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A264850
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a(n) = n*(n + 1)*(n + 2)*(7*n - 5)/12.
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3
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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
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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Table of n, a(n) for n=0..36.
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).
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FORMULA
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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
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MATHEMATICA
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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 *)
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PROG
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(MAGMA) [n*(n+1)*(n+2)*(7*n-5)/12: n in [0..50]]; // Vincenzo Librandi, Nov 27 2015
(PARI) a(n)=n*(n+1)*(n+2)*(7*n-5)/12 \\ Charles R Greathouse IV, Jul 26 2016
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CROSSREFS
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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).
Sequence in context: A039453 A219144 A063495 * A039408 A043231 A044011
Adjacent sequences: A264847 A264848 A264849 * A264851 A264852 A264853
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KEYWORD
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nonn,easy
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AUTHOR
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Ilya Gutkovskiy, Nov 26 2015
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STATUS
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approved
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