*x)/(1 - x)^5.

a(n) = Sum_{k = 0..n} A172076(k).

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