A139098 a(n) = 8*n^2.

0, 8, 32, 72, 128, 200, 288, 392, 512, 648, 800, 968, 1152, 1352, 1568, 1800, 2048, 2312, 2592, 2888, 3200, 3528, 3872, 4232, 4608, 5000, 5408, 5832, 6272, 6728, 7200, 7688, 8192, 8712, 9248, 9800, 10368, 10952, 11552, 12168, 12800, 13448, 14112, 14792, 15488, 16200

Offset: 0

Author

Omar E. Pol, Apr 25 2008

Keywords

• nonn
• easy

Comments

Opposite numbers to the centered 16-gonal numbers (A069129) in the square spiral whose vertices are the triangular numbers (A000217).

8 times the squares. - Omar E. Pol, Dec 09 2008

a(n-1) is the molecular topological index of the n-wheel graph W_n. - Eric W. Weisstein, Jul 11 2011

An n X n pandiagonal magic square has a(n) orientations. - Kausthub Gudipati, Sep 15 2011

Area of a square with diagonal 4n. - Wesley Ivan Hurt, Jun 19 2014

Sum of all the parts in the partitions of 4n into exactly two parts. - Wesley Ivan Hurt, Jul 23 2014

Equivalently: integers k such that k$ / (k/2-1)! and k$ / (k/2)! are both squares when A000178 (k) = k$ = 1!*2!*...*k! is the superfactorial of k (see A348692 for further information). - Bernard Schott, Dec 02 2021

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..800
• Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
• Eric Weisstein's World of Mathematics, Molecular Topological Index.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Subsequence of A008586 and of A349081.

Cf. A000217, A000290, A000178, A001105, A016742, A016766, A033582, A069129, A245058.

Cf. A008590, A348692.

Programs

Magma

[8*n^2: n in [0..50]]; // Vincenzo Librandi, Apr 26 2011

Maple

A139098:=n->8*n^2; seq(A139098(n), n=0..50); # Wesley Ivan Hurt, Jun 19 2014

Mathematica

8 Range[0, 50]^2 (* Wesley Ivan Hurt, Jun 19 2014 *)
LinearRecurrence[{3,-3,1},{0,8,32},50] (* Harvey P. Dale, Oct 05 2023 *)

PARI

a(n)=8*n^2 \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 8*A000290(n) = 4*A001105(n) = 2*A016742(n). - Omar E. Pol, Dec 13 2008

G.f.: -8*x*(1+x)/(x-1)^3. - R. J. Mathar, Nov 27 2015

From Amiram Eldar, Feb 03 2021: (Start)

Sum_{n>=1} 1/a(n) = Pi^2/48 (A245058).

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/96.

Product_{n>=1} (1 + 1/a(n)) = sqrt(8)*sinh(Pi/sqrt(8))/Pi.

Product_{n>=1} (1 - 1/a(n)) = sqrt(8)*sin(Pi/sqrt(8))/Pi. (End)

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Wesley Ivan Hurt, Dec 03 2021

From Elmo R. Oliveira, Dec 01 2024: (Start)

E.g.f.: 8*x*(1 + x)*exp(x).

a(n) = n*A008590(n) = A001105(2*n). (End)