A052549 -id:A052549 - cp's OEIS frontend

A081254 Numbers k such that A081252(m)/m^2 has a local maximum for m = k.

1, 3, 6, 13, 26, 53, 106, 213, 426, 853, 1706, 3413, 6826, 13653, 27306, 54613, 109226, 218453, 436906, 873813, 1747626, 3495253, 6990506, 13981013, 27962026, 55924053, 111848106, 223696213, 447392426, 894784853, 1789569706, 3579139413

Offset: 1

Klaus Brockhaus, Mar 17 2003

The limit of the local maxima, lim_{m->inf} A081252(m)/m^2 = 1/10. For local minima cf. A081253.

Row sums of the triangle A181971. - Reinhard Zumkeller, Jul 09 2012

			13 is a term since A081252(12)/12^2 = 15/144 = 0.104..., A081252(13)/13^2 = 18/169 = 0.106..., A081252(14)/14^2 = 20/196 = 0.102....

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Thomas Baruchel, Properties of the cumulated deficient binary digit sum, arXiv:1908.02250 [math.NT], 2019.
Klaus Brockhaus, Illustration for A053646, A081252, A081253 and A081254
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A000975, A020714, A020989, A048573, A052549, A053646, A072197, A081252, A081253, A266219 (binary).

[Floor(2^(n-1)*5/3): n in [1..40]]; // Vincenzo Librandi, Apr 04 2012

seq(floor(2^(n-1)*5/3),n=1..35); # Muniru A Asiru, Sep 20 2018

Rest@CoefficientList[Series[-(x^2 - x - 1)*x/((x - 1)*(x + 1)*(2*x - 1)), {x, 0, 32}], x] (* Vincenzo Librandi, Apr 04 2012 *)
a[n_]:=Floor[2^(n-1)*5/3]; Array[a,33,1] (* Stefano Spezia, Sep 01 2018 *)

a(n) = 2^(n-1)*5\3; \\ Altug Alkan, Sep 21 2018

a(n) = floor(2^(n-1)*5/3). [corrected by Michel Marcus, Sep 21 2018]
a(n) = a(n-2) + 5*2^(n-3) for n > 2;
a(n+2) - a(n) = A020714(n-1);
a(n) + a(n-1) = A052549(n-1) for n > 1;
a(2*n+1) = A020989(n); a(2n) = A072197(n-1);
a(n+1) - a(n) = A048573(n-1).
G.f.: -(x^2 - x - 1)*x/((x - 1)*(x + 1)*(2*x - 1)).
a(n) = 5*2^(n-1)/3 + (-1)^n/6-1/2. a(n) = 2*a(n-1) + (1+(-1)^n)/2, a(1)=1. - Paul Barry, Mar 24 2003
a(2n) = 2*a(2*n-1) + 1, a(2*n+1) = 2*a(2*n), a(1)=1. a(n) = A000975(n-1) + 2^(n-1). - Philippe Deléham, Oct 15 2006
a(n) = A005578(n) + A000225(n-1). - Yuchun Ji, Sep 21 2018
a(n) - a(n-2) = 2 * (a(n-1) - a(n-3)), with a(0..2)=[1,3,6]. - Yuchun Ji, Mar 18 2020

A153894 a(n) = 5*2^n - 1.

Original entry on oeis.org

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

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

a(n) is the total number of symbols required in the fully-expanded von Neumann definition of ordinal n + 1, where the string "{}" is used to represent the empty set and spaces are ignored. - Ely Golden, Nov 14 2019

a(n) converted to binary is 100 followed by n ones. - Alexandre Herrera, Oct 06 2023

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Bernard Monjardet, Acyclic domains of linear orders: a survey, in "The Mathematics of Preference, Choice and Order: Essays in Honor of Peter Fishburn", edited by Steven Brams, William V. Gehrlein and Fred S. Roberts, Springer, 2009, pp. 139-160. This version: . - _N. J. A. Sloane_, Feb 07 2009
Gennady Eremin, Partitioning the set of natural numbers into Mersenne trees and into arithmetic progressions; Natural Matrix and Linnik's constant, arXiv:2405.16143 [math.CO], 2024. See pp. 3-5, 14.
Gennady Eremin, Infinite matrix of odd natural numbers. A bit about Sophie Germain prime numbers, arXiv:2501.17090 [math.GM], 2025. See pp. 3, 11.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A052549, A309779.

