A036563 a(n) = 2^n - 3.

-2, -1, 1, 5, 13, 29, 61, 125, 253, 509, 1021, 2045, 4093, 8189, 16381, 32765, 65533, 131069, 262141, 524285, 1048573, 2097149, 4194301, 8388605, 16777213, 33554429, 67108861, 134217725, 268435453, 536870909, 1073741821, 2147483645

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign
• easy

Comments

a(n+1) is the n-th number with exactly n 1's in binary representation. - Reinhard Zumkeller, Mar 06 2003

Berstein and Onn: "For every m = 3k+1, the Graver complexity of the vertex-edge incidence matrix of the complete bipirtite graph K(3,m) satisfies g(m) >= 2^(k+2)-3." - Jonathan Vos Post, Sep 15 2007

Row sums of triangle A135857. - Gary W. Adamson, Dec 01 2007

a(n) = A164874(n-1,n-2) for n > 2. - Reinhard Zumkeller, Aug 29 2009

Starting (1, 5, 13, ...) = eigensequence of a triangle with A016777: (1, 4, 7, 10, ...) as the left border and the rest 1's. - Gary W. Adamson, Jul 24 2010

An elephant sequence, see A175655. For the central square just one A[5] vector, with decimal value 186, leads to this sequence (n >= 2). For the corner squares this vector leads to the companion sequence A123203. - Johannes W. Meijer, Aug 15 2010

First differences of A095264: A095264(n+1) - A095264(n) = a(n+2). - J. M. Bergot, May 13 2013

a(n+2) is given by the sum of n-th row of triangle of powers of 2: 1; 2 1 2; 4 2 1 2 4; 8 4 2 1 2 4 8; ... - Philippe Deléham, Feb 24 2014

Also, the decimal representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 643", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. See A283508. - Robert Price, Mar 09 2017

a(n+3) is the value of the Ackermann function A(3,n) or ack(3,n). - Olivier Gérard, May 11 2018

Examples

			a(2) = 1;
a(3) = 2 + 1 + 2 = 5;
a(4) = 4 + 2 + 1 + 2 + 4 = 13;
a(5) = 8 + 4 + 2 + 1 + 2 + 4 + 8 = 29; etc. - _Philippe Deléham_, Feb 24 2014
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..500
• Paul Barry, Conjectures and results on some generalized Rueppel sequences, arXiv:2107.00442 [math.CO], 2021.
• Yael Berstein and Shmuel Onn, The Graver complexity of integer programming, Annals of Combinatorics, Vol. 13, No. 3 (2009), pp. 289-296; arXiv preprint, arXiv:0709.1500 [math.CO], 2007.
• L' Education Mathématique, Problème 8907, 49e Annee, No 14, 15 Avril 1947, p. 113
• Irving Kaplansky and John Riordan, The problem of the rooks and its applications, in Combinatorics, Duke Mathematical Journal, 13.2 (1946): 259-268. [Annotated scanned copy]
• Irving Kaplansky and John Riordan, The problem of the rooks and its applications, Duke Mathematical Journal 13.2 (1946): 259-268. Sequence is on page 267.
• Index entries for linear recurrences with constant coefficients, signature (3,-2).

Crossrefs

Row sums of triangular array A027960. A column of A119725.

Cf. A081118, A130459, A131112, A050414, A050415, A135857, A000079, A016777, A283508.

Cf. A074877, A304370, A304371, A331372.

Programs

GAP

List([0..40], n-> 2^n -3); # G. C. Greubel, Nov 18 2019

Magma

[2^n-3: n in [0..40]]; // Vincenzo Librandi, May 09 2011

Maple

A036563:=n->2^n-3; seq(A036563(n), n=0..40); # Wesley Ivan Hurt, Jun 26 2014

Mathematica

Table[2^n - 3, {n, 0, 40}] (* Wesley Ivan Hurt, Jun 26 2014 *)
LinearRecurrence[{3,-2},{-2,-1},40] (* Harvey P. Dale, Sep 26 2018 *)

PARI

a(n)= 2^n-3 \\ Charles R Greathouse IV, Dec 22 2011

Python

def A036563(n): return (1<Chai Wah Wu, Sep 27 2024

Sage

[gaussian_binomial(n,1,2)-2 for n in range(0,40)] # Zerinvary Lajos, May 31 2009

Formula

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

The sequence 1, 5, 13, ... has a(n) = 4*2^n-3. These are the partial sums of A151821. - Paul Barry, Aug 25 2003

a(n) = A118654(n-3, 6), for n > 2. - N. J. A. Sloane, Sep 29 2006

Row sums of triangle A130459 starting (1, 5, 13, 29, 61, ...). - Gary W. Adamson, May 26 2007

Row sums of triangle A131112. - Gary W. Adamson, Jun 15 2007

Binomial transform of [1, 4, 4, 4, ...] = (1, 5, 13, 29, 61, ...). - Gary W. Adamson, Sep 20 2007

a(n) = 2*StirlingS2(n,2) - 1, for n > 0. - Ross La Haye, Jul 05 2008

a(n) = A000079(n) - 3. - Omar E. Pol, Dec 21 2008

From Mohammad K. Azarian, Jan 14 2009: (Start)

G.f.: 1/(1-2*x) - 3/(1-x).

E.g.f.: exp(2*x) - 3*exp(x). (End)

For n >= 3, a(n) = 2<+>n, where operation <+> is defined in A206853. - Vladimir Shevelev, Feb 17 2012

a(n) = 3*a(n-1) - 2*a(n-2) for n > 1, a(0)=-2, a(1)=-1. - Philippe Deléham, Dec 23 2013

Sum_{n>=1} 1/a(n) = A331372. - Amiram Eldar, Nov 18 2020