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A234038 Smallest positive integer solution x of 9*x - 2^n*y = 1. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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].

LINKS

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)

FORMULA

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).

EXAMPLE

n = 0:  9*1 - 1*8 = 1; n = 3:  9*1 - 8*1  = 1.

a(4) = (1 + 2^4*5)/9 = 9.

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Feb 17 2014

STATUS

approved

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Last modified June 24 21:50 EDT 2021. Contains 345433 sequences. (Running on oeis4.)