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

0,2

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

Table of n, a(n) for n=0..36.

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).

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

Mathematica

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

Prog

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

Crossrefs

Cf. A000035, A001045.

Sequence in context: A138205 A137224 A191830 * A350087 A091232 A336398

Adjacent sequences: A155941 A155942 A155943 * A155945 A155946 A155947

Keyword

nonn,easy

Author

Paul Curtz, Jan 31 2009

Extensions

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

Status

approved