A168077 a(2n) = A129194(2n)/2; a(2n+1) = A129194(2n+1).

0, 1, 1, 9, 4, 25, 9, 49, 16, 81, 25, 121, 36, 169, 49, 225, 64, 289, 81, 361, 100, 441, 121, 529, 144, 625, 169, 729, 196, 841, 225, 961, 256, 1089, 289, 1225, 324, 1369, 361, 1521, 400, 1681, 441, 1849, 484, 2025, 529, 2209, 576, 2401, 625, 2601

Offset: 0

Author

Paul Curtz, Nov 18 2009

Keywords

• nonn
• easy
• mult

Comments

From Paul Curtz, Mar 26 2011: (Start)

Successive A026741(n) * A026741(n+p):

p=0: 0, 1, 1, 9, 4, 25, 9, a(n),

p=1: 0, 1, 3, 6, 10, 15, 21, A000217,

p=2: 0, 3, 2, 15, 6, 35, 12, A142705,

p=3: 0, 2, 5, 9, 14, 20, 27, A000096,

p=4: 0, 5, 3, 21, 8, 45, 15, A171621,

p=5: 0, 3, 7, 12, 18, 25, 33, A055998,

p=6: 0, 7, 4, 27, 10, 55, 18,

p=7: 0, 4, 9, 15, 22, 30, 39, A055999,

p=8: 0, 9, 5, 33, 12, 65, 21, (see A061041),

p=9: 0, 5, 11, 18, 26, 35, 45, A056000. (End)

The moment generating function of p(x, m=2, n=1, mu=2) = 4*x*E(x, 2, 1), see A163931 and A274181, is given by M(a) = (-4 * log(1-a) - 4 * polylog(2, a))/a^2. The series expansion of M(a) leads to the sequence given above. - Johannes W. Meijer, Jul 03 2016

Multiplicative because both A129194 and A040001 are. - Andrew Howroyd, Jul 26 2018

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• 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. A040001, A168068, A181318.

Programs

Magma

I:=[0,1,1,9,4,25]; [n le 6 select I[n] else 3*Self(n-2)-3*Self(n-4)+Self(n-6): n in [1..60]]; // Vincenzo Librandi, Jul 10 2016

Maple

a := proc(n): n^2*(5-3*(-1)^n)/8 end: seq(a(n), n=0..46); # Johannes W. Meijer, Jul 03 2016

Mathematica

LinearRecurrence[{0,3,0,-3,0,1},{0,1,1,9,4,25},60] (* Harvey P. Dale, May 14 2011 *)
f[n_] := Numerator[(n/2)^2]; Array[f, 60, 0] (* Robert G. Wilson v, Dec 18 2012 *)
CoefficientList[Series[x(1+x+6x^2+x^3+x^4)/((1-x)^3(1+x)^3), {x,0,60}], x] (* Vincenzo Librandi, Jul 10 2016 *)

PARI

concat(0, Vec(x*(1+x+6*x^2+x^3+x^4)/((1-x)^3*(1+x)^3) + O(x^60))) \\ Altug Alkan, Jul 04 2016

PARI

a(n) = lcm(4, n^2)/4; \\ Andrew Howroyd, Jul 26 2018

Sage

(x*(1+x+6*x^2+x^3+x^4)/(1-x^2)^3).series(x, 60).coefficients(x, sparse=False) # G. C. Greubel, Feb 20 2019

Formula

From R. J. Mathar, Jan 22 2011: (Start)

G.f.: x*(1 + x + 6*x^2 + x^3 + x^4) / ((1-x)^3*(1+x)^3).

a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6).

a(n) = n^2*(5 - 3*(-1)^n)/8. (End)

a(n) = A026741(n)^2.

a(2*n) = A000290(n); a(2*n+1) = A016754(n).

a(n) - a(n-4) = 4*A064680(n+2). - Paul Curtz, Mar 27 2011

4*a(n) = A061038(n) * A010121(n+2) = A109043(n)^2, n >= 2. - Paul Curtz, Apr 07 2011

a(n) = A129194(n) / A040001(n). - Andrew Howroyd, Jul 26 2018

From Peter Bala, Feb 19 2019: (Start)

a(n) = numerator(n^2/(n^2 + 4)) = n^2/(gcd(n^2,4)) = (n/gcd(n,2))^2.

a(n) = n^2/b(n), where b(n) = [1, 4, 1, 4, ...] is a purely periodic sequence of period 2. Thus a(n) is a quasi-polynomial in n.

O.g.f.: x*(1 + x)/(1 - x)^3 - 3*x^2*(1 + x^2)/(1 - x^2)^3.

Cf. A181318. (End)

From Werner Schulte, Aug 30 2020: (Start)

Multiplicative with a(2^e) = 2^(2*e-2) for e > 0, and a(p^e) = p^(2*e) for prime p > 2.

Dirichlet g.f.: zeta(s-2) * (1 - 3/2^s).

Dirichlet convolution with A259346 equals A000290.

Sum_{n>0} 1/a(n) = Pi^2 * 7 / 24. (End)

Sum_{k=1..n} a(k) ~ (5/24) * n^3. - Amiram Eldar, Nov 28 2022