A155944 Jacobsthal numbers A001045, every second term incremented by 1.

0, 2, 1, 4, 5, 12, 21, 44, 85, 172, 341, 684, 1365, 2732, 5461, 10924, 21845, 43692, 87381, 174764, 349525, 699052, 1398101, 2796204, 5592405, 11184812, 22369621, 44739244, 89478485, 178956972, 357913941, 715827884, 1431655765, 2863311532, 5726623061, 11453246124, 22906492245

Offset: 0

Author

Paul Curtz, Jan 31 2009

Keywords

• nonn
• easy

Comments

Constructed from A001045 with periodic overlay, similar to A154890.

It appears that, except for term a(1)=2, these are the indices for which the Hankel transform of the coefficients of (1 - x)^(1/3) on F2[x] are non vanishing. See example 2.3 p. 8 of Han paper. - Michel Marcus, May 17 2020

Links

• Guo-Niu Han, Hankel continued fraction and its applications, Advances in Mathematics, Elsevier, 2016, 303, pp.295-321. 10.1016/j.aim.2016.08.013. hal-02125293.
• Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Crossrefs

Cf. A000035, A001045.

Programs

Mathematica

LinearRecurrence[{2,1,-2},{0,2,1},40] (* Harvey P. Dale, Mar 14 2014 *)

Python

print([(2**n + 1)//3 + n%2 for n in range(40)]) # Karl V. Keller, Jr., Aug 15 2021

Formula

a(n) = A001045(n) + A000035(n).

a(n+1) = 2^n + 1 - a(n).

From R. J. Mathar, Feb 10 2009: (Start)

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3).

a(n) = 1/2 + 2^n/3 - 5*(-1)^n/6.

G.f.: x(2-3x)/((1+x)(1-x)(1-2x)). (End)

a(n) = floor((2^n + 1)/3) + n mod 2. - Karl V. Keller, Jr., Aug 15 2021

Extensions

Definition rephrased, more terms from R. J. Mathar, Feb 10 2009