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A081200
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6th binomial transform of (0,1,0,1,0,1,...), A000035.
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12
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0, 1, 12, 109, 888, 6841, 51012, 372709, 2687088, 19200241, 136354812, 964249309, 6798573288, 47834153641, 336059778612, 2358521965909, 16540171339488, 115933787267041, 812299450322412, 5689910849522509, 39848449432985688
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OFFSET
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0,3
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COMMENTS
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Conjecture (verified up to a(9)): Number of collinear 5-tuples of points in a 5 X 5 X 5 X ... n-dimensional cubic grid. - Ron Hardin, May 24 2010
a(n) is also the total number of words of length n, over an alphabet of seven letters, of which one of them appears an odd number of times. See the Lekraj Beedassy, Jul 22 2003, comment on A006516 (4-letter case), and the Balakrishnan reference there. For the 2-, 3-, 5-, 6- and 8-letter case analogs see A131577, A003462, A005059, A081199, A081201 respectively. - Wolfdieter Lang, Jul 17 2017
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LINKS
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FORMULA
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a(n) = 12*a(n-1) - 35*a(n-2), a(0) = 0, a(1) = 1.
G.f.: x/((1-5*x)*(1-7*x)).
a(n) = 7^n/2 - 5^n/2.
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EXAMPLE
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The a(2) = 12 words of length 2 over {A ,B, C, D, E, F, G} with say, A, appearing an odd number of times (that is once) are: AB, AC, AD, AE, AF, AG; BA, CA, DA, EA, FA, GA. - Wolfdieter Lang, Jul 17 2017
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MATHEMATICA
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CoefficientList[Series[x / ((1 - 5 x) (1 - 7 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 07 2013 *)
LinearRecurrence[{12, -35}, {0, 1}, 30] (* Harvey P. Dale, Feb 07 2014 *)
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PROG
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(Sage) [lucas_number1(n, 12, 35) for n in range(0, 21)] # Zerinvary Lajos, Apr 27 2009
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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