A211422 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w^2 + x*y = 0.
1, 9, 17, 25, 41, 49, 57, 65, 81, 105, 113, 121, 137, 145, 153, 161, 193, 201, 225, 233, 249, 257, 265, 273, 289, 329, 337, 361, 377, 385, 393, 401, 433, 441, 449, 457, 505, 513, 521, 529, 545, 553, 561, 569, 585, 609, 617, 625, 657, 713, 753, 761
Offset: 0
Keywords
A108579 Number of symmetry classes of 3 X 3 magic squares (with distinct positive entries) having magic sum 3n.
0, 0, 0, 0, 1, 3, 4, 7, 10, 13, 17, 22, 26, 32, 38, 44, 51, 59, 66, 75, 84, 93, 103, 114, 124, 136, 148, 160, 173, 187, 200, 215, 230, 245, 261, 278, 294, 312, 330, 348, 367, 387, 406, 427, 448, 469, 491, 514, 536, 560, 584, 608, 633, 659, 684, 711, 738, 765, 793, 822, 850
Offset: 1
Comments
A magic square has distinct positive integers in its cells, whose sum is the same (the "magic sum") along any row, column, or main diagonal.
a(n) is given by a quasipolynomial of period 6.
It appears that A108579(n) is the number of ordered triples (w,x,y) with components all in {1,...,n} and w+n=2x+3y, as in the Mathematica section. For related sequences, see A211422. - Clark Kimberling, Apr 15 2012
Examples
a(5) = 1 because there is a unique 3 X 3 magic square, up to symmetry, using the first 9 positive integers.
Links
- T. Zaslavsky, Table of n, a(n) for n = 1..10000
- Matthias Beck and Thomas Zaslavsky, Six little squares and how their numbers grow, Journal of Integer Sequences, 13 (2010), Article 10.6.2.
- Matthias Beck and Thomas Zaslavsky, Six little squares and how their numbers grow: Maple Worksheets and Supporting Documentation.
- Yu. V. Chebrakov, Section 2.6.3 in "Theory of Magic Matrices. Issue TMM-1.", 2008. (in Russian)
- Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).
Programs
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Mathematica
(* This program generates a sequence described in the Comments section *) t[n_] := t[n] = Flatten[Table[-w^2 + x*y + n, {w, 1, n}, {x, 1, n}, {y, 1, n}]] c[n_] := Count[t[n], 0] t = Table[c[n], {n, 1, 80}] (* A211506 *) (* Clark Kimberling, Apr 15 2012 *)
Formula
a(n) = floor((1/4)*(n-2)^2)-floor((1/3)*(n-1)). - Mircea Merca, Oct 08 2013
G.f.: x^5*(1+2*x)/((1-x)*(1-x^2)*(1-x^3)).
Extensions
Edited by N. J. A. Sloane, Oct 04 2010
Comments
Examples
Links
Crossrefs
Programs
Mathematica
t[n_] := t[n] = Flatten[Table[w^2 + x*y, {w, -n, n}, {x, -n, n}, {y, -n, n}]] c[n_] := Count[t[n], 0] t = Table[c[n], {n, 0, 70}] (* A211422 *) (t - 1)/8 (* A120486 *)