[5*2^n-1: n in [0..30]]; // Vincenzo Librandi, Oct 28 2011

a=4;lst={a};Do[a=a*2+1;AppendTo[lst,a],{n,5!}];lst
LinearRecurrence[{3,-2},{4,9}, 25] (* or *) Table[5*2^n - 1, {n,0,25}] (* G. C. Greubel, Sep 01 2016 *)

a(n)=5*2^n-1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 2*a(n-1) + 1, n>0.
a(n) = A052549(n+1).
G.f.: (4 - 3*x) / ( (2*x-1)*(x-1) ). - R. J. Mathar, Oct 22 2011
a(n) + a(n-1)^2 = A309779(n), a perfect square. - Vincenzo Librandi, Oct 28 2011
From G. C. Greubel, Sep 01 2016: (Start)
a(n) = 3*a(n-1) - 2*a(n-2).
E.g.f.: 5*exp(2*x) - exp(x). (End)

Edited by N. J. A. Sloane, Feb 07 2009

Definition corrected by Franklin T. Adams-Watters, Apr 22 2009

A250656 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with nondecreasing x(i,j)-x(i,j-1) in the i direction and nondecreasing min(x(i,j),x(i-1,j)) in the j direction.

Original entry on oeis.org

9, 16, 19, 25, 34, 39, 36, 53, 70, 79, 49, 76, 109, 142, 159, 64, 103, 156, 221, 286, 319, 81, 134, 211, 316, 445, 574, 639, 100, 169, 274, 427, 636, 893, 1150, 1279, 121, 208, 345, 554, 859, 1276, 1789, 2302, 2559, 144, 251, 424, 697, 1114, 1723, 2556, 3581

Offset: 1

R. H. Hardin, Nov 26 2014

Table starts
....9...16....25....36....49....64....81...100...121...144...169....196....225
...19...34....53....76...103...134...169...208...251...298...349....404....463
...39...70...109...156...211...274...345...424...511...606...709....820....939
...79..142...221...316...427...554...697...856..1031..1222..1429...1652...1891
..159..286...445...636...859..1114..1401..1720..2071..2454..2869...3316...3795
..319..574...893..1276..1723..2234..2809..3448..4151..4918..5749...6644...7603
..639.1150..1789..2556..3451..4474..5625..6904..8311..9846.11509..13300..15219
.1279.2302..3581..5116..6907..8954.11257.13816.16631.19702.23029..26612..30451
.2559.4606..7165.10236.13819.17914.22521.27640.33271.39414.46069..53236..60915
.5119.9214.14333.20476.27643.35834.45049.55288.66551.78838.92149.106484.121843

			Some solutions for n=4 k=4
..1..1..0..1..1....0..0..0..0..0....0..0..0..0..0....1..1..1..0..0
..0..0..0..1..1....1..1..1..1..1....1..1..1..1..1....0..0..0..0..0
..0..0..0..1..1....1..1..1..1..1....0..0..0..0..0....0..0..0..0..0
..0..0..0..1..1....0..0..0..0..0....1..1..1..1..1....1..1..1..1..1
..0..0..0..1..1....0..1..1..1..1....1..1..1..1..1....0..0..0..1..1

R. H. Hardin, Table of n, a(n) for n = 1..880

Column 1 is A052549(n+1)
Column 2 is A176449
Column 3 is A156127(n+1)
Column 4 is A048487(n+2)
Row 1 is A000290(n+2)
Row 2 is A168244(n+3)

Empirical: T(n,k) = 2^(n-1)*k^2 + (5*2^(n-1)-1)*k + 2^(n+1)
Empirical for column k:
k=1: a(n) = 3*a(n-1) -2*a(n-2); also a(n) = 2^(n-1) +(5*2^(n-1) -1) +2^(n+1)
k=2: a(n) = 3*a(n-1) -2*a(n-2); also a(n) = 2^(n-1)*4 +(5*2^(n-1) -1)*2 +2^(n+1)
k=3: a(n) = 3*a(n-1) -2*a(n-2); also a(n) = 2^(n-1)*9 +(5*2^(n-1) -1)*3 +2^(n+1)
k=4: a(n) = 3*a(n-1) -2*a(n-2); also a(n) = 2^(n-1)*16 +(5*2^(n-1) -1)*4 +2^(n+1)
k=5: a(n) = 3*a(n-1) -2*a(n-2); also a(n) = 2^(n-1)*25 +(5*2^(n-1) -1)*5 +2^(n+1)
k=6: a(n) = 3*a(n-1) -2*a(n-2); also a(n) = 2^(n-1)*36 +(5*2^(n-1) -1)*6 +2^(n+1)
k=7: a(n) = 3*a(n-1) -2*a(n-2); also a(n) = 2^(n-1)*49 +(5*2^(n-1) -1)*7 +2^(n+1)
Empirical for row n:
n=1: a(n) = 1*n^2 + 4*n + 4
n=2: a(n) = 2*n^2 + 9*n + 8
n=3: a(n) = 4*n^2 + 19*n + 16
n=4: a(n) = 8*n^2 + 39*n + 32
n=5: a(n) = 16*n^2 + 79*n + 64
n=6: a(n) = 32*n^2 + 159*n + 128
n=7: a(n) = 64*n^2 + 319*n + 256

