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

Author

Alexander R. Povolotsky, Nov 09 2008

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000578.

Cf. A027480, A006003.

Programs

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

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 *)

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

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