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A015453
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Generalized Fibonacci numbers.
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5
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1, 1, 8, 57, 407, 2906, 20749, 148149, 1057792, 7552693, 53926643, 385039194, 2749201001, 19629446201, 140155324408, 1000716717057, 7145172343807, 51016923123706, 364263634209749, 2600862362591949, 18570300172353392
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OFFSET
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0,3
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COMMENTS
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For n >= 1, row sums of triangle for numbers 7^k*C(m,k) with duplicated diagonals. - Vladimir Shevelev, Apr 13 2012
For n >= 1, a(n) equals the numbers of words of length n-1 on alphabet {0,1,2,3,5,6,7} containing no subwords ii, (i=0,1,...,6). - Milan Janjic, Jan 31 2015
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LINKS
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FORMULA
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a(n) = 7*a(n-1) + a(n-2).
For n >= 2, a(n) = F_(n)(7) + F_(n+1)(7), where F_(n)(x) is Fibonacci polynomial (cf. A049310): F_(n)(x) = Sum_{i=0..floor((n-1)/2)} C(n-i-1,i)*x^(n-2*i-1). - Vladimir Shevelev, Apr 13 2012
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MATHEMATICA
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CoefficientList[Series[(1-6*x)/(1-7*x-x^2), {x, 0, 50}], x] (* G. C. Greubel, Dec 19 2017 *)
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PROG
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(Magma) [n le 2 select 1 else 7*Self(n-1) + Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 08 2012
(PARI) my(x='x+O('x^30)); Vec((1-6*x)/(1-7*x-x^2)) \\ G. C. Greubel, Dec 19 2017
(Sage) [lucas_number1(n+1, 7, -1) - 6*lucas_number1(n, 7, -1) for n in (0..30)] # G. C. Greubel, Dec 24 2021
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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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