A054135 a(n) = T(n,1), array T as in A054134.

Original entry on oeis.org

2, 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

Offset: 1

Clark Kimberling

nonn

From Jianing Song, May 25 2025: (Start)
As Ely Golden noted in A153894, a(n) is the total number of symbols required in the fully-expanded von Neumann definition of ordinal n - 1, where the string "{}" is used to represent the empty set and spaces are ignored. First examples:
  0 = {};
  1 = {0} = {{}};
  2 = {0,1} = {{},{{}}};
  3 = {0,1,2} = {{},{{}},{{},{{}}}};
  4 = {0,1,2,3} = {{},{{}},{{},{{}}},{{},{{}},{{},{{}}}}}.
(End)

Index entries for linear recurrences with constant coefficients, signature (3,-2).

Identical to A052549 and A153894 except for initial term.

Cf. A267524.

print([2]+[(5*2**(n-2) - 1) for n in range(2, 50)]) # Karl V. Keller, Jr., Jun 12 2022

For n > 2, a(n) = 10*A000225(n-3) + 9 = 10*(2^(n-3) - 1) + 9 = 10*2^(n-3) - 1. - Gerald McGarvey, Aug 25 2004
a(1)=1, a(n) = n + Sum_{i=1..n-1} a(i) for n > 1. - Gerald McGarvey, Sep 06 2004
a(n) = 5*2^(n-2) - 1 for n > 1. - Karl V. Keller, Jr., Jun 12 2022

A133601 A007318 * (A097806 + A133080 - I), I = Identity matrix.

Original entry on oeis.org

1, 3, 1, 5, 3, 1, 7, 6, 5, 1, 9, 10, 14, 5, 1, 11, 15, 30, 15, 7, 1, 13, 21, 55, 35, 27, 7, 1, 15, 28, 91, 70, 77, 28, 9, 1, 17, 36, 140, 126, 182, 84, 44, 9, 1, 19, 45, 204, 210, 378, 210, 156, 45, 11, 1

Offset: 0

Gary W. Adamson, Sep 18 2007

			First few rows of the triangle are:
1;
3, 1;
5, 3, 1;
7, 6, 5, 1;
9, 10, 14, 5, 1;
11, 15, 30, 15, 7, 1;
13, 21, 55, 35, 27, 7, 1;
15, 28, 91, 70, 77, 28, 9, 1;
...

Cf. A097806, A133080, A052549 (row sums).

A133601aux := proc(n,k)
    if n <> k then
        A097806(n,k)+A133080(n,k) ;
    else
        A097806(n,k)+A133080(n,k)-1 ;
    end if;
end proc:
A133601 := proc(n,k)
    add( A007318(n,j)*A133601aux(j+1,k+1),j=k..n) ;
end proc: # R. J. Mathar, Jun 20 2015

A007318 * (A097806 + A133080 - I), I = Identity matrix. Binomial transform of an infinite lower triangular matrix with (1,1,1,...) in the main diagonal and (2,1,2,1,2,...) in the subdiagonal; and the rest zeros.

A052617 E.g.f. (1+x-x^2)/((1-x)(1-2x)).

