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A068293
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a(1) = 1; thereafter a(n) = 6*(2^(n-1) - 1).
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10
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1, 6, 18, 42, 90, 186, 378, 762, 1530, 3066, 6138, 12282, 24570, 49146, 98298, 196602, 393210, 786426, 1572858, 3145722, 6291450, 12582906, 25165818, 50331642, 100663290, 201326586, 402653178, 805306362, 1610612730, 3221225466, 6442450938, 12884901882
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OFFSET
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1,2
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COMMENTS
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1/4 the number of colorings of an n X n octagonal array with 4 colors.
Consider the planar net 3^6 (as in the top left figure in the uniform planar nets link). Then a(n) is the total number of ways that a spider starting at a point P can reach any point n steps away by using a path of length n. - N. J. A. Sloane, Feb 20 2016
Equals inverse binomial transform of A091344: (1, 7, 31, 115, 391, ...).
Equals binomial transform of (1, 5, 7, 5, 7, 5, ...). (End)
For n > 1, number of ternary strings of length n with exactly 2 different digits. - Enrique Navarrete, Nov 20 2020
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LINKS
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FORMULA
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a(n) = 3*a(n-1) - 2*a(n-2); a(1)=1, a(2)=6, a(3)=18. - Harvey P. Dale, Nov 27 2011
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MATHEMATICA
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Transpose[NestList[{First[#]+1, 6(2^First[#]-1)}&, {1, 1}, 30]][[2]] (* or *) Join[{1}, LinearRecurrence[{3, -2}, {6, 18}, 30]] (* Harvey P. Dale, Nov 27 2011 *)
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PROG
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(PARI) a(n)=polcoeff(prod(i=1, 2, (1+i*x))/(prod(i=1, 2, (1-i*x))+x*O(x^n)), n)
for(n=0, 50, print1(a(n), ", "))
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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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Old definition (which is now a comment) replaced with explicit formula by N. J. A. Sloane, May 12 2010
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STATUS
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approved
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