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A005059
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a(n) = (5^n - 3^n)/2.
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21
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0, 1, 8, 49, 272, 1441, 7448, 37969, 192032, 966721, 4853288, 24325489, 121804592, 609554401, 3049366328, 15251614609, 76272421952, 381405156481, 1907154922568, 9536162033329, 47681972428112, 238413348924961, 1192077204978008, 5960417405949649
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OFFSET
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0,3
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COMMENTS
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Number of lines passing through 3 points of an n-dimensional grid of points of side 3. - David W. Wilson, c. 1999
a(n) is also the total number of words of length n, over an alphabet of five letters, one of them appearing an odd number of times. See the Lekraj Beedassy, Jul 22 2003, comment under A006516 (4-letter words), and the Balakrishnan reference there. See A003462 for the analogous 3-letter words problem. - Wolfdieter Lang, Jul 16 2017
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LINKS
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M. A. Alekseyev and T. Berger, Solving the Tower of Hanoi with Random Moves. In: J. Beineke, J. Rosenhouse (eds.) The Mathematics of Various Entertaining Subjects: Research in Recreational Math, Princeton University Press, 2016, pp. 65-79. ISBN 978-0-691-16403-8
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FORMULA
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a(n) = 8*a(n-1) - 15*a(n-2). - Paul Barry, Mar 03 2003
G.f.: x/((1-3*x)*(1-5*x)). - Paul Barry, Mar 03 2003
a(n) = Sum_{k=1..n} 2^(k-1)*3^(n-k)*binomial(n,k). - Zerinvary Lajos, Sep 24 2006
a(n) = (r^n-s^n)/(r-s) with r=5 and s=3. - Sture Sjöstedt, Oct 17 2012
a(n) = Sum_{k=0..n-1} 3^k*5^(n-k-1) for n>0, a(0)=0. - Bruno Berselli, Aug 07 2013
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EXAMPLE
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For the fifth formula: a(4) = 1*125 + 3*25 + 9*5 + 27*1 = 272. [Bruno Berselli, Aug 07 2013]
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MAPLE
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MATHEMATICA
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LinearRecurrence[{8, -15}, {0, 1}, 50] (* Sture Sjöstedt, Oct 17 2012 *)
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PROG
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(Sage) [lucas_number1(n, 8, 15) for n in range(0, 21)] # Zerinvary Lajos, Apr 23 2009
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CROSSREFS
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Cf. A081199 (binomial transform), A006516 (inverse binomial transform, and special 4-letter words). A003462 (special 3-letter words).
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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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