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A016127
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Expansion of 1/((1-2*x)*(1-5*x)).
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18
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1, 7, 39, 203, 1031, 5187, 25999, 130123, 650871, 3254867, 16275359, 81378843, 406898311, 2034499747, 10172515119, 50862608363, 254313107351, 1271565667827, 6357828601279, 31789143530683
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OFFSET
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0,2
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COMMENTS
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The sequence of fractions a(n)/(n+1) is the 3rd binomial transform of the sequence of fractions J(n+1)/(n+1) where J(n) is A001045(n). - Paul Barry, Aug 05 2005
Equals term (1,2) in M^n, M = the 3 X 3 matrix [1, 1, 3; 1, 3, 1; 3, 1, 1]. a(n)/ a(n-1) tends to 5, a root to the charpoly x^3 - 5x^2 -4x + 20. - Gary W. Adamson, Mar 12 2009
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LINKS
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FORMULA
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a(n) = (5^(n+1) - 2^(n+1))/3 = Sum_{i=0..n} 5^i*2^(n-1) = 5*a(n-1) + 2^n =2*a(n-1) + 5^n. - Henry Bottomley, Apr 07 2003
a(n) = Sum_{k=0..n} Sum_{j=0..n} 5^(n-j)*binomial(j, k);
a(n) = Sum_{k=0..n} 2^k*5^(n-k) = Sum_{k=0..n} 5^k*2^(n-k). (End)
For n > 2, a(n) = 9*a(n-1) - 24*a(n-2) + 20*a(n-3). - Gary W. Adamson, Dec 26 2007
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MATHEMATICA
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CoefficientList[Series[1/((1 - 2 x) (1 - 5 x)), {x, 0, 19}], x] (* Michael De Vlieger, Jan 31 2018 *)
LinearRecurrence[{7, -10}, {1, 7}, 30] (* Harvey P. Dale, Aug 18 2020 *)
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PROG
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(Sage) [lucas_number1(n, 7, 10) for n in range(1, 21)] # Zerinvary Lajos, Apr 23 2009
(Sage) [(5^n - 2^n)/3 for n in range(1, 21)] # Zerinvary Lajos, Jun 05 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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