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A234038
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Smallest positive integer solution x of 9*x - 2^n*y = 1.
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1
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1, 1, 1, 1, 9, 25, 57, 57, 57, 57, 569, 1593, 3641, 3641, 3641, 3641, 36409, 101945, 233017, 233017, 233017, 233017, 2330169, 6524473, 14913081, 14913081, 14913081, 14913081, 149130809, 417566265, 954437177, 954437177, 954437177
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OFFSET
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0,5
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COMMENTS
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The solution of the linear Diophantine equation 9*x - 2^n*y = 1 with smallest positive x is x=a(n), n>= 0, and the corresponding y is given by y(n) = 5^(n+3) (mod 9) = A070366(n+3) with o.g.f. (8-4*x-2*x^2+7*x^3)/((1-x+x^2)*(1-x)*(1+x)) (derived from the one given in A070366). This is the periodic sequence with period [8, 4, 2, 1, 5, 7].
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LINKS
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Table of n, a(n) for n=0..32.
W. Lang, On Collatz' Words, Sequences and Trees, arXiv preprint arXiv:1404.2710, 2014 and J. Int. Seq. 17 (2014) # 14.11.7.
Index entries for linear recurrences with constant coefficients, signature (3,-2,-8,24,-16)
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FORMULA
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a(n) = (1 + 2^n*(5^(n-3)(mod 9)))/3^2, n >= 0.
O.g.f.: (1-2*x+8*x^3-8*x^4)/((1-x)*(1-4*x^2)*(1-2*x+4*x^2) (derived from the one for y(n) given above in a comment).
a(n) = 2*(a(n-1) - 4*a(n-3) + 8*a(n-4)) - 1, n >= 4, a(0)=a(1)=a(2)=a(3) = 1 (from the y(n) recurrence given in A070366).
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EXAMPLE
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n = 0: 9*1 - 1*8 = 1; n = 3: 9*1 - 8*1 = 1.
a(4) = (1 + 2^4*5)/9 = 9.
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CROSSREFS
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A070366, A007583 (see the Feb 15 2013 comment).
Sequence in context: A336669 A239745 A269440 * A304033 A048490 A113828
Adjacent sequences: A234035 A234036 A234037 * A234039 A234040 A234041
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KEYWORD
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nonn,easy
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AUTHOR
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Wolfdieter Lang, Feb 17 2014
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STATUS
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approved
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