A155944 - OEIS

%I #27 Jan 03 2022 20:45:07

%S 0,2,1,4,5,12,21,44,85,172,341,684,1365,2732,5461,10924,21845,43692,

%T 87381,174764,349525,699052,1398101,2796204,5592405,11184812,22369621,

%U 44739244,89478485,178956972,357913941,715827884,1431655765,2863311532,5726623061,11453246124,22906492245

%N Jacobsthal numbers A001045, every second term incremented by 1.

%C Constructed from A001045 with periodic overlay, similar to A154890.

%C 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

%H Guo-Niu Han, <a href="https://hal.archives-ouvertes.fr/hal-02125293">Hankel continued fraction and its applications</a>, Advances in Mathematics, Elsevier, 2016, 303, pp.295-321. 10.1016/j.aim.2016.08.013. hal-02125293.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2).

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

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

%F From _R. J. Mathar_, Feb 10 2009: (Start)

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

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

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

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

%t LinearRecurrence[{2,1,-2},{0,2,1},40] (* _Harvey P. Dale_, Mar 14 2014 *)

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

%Y Cf. A000035, A001045.

%K nonn,easy

%O 0,2

%A _Paul Curtz_, Jan 31 2009

%E Definition rephrased, more terms from _R. J. Mathar_, Feb 10 2009