cp's OEIS Frontend

A234038 Smallest positive integer solution x of 9*x - 2^n*y = 1.

%I A234038 #26 Nov 05 2024 08:05:08
%S A234038 1,1,1,1,9,25,57,57,57,57,569,1593,3641,3641,3641,3641,36409,101945,
%T A234038 233017,233017,233017,233017,2330169,6524473,14913081,14913081,
%U A234038 14913081,14913081,149130809,417566265,954437177,954437177,954437177
%N A234038 Smallest positive integer solution x of 9*x - 2^n*y = 1.
%C A234038 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].
%H A234038 Paolo Xausa, <a href="/A234038/b234038.txt">Table of n, a(n) for n = 0..1000</a>
%H A234038 Wolfdieter Lang, <a href="http://arxiv.org/abs/1404.2710">On Collatz' Words, Sequences and Trees</a>, arXiv preprint arXiv:1404.2710, 2014 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Lang/lang6.html">J. Int. Seq. 17 (2014) # 14.11.7</a>.
%H A234038 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-8,24,-16)
%F A234038 a(n) = (1 + 2^n*(5^(n-3)(mod 9)))/3^2, n >= 3.
%F A234038 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).
%F A234038 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).
%e A234038 n = 0:  9*1 - 1*8 = 1; n = 3:  9*1 - 8*1  = 1.
%e A234038 a(4) = (1 + 2^4*5)/9 = 9.
%t A234038 A234038[n_] := Ceiling[(1 + 2^n*Mod[5^(n - 3), 9])/9]; Array[A234038, 50, 0] (* or *)
%t A234038 LinearRecurrence[{3, -2, -8, 24, -16}, {1, 1, 1, 1, 9}, 50] (* _Paolo Xausa_, Nov 05 2024 *)
%Y A234038 Cf. A070366, A007583 (see the Feb 15 2013 comment).
%K A234038 nonn,easy
%O A234038 0,5
%A A234038 _Wolfdieter Lang_, Feb 17 2014