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A104585
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a(n) = (1/2) * ( 3*n^2 - n*(-1)^n ).
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4
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0, 2, 5, 15, 22, 40, 51, 77, 92, 126, 145, 187, 210, 260, 287, 345, 376, 442, 477, 551, 590, 672, 715, 805, 852, 950, 1001, 1107, 1162, 1276, 1335, 1457, 1520, 1650, 1717, 1855, 1926, 2072, 2147, 2301, 2380, 2542, 2625, 2795, 2882, 3060, 3151, 3337, 3432, 3626
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OFFSET
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0,2
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COMMENTS
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Previous name was: Pentagonal wave sequence of the second kind.
Even-indexed terms are pentagonal numbers with even index in A000326. Odd-indexed terms are second pentagonal numbers with odd index in A005449.
A104584, pentagonal wave sequence of the first kind; switches odd and even applications and vice versa in A104585. The pentagonal wave triangle, A104586, has A104584 in odd columns and A104585 in even columns.
Integer values of (n+1)(2n+1)/3 in order of appearance. - Wesley Ivan Hurt, Sep 17 2013
Exponents of q in the identity 1 - Sum_{n >= 0} ( q^(3*n+2)*Product_{k = 1..n} (1 - q^(4*k-1)) ) = 1 - q^2 - q^5 + q^15 + q^22 - q^40 - q^51 + + - - .... Compare with Euler's pentagonal number theorem: Product_{n >= 1} (1 - q^n) = 1 - Sum_{n >= 1} ( q^n*Product_{k = 1..n-1} (1 - q^k) ) = 1 - q - q^2 + q^5 + q^7 - q^12 - q^15 + + - - .... - Peter Bala, Dec 03 2020
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LINKS
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FORMULA
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a(n) = (1/2) * ( 3*n^2 - n*(-1)^n ). - Ralf Stephan, Nov 13 2010
G.f.: x*(2+3*x+6*x^2+x^3)/(1-x)^3/(1+x)^2. - Colin Barker, Feb 13 2012
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). - Vincenzo Librandi, Apr 04 2013
Sum_{n>=1} 1/a(n) = 4*log(2) - Pi/sqrt(3).
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*Pi/sqrt(3) - 3*log(3). (End)
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EXAMPLE
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a(5) = 40 = A005449(5), a second pentagonal number.
a(6) = 51 = A000326(6), a pentagonal number.
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MATHEMATICA
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LinearRecurrence[{1, 2, -2, -1, 1}, {0, 2, 5, 15, 22}, 50] (* Harvey P. Dale, Sep 14 2015 *)
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PROG
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(Magma) I:=[0, 2, 5, 15, 22]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..60]]; // Vincenzo Librandi, Apr 04 2013
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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