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%I #124 Feb 08 2022 08:07:29
%S -2,-1,1,5,13,29,61,125,253,509,1021,2045,4093,8189,16381,32765,65533,
%T 131069,262141,524285,1048573,2097149,4194301,8388605,16777213,
%U 33554429,67108861,134217725,268435453,536870909,1073741821,2147483645
%N a(n) = 2^n - 3.
%C a(n+1) is the n-th number with exactly n 1's in binary representation. - _Reinhard Zumkeller_, Mar 06 2003
%C 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
%C Row sums of triangle A135857. - _Gary W. Adamson_, Dec 01 2007
%C a(n) = A164874(n-1,n-2) for n > 2. - _Reinhard Zumkeller_, Aug 29 2009
%C 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
%C 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
%C First differences of A095264: A095264(n+1) - A095264(n) = a(n+2). - _J. M. Bergot_, May 13 2013
%C 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
%C 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
%C a(n+3) is the value of the Ackermann function A(3,n) or ack(3,n). - _Olivier Gérard_, May 11 2018
%H Vincenzo Librandi, <a href="/A036563/b036563.txt">Table of n, a(n) for n = 0..500</a>
%H Paul Barry, <a href="https://arxiv.org/abs/2107.00442">Conjectures and results on some generalized Rueppel sequences</a>, arXiv:2107.00442 [math.CO], 2021.
%H Yael Berstein and Shmuel Onn, <a href="https://doi.org/10.1007/s00026-009-0029-6">The Graver complexity of integer programming</a>, Annals of Combinatorics, Vol. 13, No. 3 (2009), pp. 289-296; <a href="https://arxiv.org/abs/0709.1500">arXiv preprint</a>, arXiv:0709.1500 [math.CO], 2007.
%H Irving Kaplansky and John Riordan, <a href="/A274105/a274105.pdf">The problem of the rooks and its applications</a>, in Combinatorics, Duke Mathematical Journal, 13.2 (1946): 259-268. [Annotated scanned copy]
%H Irving Kaplansky and John Riordan, <a href="http://projecteuclid.org/euclid.dmj/1077473616">The problem of the rooks and its applications</a>, Duke Mathematical Journal 13.2 (1946): 259-268. Sequence is on page 267.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2).
%F a(n) = 2*a(n-1) + 3.
%F The sequence 1, 5, 13, ... has a(n) = 4*2^n-3. These are the partial sums of A151821. - _Paul Barry_, Aug 25 2003
%F a(n) = A118654(n-3, 6), for n > 2. - _N. J. A. Sloane_, Sep 29 2006
%F Row sums of triangle A130459 starting (1, 5, 13, 29, 61, ...). - _Gary W. Adamson_, May 26 2007
%F Row sums of triangle A131112. - _Gary W. Adamson_, Jun 15 2007
%F Binomial transform of [1, 4, 4, 4, ...] = (1, 5, 13, 29, 61, ...). - _Gary W. Adamson_, Sep 20 2007
%F a(n) = 2*StirlingS2(n,2) - 1, for n > 0. - _Ross La Haye_, Jul 05 2008
%F a(n) = A000079(n) - 3. - _Omar E. Pol_, Dec 21 2008
%F From _Mohammad K. Azarian_, Jan 14 2009: (Start)
%F G.f.: 1/(1-2*x) - 3/(1-x).
%F E.g.f.: exp(2*x) - 3*exp(x). (End)
%F For n >= 3, a(n) = 2<+>n, where operation <+> is defined in A206853. - _Vladimir Shevelev_, Feb 17 2012
%F a(n) = 3*a(n-1) - 2*a(n-2) for n > 1, a(0)=-2, a(1)=-1. - _Philippe Deléham_, Dec 23 2013
%F Sum_{n>=1} 1/a(n) = A331372. - _Amiram Eldar_, Nov 18 2020
%e a(2) = 1;
%e a(3) = 2 + 1 + 2 = 5;
%e a(4) = 4 + 2 + 1 + 2 + 4 = 13;
%e a(5) = 8 + 4 + 2 + 1 + 2 + 4 + 8 = 29; etc. - _Philippe Deléham_, Feb 24 2014
%p A036563:=n->2^n-3; seq(A036563(n), n=0..40); # _Wesley Ivan Hurt_, Jun 26 2014
%t Table[2^n - 3, {n, 0, 40}] (* _Wesley Ivan Hurt_, Jun 26 2014 *)
%t LinearRecurrence[{3,-2},{-2,-1},40] (* _Harvey P. Dale_, Sep 26 2018 *)
%o (Sage) [gaussian_binomial(n,1,2)-2 for n in range(0,40)] # _Zerinvary Lajos_, May 31 2009
%o (Magma) [2^n-3: n in [0..40]]; // _Vincenzo Librandi_, May 09 2011
%o (PARI) a(n)= 2^n-3 \\ _Charles R Greathouse IV_, Dec 22 2011
%o (GAP) List([0..40], n-> 2^n -3); # _G. C. Greubel_, Nov 18 2019
%Y Row sums of triangular array A027960. A column of A119725.
%Y Cf. A081118, A130459, A131112, A050414, A050415, A135857, A000079, A016777, A283508.
%Y Cf. A074877, A304370, A304371, A331372.
%K sign,easy
%O 0,1
%A _N. J. A. Sloane_
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