A117560 a(n) = n*(n^2 - 1)/2 - 1.

2, 11, 29, 59, 104, 167, 251, 359, 494, 659, 857, 1091, 1364, 1679, 2039, 2447, 2906, 3419, 3989, 4619, 5312, 6071, 6899, 7799, 8774, 9827, 10961, 12179, 13484, 14879, 16367, 17951, 19634, 21419, 23309, 25307, 27416, 29639, 31979, 34439, 37022, 39731, 42569

Offset: 2

Author

Joseph Biberstine (jrbibers(AT)indiana.edu), Mar 29 2006

Keywords

• easy
• nonn

Comments

a(n-1) is an approximation for the lower bound of the "antimagic constant" of an antimagic square of order n. The antimagic constant here is defined as the least integer in the set of consecutive integers to which the rows, columns and diagonals of the square sum. By analogy with the magic constant. This approximation follows from the observation that (2*Sum_{k=1..n^2} k) + (m) + (m+1) <= Sum_{k=0..2*n+1} (m + k) where m is the antimagic constant for an antimagic square of order n. a(n) = A027480(n+1) - 1. Stricter bounds seem likely to exist. See A117561 for the upper bounds. Note there exist no antimagic squares of order two or three, but the values are indexed here for completeness.

Examples

			a(3) = 29 because the antimagic constant of an antimagic square of order 4 must be at least 29 (see comments).
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 2..5000
• Eric Weisstein's World of Mathematics, Antimagic Square.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A027480, A049475, A050257, A117561.

Programs

Magma

[n*(n^2-1)/2 - 1: n in [2..50]]; // Vincenzo Librandi, Jun 20 2011

Mathematica

Table[n*(n^2-1)/2 - 1, {n, 2, 50}]

Formula

a(n) = n*(n^2 - 1)/2 - 1.

G.f.: x^2*(2 + 3*x - 3*x^2 + x^3)/(1-x)^4. - Colin Barker, Mar 29 2012

From Elmo R. Oliveira, Aug 19 2025: (Start)

E.g.f.: 1 + x + (1 + x)*(-2 + 2*x + x^2)*exp(x)/2.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n >= 6. (End)