A034828 a(n) = floor(n^2/4)*(n/2).

0, 0, 1, 3, 8, 15, 27, 42, 64, 90, 125, 165, 216, 273, 343, 420, 512, 612, 729, 855, 1000, 1155, 1331, 1518, 1728, 1950, 2197, 2457, 2744, 3045, 3375, 3720, 4096, 4488, 4913, 5355, 5832, 6327, 6859, 7410, 8000, 8610, 9261, 9933, 10648, 11385, 12167, 12972, 13824

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Wiener index of cycle of length n.

a(n+1) is the sum of labeled number of boxes arranged as pyramid with base n. The sum of boxes is A002620(n+1). See the illustration in links. - Kival Ngaokrajang, Jul 02 2013

Examples

			G.f.: x^2 + 3*x^3 + 8*x^4 + 15*x^5 + 27*x^6 + 42*x^7 + 64*x^8 + 90*x^9 + ...
		

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
• Kival Ngaokrajang, Illustration for n = 1..10.
• Eric Weisstein's World of Mathematics, Wiener Index.
• H. J. Wiener, Structural Determination of Paraffin Boiling Points, J. Amer. Chem. Soc. 69 (1947), 17-20.
• J. Zerovnik, Szeged index of symmetric graphs, J. Chem. Inf. Comput. Sci., 39 (1999), 77-80.
• Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Crossrefs

Equals A005996/2.

Partial sums of A001318.

Cf. A107231.

Cf. A000578, A000330, A002620.

Cf. A002620, A007310.

Cf. A062717.

Programs

Magma

[Floor(n^2/4)*(n/2): n in [0..50]]; // G. C. Greubel, Feb 23 2018

Maple

A034828:=n->n*floor(n^2/4)/2; seq(A034828(k), k=0..100); # Wesley Ivan Hurt, Nov 05 2013

Mathematica

Table[Floor[n^2/4] n/2, {n, 0, 50}] (* Harvey P. Dale, Jun 10 2011 *)
LinearRecurrence[{2, 1, -4, 1, 2, -1}, {0, 0, 1, 3, 8, 15}, 50] (* Harvey P. Dale, Jun 10 2011 *)

PARI

{a(n) = (n^2 \ 4) * n / 2} /* Michael Somos, Sep 06 2008 */

PARI

{a(n) = if( n<0, -a(-n), polcoeff( x^2 * (1 + x + x^2) / ((1 - x)^2 * (1 - x^2)^2) + x * O(x^n), n))} /* Michael Somos, Sep 06 2008 */

Formula

a(n) = (n^2-1)*n/8 if n is odd, otherwise n^3/8.

From Paul Barry, May 13 2005: (Start)

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

a(n) = 2*a(n-1) +a(n-2) -4*a(n-3) +a(n-4) +2*a(n-5) -a(n-6).

a(n) = (2*n^3 +12*n^2 +23*n +14)/16 +(n+2)*(-1)^n/16.

a(n) = Sum_{k=0..floor((n+2)/2)} ((n+2)/(n+2-k))(-1)^k*C(n+2-k, k)* C(n-2*k+2, 2)*C(n-2*k, floor((n-2*k)/2)). [Typo corrected by R. J. Mathar, Aug 18 2008] (End)

a(n) = (2*n^2 - 1 + (-1)^n) * n / 16. - Michael Somos, Sep 06 2008

Euler transform of length 3 sequence [3, 2, -1]. - Michael Somos, Sep 06 2008

a(-n) = -a(n). - Michael Somos, Sep 06 2008

a(2n) = A000578(n). a(2n+1) = 3*A000330(n). a(n) = n*A002620(n)/2. - Michael Somos, Sep 06 2008

a(n) = (-n + Sum_{k=1..n} A007310(k)^2)/24. - Jesko Matthes, Feb 19 2021

Sum_{n>=2} 1/a(n) = 6 - 8*log(2) + zeta(3). - Amiram Eldar, Apr 16 2022

a(n) = Sum_{k=1..n} A062717(k)/4. - Sela Fried, Jun 27 2022

Extensions

Definition reworded by Michael Somos, Sep 06 2008