A308951 Number of principal reversible magic squares of order 4n.

1, 3, 10, 42, 35, 42, 230, 42, 126, 393, 230, 42, 1190, 42, 230, 1158, 462, 42, 3030, 42, 1190, 1158, 230, 42, 5922, 393, 230, 3030, 1190, 42, 9350, 42, 1716, 1158, 230, 1158, 20790, 42, 230, 1158, 5922, 42, 9350, 42, 1190, 16782, 230, 42, 28644, 393, 3030, 1158, 1190, 42, 30670, 1158, 5922, 1158, 230, 42, 66290, 42, 230

Offset

0,2

Comments

The reversible magic squares are in a one-to-one correspondence with the most-perfect pandiagonal magic squares (cf. A051235).

A reversible magic square composed of integers {0,1,...,n^2-1} is principal if its rows and columns form increasing sequences and the first row starts with 0, 1.

Every reversible magic square can be transformed into a unique principal square by (i) interchanging a row/column with the symmetric row/column; and (ii) interchanging two rows/columns in one half of the square and simultaneously interchange their symmetric rows/columns in the other half.

References

K. Ollerenshaw and D. S. Bree, Most-perfect Pan-diagonal Magic Squares: Their Construction and Enumeration, Inst. Math. Applic., Southend-on-Sea, England, 1998.

I. Stewart, Most-perfect magic squares, Sci. Amer., Vol. 281, No. 5 (Nov. 1999), pp. 122-123.

Links

Max Alekseyev, Table of n, a(n) for n = 0..10000

Steve Abbott, Review of Most-perfect Pan-diagonal Magic Squares: Their Construction and Enumeration by Kathleen Ollerenshaw and David Brée, The Mathematical Gazette, Volume 82, Issue 495 November 1998, pp. 535-536.

Formula

For n>=1, let N := 4n = Product_{g} (p_g)^(s_g), where p_g are distinct primes, and W_v(n) := Sum_{i=0..v} (-1)^(v+i) * binomial(v+1,i+1) * Product_{g} binomial(s_g+i,i). Then a(n) = Sum_{v=0..Sum_{g} s_g} W_v(N)*(W_v(N)+W_{v+1}(N)).

For n>=1, a(n) = A051235(n) / 2^(4*n-2) / (2n)!^2.

Mathematica

a[n_] := Module[{s, W}, If[n == 0, Return[1]]; s = FactorInteger[4 n][[All, 2]]; W = Table[Sum[(-1)^(V - i - 1) Binomial[V, i + 1]  Product[ Binomial[ s[[g]] + i, i], {g, 1, Length[s]}], {i, 0, V - 1}], {V, 1, Total[s] + 1}]; Sum[W[[V]] (W[[V]] + W[[V+1]]), {V, 1, Length[W]-1}]];
Table[a[n], {n, 0, 62}] (* Jean-François Alcover, Jul 03 2019, translated from PARI *)

Prog

(PARI) { A308951(n) = if(n==0, return(1)); my(s=factor(4*n)[, 2], W=vector(vecsum(s)+1, V, sum(i=0, V-1, (-1)^(V-i-1) * binomial(V, i+1) * prod(g=1, #s, binomial(s[g]+i, i)) ))); sum(V=1, #W-1, W[V]*(W[V]+W[V+1])); }

Crossrefs

Cf. A051235.

Sequence in context: A245502 A009329 A009364 * A295236 A370537 A214835

Adjacent sequences: A308948 A308949 A308950 * A308952 A308953 A308954

Keyword

nonn,easy

Author

Max Alekseyev, Jul 02 2019

Status

approved