A156430 Number of n X n arrays of squares of integers, symmetric about both diagonal and antidiagonal, with all rows summing to 2.
1, 2, 10, 12, 84, 132, 954, 1728, 13290, 26820, 217500, 481320, 4086600, 9783480, 86549820, 221921280, 2037627900, 5552479800, 52745205240, 151802154000, 1487961422640, 4500041903280, 45412066438200, 143712079822080, 1490217165997560, 4917227802767280
Offset: 2
Keywords
Links
- Albert Zhou, Proof of recurrence
Programs
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Python
# Even-dim bisymmetric A = [1, 1, 10] B = [0, 2, 6] C = [0, 1, 6] for n in range(3, 13): a_next = A[-1] + (n-1)*A[-2] + 4*(n-1)*B[-1] + 2*(n-1)*(n-2)*C[-1] b_next = 2*A[-1] + 2*(n-1)*B[-1] c_next = 4*B[-1] - 2*A[-2] + 4*(n-2)*A[-3] + 4*(n-2)*(n-3)*C[-2] A.append(a_next) B.append(b_next) C.append(c_next) # Odd-dim bisymmetric A_odd = [B[n]*n for n in range(len(B))] # Albert Zhou, Jan 26 2025
Formula
From Albert Zhou, Jan 26 2025: (Start)
a(2*n) = a(2*(n-1)) + (n-1)*a(2*(n-2)) + 4*(n-1)*b(2*(n-1)) + 2*(n-1)*(n-2)*c(2*(n-1)), where
b(2*n) = 2*a(2*(n-1)) + 2*(n-1)*b(2*(n-1)), and
c(2*n) = 4*b(2*(n-1)) - 2*a(2*(n-2)) + 4*(n-2)*a(2*(n-3)) + 4*(n-2)*(n-3)*c(2*(n-2)), with
a(0) = 1, a(2) = 1, a(4) = 10, and
b(0) = 0, b(2) = 2, b(4) = 6, and
c(0) = 0, c(2) = 1, c(6) = 6.
a(2*n+1) = n*b(2*n).
Proof attached. (End)
Extensions
a(26)-a(27) from Albert Zhou, Jan 26 2025