A129194 a(n) = (n/2)^2*(3 - (-1)^n).

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

Offset: 0

Author

Paul Barry, Apr 02 2007

Keywords

• easy
• frac
• nonn
• mult

Comments

The numerator of the integral is 2,1,2,1,2,1,...; the moments of the integral are 2/(n+1)^2. See 2nd formula.

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. Pólya and G. Szegő, 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
• Olivier Bordelles, A Multidimensional Cesaro Type Identity and Applications, J.
• Int. Seq. 18 (2015) # 15.3.7.
• 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).

Crossrefs

Cf. A000290, A001105, A007434, A010713, A016742, A016754, A040001.

Cf. A061038, A061040, A061050, A105398, A129204, A152020, A222171, A309337.

Programs

Magma

[n^2*(3-(-1)^n)/4: n in [0..60]]; // Vincenzo Librandi, Apr 26 2011

Maple

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

Mathematica

Table[n^2*(3-(-1)^n)/4, {n,0,60}] (* Wesley Ivan Hurt, Jul 11 2016 *)
LinearRecurrence[{0,3,0,-3,0,1},{0,1,2,9,8,25},60] (* Harvey P. Dale, Dec 27 2023 *)

PARI

a(n) = lcm(2, n^2)/2; \\ Andrew Howroyd, Jul 25 2018

SageMath

[n^2*(1+(n%2))/2 for n in range(61)] # G. C. Greubel, Apr 04 2023

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))*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

From Peter Bala, Jan 16 2024: (Start)

a(n) = Sum_{1 <= i, j <= n} (-1)^(1 + gcd(i,j,n)) = Sum_{d | n} (-1)^(d+1) * J_2(n/d), that is, the Dirichlet convolution of the pair of multiplicative functions f(n) = (-1)^(n+1) and the Jordan totient function J_2(n) = A007434(n). Hence this sequence is multiplicative. Cf. A193356 and A309337.

Dirichlet g.f.: (1 - 2/2^s)*zeta(s-2). (End)

a(n) = Sum_{1 <= i, j <= n} (-1)^(n + gcd(i, n)*gcd(j, n)) = Sum_{d|n, e|n} (-1)^(n+e*d) * phi(n/d)*phi(n/e). - Peter Bala, Jan 22 2024

Extensions

More terms from Michel Marcus, Dec 28 2013