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A172078
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a(n) = n*(16*n^2 + 3*n - 13)/6.
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7
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0, 1, 19, 70, 170, 335, 581, 924, 1380, 1965, 2695, 3586, 4654, 5915, 7385, 9080, 11016, 13209, 15675, 18430, 21490, 24871, 28589, 32660, 37100, 41925, 47151, 52794, 58870, 65395, 72385, 79856, 87824, 96305, 105315, 114870, 124986, 135679
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OFFSET
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0,3
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COMMENTS
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Generated by the formula n*(n+1)*(2*d*n-(2*d-3))/6 for d=8.
In fact, the sequence is related to A001107 by a(n) = n*A001107(n) - Sum_{k=0..n-1} A001107(k), and this is the case d=8 in the identity n*(n*(d*n-d+2)/2) - Sum_{k=0..n-1} k*(d*k-d+2)/2 = n*(n+1)*(2*d*n-2*d+3)/6. - Bruno Berselli, Dec 14 2010
Inverse binomial transform of this sequence: 0, 1, 17, 16, 0, 0 (0 continued). - Bruno Berselli, Dec 14 2010
Principal diagonal of the convolution array A213835. - Clark Kimberling, Jul 04 2012
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REFERENCES
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E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93. - Bruno Berselli, Feb 13 2014
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
B. Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian), 2008.
Index to sequences related to pyramidal numbers
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
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FORMULA
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a(n) = n*(n+1)*(16*n-13)/6.
G.f.: x*(1+15*x)/(1-x)^4. - Bruno Berselli, Dec 14 2010
a(n) = Sum_{i=0..n-1} (n-i)*(16*i+1), with a(0)=0. - Bruno Berselli, Feb 10 2014
E.g.f.: x*(6 +51*x +16*x^2)*exp(x)/6. - G. C. Greubel, Aug 30 2019
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MAPLE
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A172078:=n->(16*n^3+3*n^2-13*n)/6: seq(A172078(n), n=0..50); # Wesley Ivan Hurt, Jan 21 2017
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MATHEMATICA
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LinearRecurrence[{4, -6, 4, -1}, {0, 1, 19, 70}, 50] (* Vincenzo Librandi, Mar 01 2012 *)
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PROG
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(PARI) a(n)=n*(16*n^2+3*n-13)/6 \\ Charles R Greathouse IV, Oct 07 2015
(Magma) [n*(n+1)*(16*n-13)/6: n in [0..40]]; // G. C. Greubel, Aug 30 2019
(Sage) [n*(n+1)*(16*n-13)/6 for n in (0..40)] # G. C. Greubel, Aug 30 2019
(GAP) List([0..40], n-> n*(n+1)*(16*n-13)/6); # G. C. Greubel, Aug 30 2019
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CROSSREFS
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Cf. A001107.
Cf. similar sequences listed in A237616.
Sequence in context: A007547 A217081 A010007 * A196136 A198002 A093350
Adjacent sequences: A172075 A172076 A172077 * A172079 A172080 A172081
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KEYWORD
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nonn,easy
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AUTHOR
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Vincenzo Librandi, Jan 25 2010
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STATUS
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approved
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