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A129194 a(n) = n^2*(3/4 - (-1)^n/4). 12
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

REFERENCES

G. Polya and G. Szego, Problems and Theorems in Analysis II (Springer 1924, reprinted 1976), Part Eight, Chap. 1, Sect. 7, Problem 73.

LINKS

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).

FORMULA

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))*int(exp(i*n*t)(-((Pi-t)/i)^2),t,0,2*Pi)), 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) = numerator of 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 Polya and Szego. - 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) = 7*Pi^2/48 = 1.43931730849219813191336327081... (End)

a(n) = A000290(n) / A040001(n). - Andrew Howroyd, Jul 25 2018

MAPLE

A129194:=n->n^2*(3/4 - (-1)^n/4): seq(A129194(n), n=0..80); # Wesley Ivan Hurt, Jul 11 2016

MATHEMATICA

Table[n^2*(3/4 - (-1)^n/4), {n, 0, 60}] (* Wesley Ivan Hurt, Jul 11 2016 *)

PROG

(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

CROSSREFS

Cf. A016742, A010713, A105398, A152020, A000290, A061038, A061040, A061050. - Paul Curtz, Nov 21 2008

Cf. A040001, A129204.

Sequence in context: A069815 A215025 A162954 * A300780 A272347 A214300

Adjacent sequences:  A129191 A129192 A129193 * A129195 A129196 A129197

KEYWORD

easy,frac,nonn,mult

AUTHOR

Paul Barry, Apr 02 2007

EXTENSIONS

More terms from Michel Marcus, Dec 28 2013

STATUS

approved

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Last modified October 16 05:52 EDT 2019. Contains 328045 sequences. (Running on oeis4.)