A087698 Triangle read by rows, giving T(n,k) = maximum number of examples (Boolean inputs) at Hamming distance 2 for symmetric Boolean functions that can have different outputs.
1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 3, 4, 4, 3, 1, 1, 4, 7, 8, 7, 4, 1, 1, 5, 11, 15, 15, 11, 5, 1, 1, 6, 16, 26, 30, 26, 16, 6, 1, 1, 7, 22, 42, 56, 56, 42, 22, 7, 1, 1, 8, 29, 64, 98, 112, 98, 64, 29, 8, 1, 1, 9, 37, 93, 162, 210, 210, 162, 93, 37, 9, 1, 1, 10, 46, 130, 255, 372, 420
Offset: 0
Examples
Triangle begins: 1 N=0 1 1 N=1 1 0 1 N=2 1 1 1 1 N=3 1 2 2 2 1 N=4
Links
- L. Franco, A measure for the complexity of Boolean functions related to their implementation in neural networks, arXiv:cond-mat/0111169 [cond-mat.dis-nn], 2001.
- L. Franco and S. A. Cannas, Non-glassy ground-state in a long-range antiferromagnetic frustrated model in the hypercubic cell, arXiv:cond-mat/0302412 [cond-mat.stat-mech], 2003; Phys. A 332 (2004), no. 1-4, 337-348.
Formula
T(n, N) = ((N-n)^2 + n^2 - N) * C(N, n) / (N^2 - N) n is the term for the series containing N+1 terms
From Peter Bala, Mar 20 2018: (Start)
Except for (n,k) = (1,0) the formula T(n,k) = C(n,k) - 2*C(n-1,n-k-1) + 2*C(n-2,n-k-2), where C(n,k) = n!/(k!*(n-k)!) for 0 <= k <= n, otherwise 0, appears to give the correct table entries.
Appears to equal A159853, the Riordan array ((1-2*x+2*x^2)/(1-x), x/(1-x)), except for the entry T(1,0). If this is correct then provided n =! 1 we have exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(1 + x + x^2/2! + x^3/3!) = 1 + 2*x + 2*x^2/2! + 4*x^3/3! + 8*x^4/4! + 15*x^5/5! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). (End)
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