A054000 a(n) = 2*n^2 - 2.

0, 6, 16, 30, 48, 70, 96, 126, 160, 198, 240, 286, 336, 390, 448, 510, 576, 646, 720, 798, 880, 966, 1056, 1150, 1248, 1350, 1456, 1566, 1680, 1798, 1920, 2046, 2176, 2310, 2448, 2590, 2736, 2886, 3040, 3198, 3360, 3526, 3696, 3870, 4048, 4230, 4416

Offset: 1

Author

Asher Auel, Jan 12 2000

Keywords

• nonn
• easy

Comments

a(n) is the number of edges in (n+1) X (n+1) square grid with all horizontal, vertical and great diagonal segments filled in.

Nonnegative X values of integer solutions to the equation 2*X^3 + 4*X^2 = Y^2. To find Y values: b(n) = 2*n*(2*n^2 - 2). - Mohamed Bouhamida, Nov 06 2007

Second term of an arithmetic progression of 5 numbers with common difference 2n+1. The sum of squares of such 5 terms equals the sum of squares of 5 consecutive numbers starting a(n) + 2n + 1. - Carmine Suriano, Oct 16 2013

For m > 2, a(m-1) = 2*m*(m-2) is the number of Hamiltonian circuits on an m-gonal bipyramid with labeled vertices. - Stanislav Sykora, Jul 22 2014

a(n+1), n >= 0, appears also as the third member of the quartet [p0(n), p1(n), a(n+1), p3(n)] of the square of [n, n+1, n+2, n+3] in the Clifford algebra Cl_2 for n >= 0. p0(n) = -A147973(n+3), p1(n) = A046092(n) and p3(n) = A139570(n). See a comment on A147973, also with a reference. - Wolfdieter Lang, Oct 15 2014

From Bui Quang Tuan, Mar 31 2015: (Start)

For n >= 2, a(n) is the total sum of all numbers on the perimeter of a square consisting of n columns, each of which contains n numbers 1, 2, 3, ..., n.

Here is an example with n = 5:

1 1 1 1 1

2 2 2 2 2

3 3 3 3 3

4 4 4 4 4

5 5 5 5 5

where 1+1+1+1+1 + 2+2 + 3+3 + 4+4 + 5+5+5+5+5 = 48 = a(5).

(End)

Nonnegative k such that k/2+1 is a square. - Bruno Berselli, Apr 10 2018

Examples

			For n=5, a(5)=48 and 37^2 + 48^2 + 59^2 + 70^2 + 81^2 = 59^2 + 60^2 + 61^2 + 62^2 + 63^2. - _Carmine Suriano_, Oct 16 2013
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000
• Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

a(n) = A100345(n+1, n-4) for n>2.

Cf. A000217, A001082, A002378, A002943, A005563, A028347, A036666, A046092, A056220, A062717, A067725, A087475.

Programs

Maple

[ seq(2*n^2 - 2, n=1..60) ];

Mathematica

2 Range[50]^2 - 2 (* or *) LinearRecurrence[{3, -3, 1}, {0, 6, 16}, 50] (* Harvey P. Dale, Feb 03 2012 *)
CoefficientList[Series[2 x (3 - x) / (1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Apr 01 2015 *)

PARI

a(n)=2*n^2-2 \\ Charles R Greathouse IV, Sep 24 2015

Formula

a(n) = 4*n + a(n-1) - 2, with n>1, a(1)=0. - Vincenzo Librandi, Aug 06 2010

a(1)=0, a(2)=6, a(3)=16; for n>3, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Feb 03 2012

a(n) = (n+i)^2 + (n-i)^2, where i=sqrt(-1). - Bruno Berselli, Jan 23 2014

a(n) = 1*A000290(n-1) + 2*A000217(n-1) + 3*A001477(n-1). - J. M. Bergot, Apr 23 2014

G.f.: 2*x^2*(3 - x)/(1 - x)^3. - Vincenzo Librandi, Apr 01 2015

E.g.f.: 2*(x^2 + x -1)*exp(x) + 2. - G. C. Greubel, Jul 13 2017

a(n) + a(n+2) = A005843(n+1)^2. - Ezhilarasu Velayutham, May 30 2019

From Amiram Eldar, Dec 09 2021: (Start)

Sum_{n>=2} 1/a(n) = 3/8.

Sum_{n>=2} (-1)^n/a(n) = 1/8. (End)