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A269657
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Number of length-4 0..n arrays with no adjacent pair x,x+1 repeated.
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3
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1, 15, 79, 253, 621, 1291, 2395, 4089, 6553, 9991, 14631, 20725, 28549, 38403, 50611, 65521, 83505, 104959, 130303, 159981, 194461, 234235, 279819, 331753, 390601, 456951, 531415, 614629, 707253, 809971, 923491, 1048545, 1185889, 1336303
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OFFSET
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0,2
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COMMENTS
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I.e., a(n) = # {x in {0..n}^4 | x[1] != x[0]+1 or x[2] != x[0] or x[3] != x[1]}. The only possibility to have an adjacent x,x+1 pair repeated in a length-4 array is to have the array (x,x+1,x,x+1), with 0 <= x <= n-1 given the restriction on the domain of coefficients. This implies a(n) = (n+1)^4 - n and previously conjectured formulas. - M. F. Hasler, Feb 29 2020
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LINKS
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FORMULA
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Empirical: a(n) = n^4 + 4*n^3 + 6*n^2 + 3*n + 1.
G.f.: (1 + 10*x + 14*x^2 - 2*x^3 + x^4) / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.
(End)
a(n) = (n+1)^4 - n, cf. comment, confirming the above conjectured formulas. - M. F. Hasler, Feb 29 2020
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EXAMPLE
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For n=0, the only length-4 0..0 array is (0,0,0,0) and it satisfies the restriction, so a(0) = 1.
For n=1, there is only one 4-tuple with coefficients in 0..1 which has a repeated pair (x,x+1), namely (0,1,0,1). Thus, a(1) = 2^4 - 1 = 15.
For n=2, there are two 4-tuples with coefficients in 0..2 which have a repeated pair (x,x+1), namely (0,1,0,1) and (1,2,1,2). Thus, a(1) = 3^4 - 2 = 79.
(End)
Some solutions for n=3 (length-4 arrays shown as columns):
1 1 0 2 0 2 2 3 0 3 2 1 0 3 1 1
1 0 0 1 3 2 0 1 2 3 2 1 2 0 0 2
1 1 2 2 1 0 0 2 2 0 2 0 0 0 0 1
3 3 0 1 0 0 1 2 2 1 3 3 2 2 0 3
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MATHEMATICA
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Denominator/@Flatten[Table[x/.Solve[m-Sqrt[-1/(1/(1/(1-x)-(m-1))-(m+1))]==0], {m, 2, 34}]] (* Ed Pegg Jr, Jan 14 2020 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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