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A140090
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a(n) = n*(3*n + 7)/2.
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35
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0, 5, 13, 24, 38, 55, 75, 98, 124, 153, 185, 220, 258, 299, 343, 390, 440, 493, 549, 608, 670, 735, 803, 874, 948, 1025, 1105, 1188, 1274, 1363, 1455, 1550, 1648, 1749, 1853, 1960, 2070, 2183, 2299, 2418, 2540, 2665, 2793, 2924
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OFFSET
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0,2
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COMMENTS
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This sequence is mentioned in the Guo-Niu Han's paper, chapter 6: Dictionary of the standard puzzle sequences, p. 19 (see link). - Omar E. Pol, Oct 28 2011
Number of cards needed to build an n-tier house of cards with a flat, one-card-wide roof. - Tyler Busby, Dec 28 2022
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LINKS
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FORMULA
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a(n) = (3*n^2 + 7*n)/2.
E.g.f.: (1/2)*(3*x^2 + 10*x)*exp(x). - G. C. Greubel, Jul 17 2017
Sum_{n>=1} 1/a(n) = 117/98 - Pi/(7*sqrt(3)) - 3*log(3)/7.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*Pi/(7*sqrt(3)) + 4*log(2)/7 - 75/98. (End)
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MATHEMATICA
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LinearRecurrence[{3, -3, 1}, {0, 5, 13}, 50] (* Harvey P. Dale, Jan 17 2022 *)
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PROG
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CROSSREFS
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The generalized pentagonal numbers b*n+3*n*(n-1)/2, for b = 1 through 12, form sequences A000326, A005449, A045943, A115067, A140090, A140091, A059845, A140672, A140673, A140674, A140675, A151542.
Cf. numbers of the form n*(d*n + 10 - d)/2: A008587, A056000, A028347, A014106, A028895, A045944, A186029, A007742, A022267, A033429, A022268, A049452, A186030, A135703, A152734, A139273.
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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