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1, 14, 60, 170, 385, 756, 1344, 2220, 3465, 5170, 7436, 10374, 14105, 18760, 24480, 31416, 39729, 49590, 61180, 74690, 90321, 108284, 128800, 152100, 178425, 208026, 241164, 278110, 319145, 364560, 414656, 469744, 530145, 596190
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OFFSET
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0,2
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COMMENTS
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4-dimensional pyramidal number, composed of consecutive 3-dimensional slices; each of which is a 3-dimensional 12-gonal (or dodecagonal) pyramidal number; which in turn is composed of consecutive 2-dimensional slices 12-gonal numbers. - Jonathan Vos Post, Mar 17 2006
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REFERENCES
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Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
Herbert John Ryser, Combinatorial Mathematics, "The Carus Mathematical Monographs", No. 14, John Wiley and Sons, 1963, pp. 1-8.
Murray R. Spiegel, Calculus of Finite Differences and Difference Equations, "Schaum's Outline Series", McGraw-Hill, 1971, pp. 10-20, 79-94.
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LINKS
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FORMULA
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a(n) = C(n+3, 3)*(5*n+2)/2 = (n+1)*(n+2)*(n+3)*(5*n+2)/12.
G.f.: (1+9*x)/(1-x)^5.
Sum_{n>=0} 1/a(n) = (125*log(5) + 10*sqrt(5*(5-2*sqrt(5)))*Pi - 50*sqrt(5)*log(phi) - 84)/104, where phi is the golden ratio (A001622).
Sum_{n>=0} (-1)^n/a(n) = (50*sqrt(5)*log(phi) + 5*sqrt(50-10*sqrt(5))*Pi - 256*log(2) + 90)/52. (End)
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MATHEMATICA
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Accumulate[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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CROSSREFS
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Cf. A093645 ((10, 1) Pascal, column m=4).
Cf. A220212 for a list of sequences produced by the convolution of the natural numbers with the k-gonal numbers.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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