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A003948
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Expansion of (1+x)/(1-5*x).
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64
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1, 6, 30, 150, 750, 3750, 18750, 93750, 468750, 2343750, 11718750, 58593750, 292968750, 1464843750, 7324218750, 36621093750, 183105468750, 915527343750, 4577636718750, 22888183593750, 114440917968750, 572204589843750, 2861022949218750, 14305114746093750
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OFFSET
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0,2
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COMMENTS
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Coordination sequence for infinite tree with valency 6.
The n-th term of the coordination sequence of the infinite tree with valency 2m is the same as the number of reduced words of size n in the free group on m generators. In the five sequences A003946, A003948, A003950, A003952, A003954, m is 2, 3, 4, 5, 6. - Avi Peretz (njk(AT)netvision.net.il), Feb 23 2001 and Ola Veshta (olaveshta(AT)my-deja.com), Mar 30 2001
Hamiltonian path in S_4 X P_2n.
For n>=1, a(n+1) is equal to the number of functions f:{1,2,...,n+1}->{1,2,3,4,5,6} such that for fixed, different x_1, x_2,...,x_n in {1,2,...,n+1} and fixed y_1, y_2,...,y_n in {1,2,3,4,5,6} we have f(x_i)<>y_i, (i=1..n). - Milan Janjic, May 10 2007
For n>=1, a(n) equals the numbers of words of length n over the alphabet {0..5} with no two adjacent letters identical. - Milan Janjic, Jan 31 2015 [Corrected by David Nacin, May 30 2017]
a(n) equals the numbers of sequences of length n on {0,...,5} where no two adjacent terms differ by three. - David Nacin, May 30 2017
It appears that these are the only n>1 for which alpha(n)=2n, where alpha(n) is the entry point of n in the Fibonacci sequence, see A001177. - Philippe Schnoebelen, Apr 11 2024
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LINKS
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FORMULA
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G.f.: (1+x)/(1-5*x).
The Hankel transform of this sequence is [1,-6,0,0,0,0,0,0,0,0,...]. - Philippe Deléham, Nov 21 2007
G.f.: 2/x - 5 - 8/(x*U(0)) where U(k)= 1 + 2/(3^k - 3^k/(2 + 1 - 12*x*3^k/(6*x*3^k + 1/U(k+1)))) ; (continued fraction, 4-step). - Sergei N. Gladkovskii, Oct 30 2012
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MAPLE
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k := 6; if n = 0 then 1 else k*(k-1)^(n-1); fi;
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MATHEMATICA
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q = 6; Join[{a = 1}, Table[If[n != 0, a = q*a - a, a = q*a], {n, 0, 25}]] (* and *) Join[{1}, 6*5^Range[0, 25]] (* Vladimir Joseph Stephan Orlovsky, Jul 11 2011 *)
CoefficientList[Series[(1+x)/(1-5x), {x, 0, 30}], x] (* Michael De Vlieger, Dec 10 2016 *)
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PROG
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(Magma) [1] cat [6*5^(n-1): n in [1..30]]; // G. C. Greubel, Sep 24 2019
(Sage) [1]+[6*5^(n-1) for n in (1..30)] # G. C. Greubel, Sep 24 2019
(GAP) Concatenation([1], List([1..30], n-> 6*5^(n-1) )); # G. C. Greubel, Sep 24 2019
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CROSSREFS
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KEYWORD
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nonn,easy,nice,walk
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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