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A007587
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12-gonal (or dodecagonal) pyramidal numbers: n(n+1)(10n-7)/6.
(Formerly M4895)
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7
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0, 1, 13, 46, 110, 215, 371, 588, 876, 1245, 1705, 2266, 2938, 3731, 4655, 5720, 6936, 8313, 9861, 11590, 13510, 15631, 17963, 20516, 23300, 26325, 29601, 33138, 36946, 41035, 45415, 50096, 55088, 60401, 66045, 72030, 78366, 85063, 92131, 99580, 107420, 115661
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OFFSET
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0,3
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COMMENTS
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Binomial transform of [1, 12, 21, 10, 0, 0, 0, ...] = (1, 13, 46, 110, ...). - Gary W. Adamson, Nov 28 2007
This sequence is related to A000566 by a(n) = n*A000566(n) - Sum_{i=0..n-1} A000566(i) and this is the case d=5 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, Oct 18 2010
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REFERENCES
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A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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B. Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian), 2008.
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FORMULA
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a(n) = (10*n-7)*binomial(n+1, 2)/3.
G.f.: x*(1+9*x)/(1-x)^4.
a(n) = Sum_{i=0..n-1} (n-i)*(10*i+1), with a(0)=0. - Bruno Berselli, Feb 10 2014
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MAPLE
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MATHEMATICA
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CoefficientList[Series[x(1+9x)/(1-x)^4, {x, 0, 45}], x] (* Vincenzo Librandi, Jun 20 2013 *)
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PROG
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(Magma) [ n eq 1 select 0 else Self(n-1)+(n-1)*(5*n-9): n in [1..45] ]; // Klaus Brockhaus, Nov 20 2008
(PARI) a(n)=if(n, ([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; -1, 4, -6, 4]^n*[0; 1; 13; 46])[1, 1], 0) \\ Charles R Greathouse IV, Oct 07 2015
(PARI) vector(45, n, n*(n-1)*(10*n-17)/6) \\ G. C. Greubel, Aug 30 2019
(Sage) [n*(n+1)*(10*n-7)/6 for n in (0..45)] # G. C. Greubel, Aug 30 2019
(GAP) List([0..45], n-> n*(n+1)*(10*n-7)/6); # G. C. Greubel, Aug 30 2019
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CROSSREFS
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See similar sequences listed in A237616.
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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