A052549 a(n) = 5*2^(n-1) - 1, n>0, with a(0)=1.

1, 4, 9, 19, 39, 79, 159, 319, 639, 1279, 2559, 5119, 10239, 20479, 40959, 81919, 163839, 327679, 655359, 1310719, 2621439, 5242879, 10485759, 20971519, 41943039, 83886079, 167772159, 335544319, 671088639, 1342177279, 2684354559

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Comments

A153894 is a better version of this sequence. - N. J. A. Sloane, Feb 07 2009

Equals binomial transform of [1, 3, 2, 3, 2, 3, 2, ...]. - Gary W. Adamson, May 11 2008

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• David Eppstein, Making Change in 2048, arXiv:1804.07396 [cs.DM], 2018.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 486
• Index entries for linear recurrences with constant coefficients, signature (3,-2).

Crossrefs

Cf. A000225, A153894.

Programs

GAP

Concatenation([1], List([1..30], n-> 5*2^(n-1) -1)); # G. C. Greubel, May 07 2019

Magma

[n eq 0 select 1 else 5*2^(n-1) -1: n in [0..30]]; // G. C. Greubel, May 07 2019

Maple

spec := [S,{S=Prod(Sequence(Union(Z,Z)),Union(Z,Sequence(Z)))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

Mathematica

{1}~Join~Array[5*2^(# -1)-1 &,30] (* Michael De Vlieger, Jul 18 2018 *)
LinearRecurrence[{3,-2}, {1,4,9}, 30] (* G. C. Greubel, May 07 2019 *)

PARI

vector(30, n, n--; if(n==0, 1, 5*2^(n-1) -1)) \\ G. C. Greubel, May 07 2019

Python

a052549 = [1] + [(5<<(n-1))-1 for n in range(1, 30)]
print(a052549)  # Gennady Eremin, Sep 10 2023

Sage

[1]+[5*2^(n-1) -1 for n in (1..30)] # G. C. Greubel, May 07 2019

Formula

G.f.: (1 + x - x^2)/((1-2*x)*(1-x)).

a(n) = 2*a(n-1) + 1, for n>1, with a(0)=1 and a(1)=4.

E.g.f.: (5*exp(2*x) - 2*exp(x) - 1)/2. - G. C. Greubel, May 07 2019

a(n) = 1 + A000225(n-1) + A000225(n+1) for n > 0. - Gennady Eremin, Sep 08 2023

Extensions

More terms from James Sellers, Jun 06 2000