A300758 a(n) = 2n*(n+1)*(2n+1).

0, 12, 60, 168, 360, 660, 1092, 1680, 2448, 3420, 4620, 6072, 7800, 9828, 12180, 14880, 17952, 21420, 25308, 29640, 34440, 39732, 45540, 51888, 58800, 66300, 74412, 83160, 92568, 102660, 113460, 124992, 137280, 150348, 164220, 178920, 194472, 210900, 228228

Offset

0,2

Comments

The altitude h(n) = a(n)/A001844(n) of the (A005408(n), A046092(n) and A001844(n)) rectangular triangle is an irreducible fraction. - Ralf Steiner, Feb 25 2020

In this case, area A = a(n)/2 = A055112(n). - Bernard Schott, Feb 27 2020

a(n) is the minimum total number of up, down, left, right moves needed to bring one token from each cell of a (2*n+1) X (2*n+1) square grid to the cell (n,n). - Maksim Kita, Dec 08 2025

Links

Table of n, a(n) for n=0..38.

Stephen P. Borgatti and Martin G. Everett, Notions of Position in Social Network Analysis, Sociological Methodology, 22 (1992), 1-35.

Christopher Purcell and Puck Rombach, Role colouring graphs in hereditary classes, arXiv:1802.10180 [math.CO], 2018.

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

Formula

a(n) = 12*A000330(n).

G.f.: 12*x*(1+x)/(1-x)^4. - Colin Barker, Mar 12 2018

a(n) = 6*A006331(n) = 4*A059270(n) = 3*A002492(n) = 2*A055112(n). - Omar E. Pol, Apr 04 2018

From Ralf Steiner, Feb 27 2020: (Start)

a(n) = 2*n*A000384(n+1).

a(n) = sqrt(A016754(n)*A060300(n)). (End)

a(n) = A005408(n) * A046092(n). - Bruce J. Nicholson, Apr 24 2020

E.g.f.: 2*x*(6 + 9*x + 2*x^2)*exp(x). - Stefano Spezia, Dec 22 2025

From Amiram Eldar, Dec 23 2025: (Start)

Sum_{n>=1} 1/a(n) = 3/2 - 2*log(2).

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/2 - 3/2. (End)

Mathematica

a[n_] := 2*n*(n+1)*(2*n+1); Array[a, 40, 0] (* Amiram Eldar, Dec 23 2025 *)
LinearRecurrence[{4, -6, 4, -1}, {0, 12, 60, 168}, 40] (* Harvey P. Dale, Jan 25 2026 *)

Crossrefs

Cf. A000330, A002492, A006331, A055112, A059270, A005408, A046092, A001844, A000384, A016754, A060300.

Cf. A027480.

Sequence in context: A321465 A000141 A328094 * A332544 A279509 A008530

Adjacent sequences: A300755 A300756 A300757 * A300759 A300760 A300761

Keyword

nonn,easy

Author

Christopher Purcell, Mar 12 2018

Extensions

Edited by N. J. A. Sloane, Aug 01 2019

Status

approved