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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
(list;
graph;
refs;
listen;
history;
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internal format)
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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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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 entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
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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_{k=0..n} k*(5*k-4). - Klaus Brockhaus, Nov 20 2008
a(n) = Sum_{i=0..n-1} (n-i)*(10*i+1), with a(0)=0. - Bruno Berselli, Feb 10 2014
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Oct 23 2014
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MAPLE
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A007587:=n->n*(n+1)*(10*n-7)/6: seq(A007587(n), n=0..50); # Wesley Ivan Hurt, Oct 23 2014
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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 *)
Table[n(n+1)(10n-7)/6, {n, 0, 50}] (* Harvey P. Dale, Nov 13 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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Cf. A093645 ((10, 1) Pascal, column m=3). Partial sums of A051624.
Cf. A000566.
See similar sequences listed in A237616.
Sequence in context: A147208 A281315 A010003 * A318035 A219905 A326163
Adjacent sequences: A007584 A007585 A007586 * A007588 A007589 A007590
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane, R. K. Guy
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STATUS
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approved
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