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A004126
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a(n) = n*(7*n^2 - 1)/6.
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21
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0, 1, 9, 31, 74, 145, 251, 399, 596, 849, 1165, 1551, 2014, 2561, 3199, 3935, 4776, 5729, 6801, 7999, 9330, 10801, 12419, 14191, 16124, 18225, 20501, 22959, 25606, 28449, 31495, 34751, 38224, 41921, 45849, 50015, 54426, 59089, 64011
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OFFSET
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0,3
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COMMENTS
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3-dimensional analog of centered polygonal numbers.
Sum of n triangular numbers starting from T(n), where T = A000217. E.g., a(4) = T(4) + T(5) + T(6) + T(7) = 10 + 15 + 21 + 28 = 74. - Amarnath Murthy, Jul 16 2004
Also as a(n) = (1/6)*(7*n^3-n), n>0: structured heptagonal diamond numbers (vertex structure 8). Cf. A100179 = alternate vertex; A000447 = structured diamonds; A100145 for more on structured numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004
Binomial transform of (0, 1, 7, 7, 0, 0, 0, ...) and third partial sum of (0, 1, 6, 7, 7, 7, ...). - Gary W. Adamson, Oct 05 2015
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REFERENCES
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E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 140.
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LINKS
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T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
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FORMULA
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E.g.f.: (x/6)*(7*x^2 + 21*x + 6)*exp(x). - G. C. Greubel, Oct 05 2015
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MAPLE
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seq(binomial(2*n+1, 3)-binomial(n+1, 3), n=0..38); # Zerinvary Lajos, Jan 21 2007
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MATHEMATICA
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PROG
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(Maxima) makelist(n*(7*n^2-1)/6, n, 0, 30); /* Martin Ettl, Jan 08 2013 */
(PARI) vector(100, n, n--; n*(7*n^2 - 1)/6) \\ Altug Alkan, Oct 06 2015
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CROSSREFS
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1/12*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.
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KEYWORD
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nonn,easy
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AUTHOR
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Albert D. Rich (Albert_Rich(AT)msn.com)
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STATUS
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approved
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