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

2,2

Links

Table of n, a(n) for n=2..27.

Albert Zhou, Proof of recurrence

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)

Prog

(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

Crossrefs

Cf. A001499, A000986.

Sequence in context: A181427 A055697 A055705 * A347336 A303356 A299317

Adjacent sequences: A156427 A156428 A156429 * A156431 A156432 A156433

Keyword

nonn

Author

R. H. Hardin, Feb 09 2009

Extensions

a(26)-a(27) from Albert Zhou, Jan 26 2025

Status

approved