

A036563


a(n) = 2^n  3.


63



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
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OFFSET

0,1


COMMENTS

a(n+1) is the nth 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 vertexedge 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(n1,n2) 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 nth 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 xaxis, from the left edge to the origin, of the nth stage of growth of the twodimensional cellular automaton defined by "Rule 643", based on the 5celled 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


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. 289296; arXiv preprint, arXiv:0709.1500 [math.CO], 2007.
Irving Kaplansky and John Riordan, The problem of the rooks and its applications, in Combinatorics, Duke Mathematical Journal, 13.2 (1946): 259268. [Annotated scanned copy]
Irving Kaplansky and John Riordan, The problem of the rooks and its applications, Duke Mathematical Journal 13.2 (1946): 259268. Sequence is on page 267.
Index entries for linear recurrences with constant coefficients, signature (3,2).


FORMULA

a(n) = 2*a(n1) + 3.
The sequence 1, 5, 13, ... has a(n) = 4*2^n3. These are the partial sums of A151821.  Paul Barry, Aug 25 2003
a(n) = A118654(n3, 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/(12*x)  3/(1x).
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(n1)  2*a(n2) 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


EXAMPLE

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


MAPLE

A036563:=n>2^n3; 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 *)


PROG

(Sage) [gaussian_binomial(n, 1, 2)2 for n in range(0, 40)] # Zerinvary Lajos, May 31 2009
(Magma) [2^n3: n in [0..40]]; // Vincenzo Librandi, May 09 2011
(PARI) a(n)= 2^n3 \\ Charles R Greathouse IV, Dec 22 2011
(GAP) List([0..40], n> 2^n 3); # G. C. Greubel, Nov 18 2019


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.
Sequence in context: A324960 A174986 A327671 * A025264 A321716 A245567
Adjacent sequences: A036560 A036561 A036562 * A036564 A036565 A036566


KEYWORD

sign,easy


AUTHOR

N. J. A. Sloane


STATUS

approved



