A209646 Number of n X 4 0..1 arrays avoiding 0 0 1 and 1 0 0 horizontally and 0 0 1 and 1 0 1 vertically.
9, 81, 270, 630, 1215, 2079, 3276, 4860, 6885, 9405, 12474, 16146, 20475, 25515, 31320, 37944, 45441, 53865, 63270, 73710, 85239, 97911, 111780, 126900, 143325, 161109, 180306, 200970, 223155, 246915, 272304, 299376, 328185, 358785, 391230
Offset: 1
Examples
Some solutions for n=4: 0 1 0 1 0 1 1 0 0 0 0 0 1 0 1 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 1 0
Links
- R. H. Hardin, Table of n, a(n) for n = 1..210
- Robert Israel, Maple-assisted proof of formula
- Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
Crossrefs
Cf. A209650.
Programs
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Maple
seq(9*n^3 + (9/2)*n^2 - (9/2)*n, n=1..100); # Robert Israel, Mar 07 2018
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PARI
Vec(9*x*(1 + 5*x) / (1 - x)^4 + O(x^40)) \\ Colin Barker, Jul 12 2018
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PARI
a(n) = 9*n^3+(9/2)*n^2-(9/2)*n; \\ Altug Alkan, Jul 12 2018
Formula
Empirical: a(n) = 9*n^3 + (9/2)*n^2 - (9/2)*n.
Formula confirmed by Robert Israel, Mar 07 2018: see link.
From Colin Barker, Jul 12 2018: (Start)
G.f.: 9*x*(1 + 5*x) / (1 - x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>4.
(End)
Comments