A105348 An indicator sequence for the Jacobsthal numbers.

1, 2, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0

Offset: 0

Author

Paul Barry, Apr 01 2005

Keywords

• easy
• nonn

Comments

a(n) is the number of solutions to the Diophantine equation 2*x^2 - (9*n+1)*x + 9*n^2 = 1 where valid solutions are restricted to powers of 4. - Hieronymus Fischer, May 17 2007

Examples

			a(1)=2 since J(1)=J(2)=1.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..87381

Crossrefs

For partial sums see A130253. Cf. A130249, A130250, A147612.

Programs

PARI

A147612aux(n,i) = if(!(n%2),n,A147612aux((n+i)/2,-i));
A147612(n) = 0^(A147612aux(n,1)*A147612aux(n,-1));
A105348(n) = if(1==n,2,A147612(n)); \\ Antti Karttunen, Nov 02 2018

Python

def A105348(n): return (m:=3*n+1).bit_length()-(m-3).bit_length() if n else 1 # Chai Wah Wu, Apr 18 2025

Formula

G.f.: Sum_{k>=0} x^A001045(k).

a(n) = 1 + floor(log_2(3n+1)) - ceiling(log_2(3n-1)) = floor(log_2(3n+1)) - floor(log_2(3n-2)) for n >= 1. Also true: a(n) = 1 + A130249(n) - A130250(n) = A130253(n) - A130250(n) = A130250(n+1) - A130250(n) for n >= 0. - Hieronymus Fischer, May 17 2007