A154890 -id:A154890 - cp's OEIS frontend

A155701 a(n) = (4^n + 8)/3.

3, 4, 8, 24, 88, 344, 1368, 5464, 21848, 87384, 349528, 1398104, 5592408, 22369624, 89478488, 357913944, 1431655768, 5726623064, 22906492248, 91625968984, 366503875928, 1466015503704, 5864062014808, 23456248059224, 93824992236888, 375299968947544

Offset: 0

Paul Curtz, Jan 25 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4).

[(4^n+8)/3: n in [0..35]]; // Vincenzo Librandi, Jul 24 2011

A155701 := proc(n) (4^n+8)/3 ; end: seq(A155701(n),n=0..80) ; # R. J. Mathar, Jul 23 2009

A155701[n_]:=(4^n+8)/3;Array[A155701,50,0] (* Paolo Xausa, Dec 05 2023 *)

a(n) = 3 + A002450(n).
a(n) = 5*a(n-1) - 4*a(n-2) = 4*a(n-1) - 8.
a(n) = A154879(2n) = A154890(2n).
a(n+1) - a(n) = A000302(n).
a(n+1) = 4*A047849(n) = 4*A078008(2n).
G.f.: (3-11*x)/((4*x-1)*(x-1)). - R. J. Mathar, Jul 23 2009

Edited and extended by R. J. Mathar, Jul 23 2009

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

Original entry on oeis.org

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

Paul Curtz, Jan 31 2009

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

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).

Cf. A000035, A001045.

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

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

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

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

cp's OEIS Frontend

A155701 a(n) = (4^n + 8)/3.

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