A147656 The arithmetic mean of the n-th and (n+1)-st cubes, rounded down.

0, 4, 17, 45, 94, 170, 279, 427, 620, 864, 1165, 1529, 1962, 2470, 3059, 3735, 4504, 5372, 6345, 7429, 8630, 9954, 11407, 12995, 14724, 16600, 18629, 20817, 23170, 25694, 28395, 31279, 34352, 37620, 41089, 44765, 48654, 52762, 57095, 61659

Offset

0,2

Comments

The terms of this sequence relate to intervals between cubes in the same fashion as terms of A002378 are related to intervals between squares.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

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

Formula

a(n) = floor((A000578(n) + A000578(n+1))/2).

From R. J. Mathar, Nov 11 2008: (Start)

a(n) = A000578(n) + A045943(n) = n*(2n^2+3n+3)/2.

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

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, May 06 2012

a(n) = A027480(n) + A006003(n). - Bruce J. Nicholson, Jun 03 2018

From A.H.M. Smeets, Sep 10 2018: (Start)

a(n) = Sum_{k=0..n-1} (n+1)^2-k for n >= 0 with empty domain of summation for n = 0.

a(n) = n*(n+1)^2 - n*(n-1)/2 for n >= 0.

Lim_{n -> inf} a(n-1)/n^3 = 1. (End)

E.g.f.: exp(x)*(8*x + 9*x^2 + 2*x^3)/2. - Stefano Spezia, Sep 12 2018

a(n) = A081435(n)-1. - R. J. Mathar, Sep 14 2018

Maple

seq(coeff(series(x*(x^2+x+4)/(1-x)^4, x, n+1), x, n), n = 0 .. 40); # Muniru A Asiru, Sep 11 2018

Mathematica

Table[(n^3+(n+1)^3-1)/2, {n, 0, 70}] (* Vladimir Joseph Stephan Orlovsky, May 04 2011 *)

Prog

(PARI) j=[]; for (n=0, 40, j=concat(j, n^3+floor(((n+1)^3 - n^3)/2))); j
(PARI) a(n) = n*(2*n^2+3*n+3)/2; \\ Altug Alkan, Sep 20 2018
(Magma) I:=[0, 4, 17, 45]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, May 06 2012

Crossrefs

Cf. A000578.

Cf. A027480, A006003.

Sequence in context: A212575 A162148 A166781 * A095667 A212577 A332863

Adjacent sequences: A147653 A147654 A147655 * A147657 A147658 A147659

Keyword

nonn,easy

Author

Alexander R. Povolotsky, Nov 09 2008

Status

approved