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A054602 a(n) = Sum_{d|3} phi(d)*n^(3/d). 16

%I #72 Nov 28 2021 12:57:55

%S 0,3,12,33,72,135,228,357,528,747,1020,1353,1752,2223,2772,3405,4128,

%T 4947,5868,6897,8040,9303,10692,12213,13872,15675,17628,19737,22008,

%U 24447,27060,29853,32832,36003,39372,42945,46728,50727,54948

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

%C Every term is the product plus the sum of 3 consecutive numbers. - _Vladimir Joseph Stephan Orlovsky_, Oct 24 2009

%C 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

%H Seiichi Manyama, <a href="/A054602/b054602.txt">Table of n, a(n) for n = 0..10000</a>

%H Thomas Oléron Evans, <a href="http://www.mathistopheles.co.uk/2015/08/22/queues-of-cubes/">Queues of Cubes</a>, Mathistopheles, August 22 2015.

%H Aleksandar Petojević, <a href="http://dx.doi.org/10.5937/MatMor0801037P">A Note about the Pochhammer Symbol</a>, Mathematica Moravica, Vol. 12-1 (2008), 37-42.

%H Michelle Rudolph-Lilith, <a href="http://arxiv.org/abs/1508.07894">On the Product Representation of Number Sequences, with Application to the Fibonacci Family</a>, arXiv preprint arXiv:1508.07894 [math.NT], 2015.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F a(n) = n^3 + 2n = A073133(n, 3). - _Henry Bottomley_, Jul 16 2002

%F G.f.: 3*x*(x^2+1)/(x-1)^4. - _Colin Barker_, Dec 21 2012

%F a(n) = ((n-1)^3 + n^3 + (n+1)^3) / 3. - _David Morales Marciel_, Aug 28 2015

%F From _Bernard Schott_, Nov 28 2021: (Start)

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

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

%t nterms=100;Table[n^3+2n,{n,0,nterms}] (* _Paolo Xausa_, Nov 25 2021 *)

%o (PARI) a(n)=n^3+2*n \\ _Charles R Greathouse IV_, Sep 01 2015

%Y Row n=3 of A185651.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Apr 16 2000

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Last modified March 28 13:42 EDT 2024. Contains 371254 sequences. (Running on oeis4.)