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A129194
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a(n) = n^2*(3/4 - (-1)^n/4).
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13
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0, 1, 2, 9, 8, 25, 18, 49, 32, 81, 50, 121, 72, 169, 98, 225, 128, 289, 162, 361, 200, 441, 242, 529, 288, 625, 338, 729, 392, 841, 450, 961, 512, 1089, 578, 1225, 648, 1369, 722, 1521, 800, 1681, 882, 1849, 968, 2025, 1058, 2209, 1152, 2401, 1250, 2601, 1352
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OFFSET
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0,3
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COMMENTS
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The numerator of the integral is 2,1,2,1,2,1....; the moments of the integral are 2/(n+1)^2.
The sequence alternates between twice a square and an odd square, A001105(n) and A016754(n).
Partial sums of the positive elements give the absolute values of A122576. - Omar E. Pol, Aug 22 2011
Partial sums of the positive elements give A212760. - Omar E. Pol, Dec 28 2013
Conjecture: denominator of 4/n - 2/n^2. - Wesley Ivan Hurt, Jul 11 2016
Multiplicative because both A000290 and A040001 are. - Andrew Howroyd, Jul 25 2018
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REFERENCES
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G. Pólya and G. Szegő, Problems and Theorems in Analysis II (Springer 1924, reprinted 1976), Part Eight, Chap. 1, Sect. 7, Problem 73.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..960
John M. Campbell, An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences, arXiv preprint arXiv:1105.3399 [math.GM], 2011.
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).
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FORMULA
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G.f.: x*(1 + 2*x + 6*x^2 + 2*x^3 + x^4)/(1-x^2)^3; a(n+1) = denominator((1/(2*Pi))*Integral_{t=0..2*Pi} exp(i*n*t)(-((Pi-t)/i)^2)), i=sqrt(-1).
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n > 5. - Paul Curtz, Mar 07 2011
a(n) is the numerator of the coefficient of x^4 in the Maclaurin expansion of exp(-n*x^2). - Francesco Daddi, Aug 04 2011
O.g.f. as a Lambert series: x*Sum_{n >= 1} J_2(n)*x^n/(1 + x^n), where J_2(n) denotes the Jordan totient function A007434(n). See Pólya and Szegő. - Peter Bala, Dec 28 2013
From Ilya Gutkovskiy, Jul 11 2016: (Start)
E.g.f.: x*((2*x + 1)*sinh(x) + (x + 2)*cosh(x))/2.
Sum_{n>=1} 1/a(n) = 5*Pi^2/24. [corrected by Amiram Eldar, Sep 11 2022] (End)
a(n) = A000290(n) / A040001(n). - Andrew Howroyd, Jul 25 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/24 (A222171). - Amiram Eldar, Sep 11 2022
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MAPLE
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A129194:=n->n^2*(3/4 - (-1)^n/4): seq(A129194(n), n=0..80); # Wesley Ivan Hurt, Jul 11 2016
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MATHEMATICA
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Table[n^2*(3/4 - (-1)^n/4), {n, 0, 60}] (* Wesley Ivan Hurt, Jul 11 2016 *)
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PROG
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(Magma) [n^2*(3/4-(-1)^n/4): n in [0..50]]; // Vincenzo Librandi, Apr 26 2011
(PARI) a(n) = lcm(2, n^2)/2; \\ Andrew Howroyd, Jul 25 2018
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CROSSREFS
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Cf. A016742, A010713, A105398, A152020, A000290, A061038, A061040, A061050. - Paul Curtz, Nov 21 2008
Cf. A040001, A129204, A222171.
Sequence in context: A215025 A162954 A357993 * A300780 A272347 A214300
Adjacent sequences: A129191 A129192 A129193 * A129195 A129196 A129197
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KEYWORD
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easy,frac,nonn,mult
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AUTHOR
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Paul Barry, Apr 02 2007
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EXTENSIONS
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More terms from Michel Marcus, Dec 28 2013
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STATUS
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approved
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