A052940 a(0) = 1; a(n) = 3*2^n - 1, for n > 0.

1, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, 196607, 393215, 786431, 1572863, 3145727, 6291455, 12582911, 25165823, 50331647, 100663295, 201326591, 402653183, 805306367, 1610612735, 3221225471, 6442450943, 12884901887

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

A simple regular expression.

Numbers k > 1 such that a(k-1)^2 + a(k) is square, e.g., 5^2 + 11 = 6^2; 11^2 + 23 = 12^2. - Vincenzo Librandi, Aug 06 2010

Numerator of the sum of terms at the n-th level of the Calkin-Wilf tree. - Carl Najafi, Jul 10 2011

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..3320
• Gennady Eremin, Dyck Numbers, III. Enumeration and bijection with symmetric Dyck paths, arXiv:2302.02765 [math.CO], 2023.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 931
• Index entries for linear recurrences with constant coefficients, signature (3,-2).

Crossrefs

Apart from initial terms, same as A055010 and A083329.

Subsequence of A036991.

Cf. A000225, A023548, A107909, A134060, A140182.

Programs

GAP

Concatenation([1], List([1..30], n-> 3*2^n -1)); # G. C. Greubel, Oct 18 2019

Magma

[1] cat [3*2^n - 1: n in [1..30]]; // Vincenzo Librandi, Dec 01 2015

Maple

spec:= [S,{S=Prod(Sequence(Union(Z,Z)),Union(Sequence(Z),Z,Z))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
seq(`if`(n=0,1,3*2^n -1), n=0..30); # G. C. Greubel, Oct 18 2019

Mathematica

Join[{1},Table[3*2^n-1,{n,30}]] (* or *) Join[{1},LinearRecurrence[{3,-2},{5,11},30]] (* Harvey P. Dale, Mar 07 2015 *)

PARI

a(n)=if(n,3*2^n-1,1) \\ Charles R Greathouse IV, Oct 07 2015

PARI

Vec((1+2*x-2*x^2)/(-1+2*x)/(-1+x) + O(x^30)) \\ Altug Alkan, Dec 01 2015

Python

print([1] + [(3<Gennady Eremin, Aug 29 2023

Sage

[1]+[3*2^n -1 for n in (1..30)] # G. C. Greubel, Oct 18 2019

Formula

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

a(n) = 3*a(n-1) - 2*a(n-2) for n > 2.

Binomial transform of 3 - 0^n - (-1)^n = (1, 4, 2, 4, 2, 4, 2, ...). - Paul Barry, Jun 30 2003

a(n) = A107909(A023548(n+1)) for n > 1. - Reinhard Zumkeller, May 28 2005

Row sums of triangle A134060. - Gary W. Adamson, Oct 05 2007

Equals row sums of triangle A140182. - Gary W. Adamson, May 11 2008

Equals M*Q, where M is a modified Pascal triangle (1,2) with first term "1" instead of 2; as an infinite lower triangular matrix. Q is the vector (1, 2, 2, 2, ...). - Gary W. Adamson, Nov 30 2015

From Gennady Eremin, Aug 29 2023: (Start)

a(n+1) = 2*a(n) + 1 for n > 0.

a(n) = (A000225(n+1) + A000225(n+2))/2 for n > 0. (End)

Extensions

More terms from James Sellers, Jun 08 2000

a(30)-a(32) from Vincenzo Librandi, Dec 01 2015