A309335 a(n) = n^3 if n odd, 7*n^3/8 if n even.

0, 1, 7, 27, 56, 125, 189, 343, 448, 729, 875, 1331, 1512, 2197, 2401, 3375, 3584, 4913, 5103, 6859, 7000, 9261, 9317, 12167, 12096, 15625, 15379, 19683, 19208, 24389, 23625, 29791, 28672, 35937, 34391, 42875, 40824, 50653, 48013, 59319, 56000, 68921, 64827, 79507, 74536, 91125

Offset

0,3

Comments

Moebius transform of A007331.

Links

Amiram Eldar, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (0,4,0,-6,0,4,0,-1).

Formula

G.f.: x * (1 + 7*x + 23*x^2 + 28*x^3 + 23*x^4 + 7*x^5 + x^6)/(1 - x^2)^4.

G.f.: Sum_{k>=1} J_3(k) * x^k/(1 - x^(2*k)), where J_3() is the Jordan function (A059376).

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

a(n) = n^3 * (15 - (-1)^n)/16.

a(n) = Sum_{d|n, n/d odd} J_3(d).

Sum_{n>=1} 1/a(n) = 57*zeta(3)/56 = 1.223522205001729897639...

Multiplicative with a(2^e) = 7*2^(3*e-3), and a(p^e) = p^(3*e) for odd primes p. - Amiram Eldar, Oct 26 2020

Euler transform is A248882. - Georg Fischer, Nov 10 2020

Mathematica

a[n_] := If[OddQ[n], n^3, 7 n^3/8]; Table[a[n], {n, 0, 45}]
nmax = 45; CoefficientList[Series[x (1 + 7 x + 23 x^2 + 28 x^3 + 23 x^4 + 7 x^5 + x^6)/(1 - x^2)^4, {x, 0, nmax}], x]
LinearRecurrence[{0, 4, 0, -6, 0, 4, 0, -1}, {0, 1, 7, 27, 56, 125, 189, 343}, 46]
Table[n^3 (15 - (-1)^n)/16, {n, 0, 45}]

Crossrefs

Cf. A000578, A007331, A016755, A026741, A059376, A248882, A303383 (partial sums), A308422, A309336.

Sequence in context: A098948 A015873 A344721 * A175367 A022271 A269449

Adjacent sequences: A309332 A309333 A309334 * A309336 A309337 A309338

Keyword

nonn,easy,mult

Author

Ilya Gutkovskiy, Jul 24 2019

Status

approved