A054602 a(n) = Sum_{d|3} phi(d)*n^(3/d).

0, 3, 12, 33, 72, 135, 228, 357, 528, 747, 1020, 1353, 1752, 2223, 2772, 3405, 4128, 4947, 5868, 6897, 8040, 9303, 10692, 12213, 13872, 15675, 17628, 19737, 22008, 24447, 27060, 29853, 32832, 36003, 39372, 42945, 46728, 50727, 54948, 59397, 64080, 69003, 74172

Offset: 0

Author

N. J. A. Sloane, Apr 16 2000

Keywords

• nonn
• easy

Comments

Every term is the product plus the sum of 3 consecutive numbers. - Vladimir Joseph Stephan Orlovsky, Oct 24 2009

Continued fraction [n,n,n] = (n^2+1)/(n^3+2n) = (n^2+1)/a(n); e.g., [7,7,7] = 50/357. - Gary W. Adamson, Jul 15 2010

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Thomas Oléron Evans, Queues of Cubes, Mathistopheles, August 22 2015.
• Aleksandar Petojević, A Note about the Pochhammer Symbol, Mathematica Moravica, Vol. 12-1 (2008), pp. 37-42.
• Michelle Rudolph-Lilith, On the Product Representation of Number Sequences, with Application to the Fibonacci Family, arXiv preprint arXiv:1508.07894 [math.NT], 2015.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Row n=3 of A185651.

Cf. A006527, A007531, A008585, A073133, A292022.

Programs

Mathematica

nterms=100;Table[n^3+2n,{n,0,nterms}] (* Paolo Xausa, Nov 25 2021 *)

PARI

a(n)=n^3+2*n \\ Charles R Greathouse IV, Sep 01 2015

Formula

a(n) = n^3 + 2*n = A073133(n,3). - Henry Bottomley, Jul 16 2002

G.f.: 3*x*(x^2+1)/(x-1)^4. - Colin Barker, Dec 21 2012

a(n) = ((n-1)^3 + n^3 + (n+1)^3)/3. - David Morales Marciel, Aug 28 2015

From Bernard Schott, Nov 28 2021: (Start)

a(n) = A007531(n+1) + A008585(n) (see 1st comment).

a(n) = 3*A006527(n). (End)

From Elmo R. Oliveira, Aug 09 2025: (Start)

E.g.f.: exp(x)*x*(3 + 3*x + x^2).

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).

a(n) = A292022(n)/4. (End)