Original entry on oeis.org

1, 4, 18, 114, 936, 9480, 114480, 1607760, 25764480, 464123520, 9286099200, 204334099200, 4904497382400, 127523158963200, 3570735629260800, 107123376552192000, 3427968972460032000, 116551300751069184000

Offset: 0

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 562

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

With[{nn=20},CoefficientList[Series[(1+x-x^2)/((1-x)(1-2x)),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 25 2018 *)

E.g.f.: -(-x+x^2-1)/(-1+2*x)/(-1+x)
Recurrence: {a(0)=1, a(1)=4, (2*n^2+6*n+4)*a(n) +(-6-3*n)*a(n+1) +a(n+2)=0, a(2)=18}
(-1+5*2^(n-1))*n!, n>0.
a(n)=n!*A052549(n). - R. J. Mathar, Jun 03 2022

A140183 Triangle read by rows, binomial transform of an infinite lower triangular matrix with (1,2,1,2,1,2,...) in the main diagonal, (1,1,1,...) in the subdiagonal and the rest zeros.

Original entry on oeis.org

1, 2, 2, 3, 5, 1, 4, 9, 4, 2, 5, 14, 10, 9, 1, 6, 20, 20, 25, 6, 2, 7, 27, 35, 55, 21, 13, 1, 8, 35, 56, 105, 56, 49, 8, 2, 9, 44, 84, 182, 126, 140, 36, 17, 1, 10, 54, 120, 294, 252, 336, 120, 81, 10, 2, 1, 65, 165, 450, 462, 714, 330, 285, 55, 21, 1

Offset: 0

Gary W. Adamson, May 11 2008

Row sums = A052549: (1, 4, 9, 19, 39, 79,...).

			First few rows of the triangle are:
1;
2, 2;
3, 5, 1;
4, 9, 4, 2;
5, 14, 10, 9, 1;
6, 20, 20, 25, 6, 2;
7, 27, 35, 55, 21, 13, 1;
...

Cf. A052549.

Triangle read by rows, A007318 as an infinite lower triangular matrix * a bidiagonal matrix with (1,2,1,2,1,2,...) in the main diagonal and (1,1,1,...) in the subdiagonal.

A141499 a(0)=0; a(1)=1; a(n) = triangular number at index 5*2^(n-2)-1.

Original entry on oeis.org

0, 1, 10, 45, 190, 780, 3160, 12720, 51040, 204480, 818560, 3275520, 13104640, 52423680, 209704960, 838840320, 3355402240, 13421690880, 53686927360, 214748037120, 858992803840, 3435972526080, 13743892725760, 54975576145920, 219902315069440, 879609281249280

Offset: 0

Roger L. Bagula, Aug 10 2008

The sequence a(n)=b(n)*(b(n)-1)/2 gives an SO(2),SO(5),SO(10),SO(20), ...

Index entries for linear recurrences with constant coefficients, signature (6,-8).

Cf. A084215.

Clear[a] a[0] = 1; a[1] = 2; a[2] = 5; a[n_] := a[n] = a[1]*a[n - 1]; Table[a[n]*(a[n] - 1)/2, {n, 0, 20}]
Join[{0,1},LinearRecurrence[{6,-8},{10,45},30]] (* Harvey P. Dale, May 23 2013 *)

a(0)=0. a(n)=A000217(A052549(n-1)), n>0. - R. J. Mathar, Oct 29 2008

a(n)=5*2^(-5+n)*(-4+5*2^n) for n>1. a(n)=6*a(n-1)-8*a(n-2) for n>3. G.f.: x*(1+4*x-7*x^2)/((1-2*x)*(1-4*x)). [Colin Barker, Aug 16 2012]

Edited by N. J. A. Sloane, Aug 16 2008
Corrected the definition, which was describing an auxiliary sequence. - R. J. Mathar, Oct 29 2008
More terms from Harvey P. Dale, May 23 2013

cp's OEIS Frontend

A081254 Numbers k such that A081252(m)/m^2 has a local maximum for m = k.

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A153894 a(n) = 5*2^n - 1.

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A250656 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with nondecreasing x(i,j)-x(i,j-1) in the i direction and nondecreasing min(x(i,j),x(i-1,j)) in the j direction.

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A054135 a(n) = T(n,1), array T as in A054134.

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A133601 A007318 * (A097806 + A133080 - I), I = Identity matrix.

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A052617 E.g.f. (1+x-x^2)/((1-x)(1-2x)).

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A140183 Triangle read by rows, binomial transform of an infinite lower triangular matrix with (1,2,1,2,1,2,...) in the main diagonal, (1,1,1,...) in the subdiagonal and the rest zeros.

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A141499 a(0)=0; a(1)=1; a(n) = triangular number at index 5*2^(n-2)-1.

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