A239130 Smallest positive integer solution x = a(n) of (3^4)*x - 2^n*y = 1 for n >= 0.

1, 1, 1, 1, 1, 17, 49, 49, 177, 177, 177, 177, 2225, 2225, 2225, 18609, 18609, 84145, 84145, 84145, 608433, 1657009, 1657009, 1657009, 1657009, 1657009, 1657009, 1657009, 135874737, 404310193, 941181105, 2014922929, 2014922929

Offset: 0

Author

Wolfdieter Lang, Mar 22 2014

Keywords

• nonn
• easy

Comments

This is instance m=4 of the m-family of smallest positive solutions [x0(m,n), y0(m,n)] of 3^m*x - 2^n*y = 1, n >= 0, m >= 0, described in a comment on A239125.

The companion sequence is y(n) = y0(4, n) = A239131(n), which is periodic with period length phi(3^4) = 54, where phi(n) = A000010(n) (Euler's totient).

The G.f. can be found from that of the periodic sequence y(n).

Examples

			n=0: 81*1 - 1*80 = 1;
n=1: 81*1 - 2*40 = 1;
n=2: 81*1 - 4*20 = 1;
n=3: 81*1 - 8*10 = 1;
n=4: 81*1 - 16*5 = 1;
n=5: 81*17 - 32*5 =1; ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Wolfdieter Lang, On Collatz' Words, Sequences and Trees, arXiv preprint arXiv:1404.2710 [math.NT], 2014 and J. Int. Seq. 17 (2014) # 14.11.7

Crossrefs

Cf. A000010, A007583 (m=1), A234038 (m=2), A239125 (m=3), A239131.

Programs

Magma

[Floor(2^n*((41^(n+27) mod 81)/81))+1: n in [0..40]]; // Vincenzo Librandi, Mar 23 2014

Mathematica

Floor[Table[(2^n Mod[(41^(n + 27)), 81])/81 + 1, {n, 0, 40}]] (* Vincenzo Librandi, Mar 23 2014 *)

Formula

a(n) = (1 + 2^n*y0(4, n))/3^4, with y0(4, n) == ((3^4+1)/2)^(n + 3^3) (mod 3^4) = A239131(n), n >= 0.

a(n + 54) = 2^(54)*a(n) - (2^(54)-1)/3^4, n >= 0, from the y0(4, n) periodicity.