A299998 -id:A299998 - cp's OEIS frontend

A006125 a(n) = 2^(n*(n-1)/2).

1, 1, 2, 8, 64, 1024, 32768, 2097152, 268435456, 68719476736, 35184372088832, 36028797018963968, 73786976294838206464, 302231454903657293676544, 2475880078570760549798248448, 40564819207303340847894502572032, 1329227995784915872903807060280344576

Offset: 0

N. J. A. Sloane

Number of graphs on n labeled nodes; also number of outcomes of labeled n-team round-robin tournaments.
Number of perfect matchings of order n Aztec diamond. [see Speyer]
Number of Gelfand-Zeitlin patterns with bottom row [1,2,3,...,n]. [Zeilberger]
For n >= 1 a(n) is the size of the Sylow 2-subgroup of the Chevalley group A_n(2) (sequence A002884). - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 30 2001
From James Propp: (Start)
a(n) is the number of ways to tile the region
           o-----o
           |.....|
        o--o.....o--o
        |...........|
     o--o...........o--o
     |.................|
  o--o.................o--o
  |.......................|
  |.......................|
  |.......................|
  o--o.................o--o
     |.................|
     o--o...........o--o
        |...........|
        o--o.....o--o
           |.....|
           o-----o
(top-to-bottom distance = 2n) with dominoes like either of
  o--o      o-----o
  |..|  or  |.....|
  |..|      o-----o
  |..|
  o--o
(End)
The number of domino tilings in A006253, A004003, A006125 is the number of perfect matchings in the relevant graphs. There are results of Jockusch and Ciucu that if a planar graph has a rotational symmetry then the number of perfect matchings is a square or twice a square - this applies to these 3 sequences. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 12 2001
Let M_n denotes the n X n matrix with M_n(i,j)=binomial(2i,j); then det(M_n)=a(n+1). - Benoit Cloitre, Apr 21 2002
Smallest power of 2 which can be expressed as the product of n distinct numbers (powers of 2), e.g., a(4) = 1024 = 2*4*8*16. Also smallest number which can be expressed as the product of n distinct powers. - Amarnath Murthy, Nov 10 2002
The number of binary relations that are both reflexive and symmetric on an n-element set. - Justin Witt (justinmwitt(AT)gmail.com), Jul 12 2005
The number of symmetric binary relations on an (n-1)-element set. - Peter Kagey, Feb 13 2021
To win a game, you must flip n+1 heads in a row, where n is the total number of tails flipped so far. Then the probability of winning for the first time after n tails is A005329 / A006125. The probability of having won before n+1 tails is A114604 / A006125. - Joshua Zucker, Dec 14 2005
a(n) = A126883(n-1)+1. - Zerinvary Lajos, Jun 12 2007
Equals right border of triangle A158474 (unsigned). - Gary W. Adamson, Mar 20 2009
a(n-1) is the number of simple labeled graphs on n nodes such that every node has even degree. - Geoffrey Critzer, Oct 21 2011
a(n+1) is the number of symmetric binary matrices of size n X n. - Nathan J. Russell, Aug 30 2014
Let T_n be the n X n matrix with T_n(i,j) = binomial(2i + j - 3, j-1); then det(T_n) = a(n). - Tony Foster III, Aug 30 2018
k^(n*(n-1)/2) is the determinant of n X n matrix T_(i,j) = binomial(k*i + j - 3, j-1), in this case k=2. - Tony Foster III, May 12 2019
Let B_n be the n+1 X n+1 matrix with B_n(i, j) = Sum_{m=max(0, j-i)..min(j, n-i)} (binomial(i, j-m) * binomial(n-i, m) * (-1)^m), 0<=i,j<=n. Then det B_n = a(n+1). Also, deleting the first row and any column from B_n results in a matrix with determinant a(n). The matrices B_n have the following property: B_n * [x^n, x^(n-1) * y, x^(n-2) * y^2, ..., y^n]^T = [(x-y)^n, (x-y)^(n-1) * (x+y), (x-y)^(n-2) * (x+y)^2, ..., (x+y)^n]^T. - Nicolas Nagel, Jul 02 2019
a(n) is the number of positive definite (-1,1)-matrices of size n X n. - Eric W. Weisstein, Jan 03 2021
a(n) is the number of binary relations on a labeled n-set that are both total and antisymmetric. - José E. Solsona, Feb 05 2023

			From _Gus Wiseman_, Feb 11 2021: (Start)
This sequence counts labeled graphs on n vertices. For example, the a(0) = 1 through a(2) = 8 graph edge sets are:
  {}  {}  {}    {}
          {12}  {12}
                {13}
                {23}
                {12,13}
                {12,23}
                {13,23}
                {12,13,23}
This sequence also counts labeled graphs with loops on n - 1 vertices. For example, the a(1) = 1 through a(3) = 8 edge sets are the following. A loop is represented as an edge with two equal vertices.
  {}  {}    {}
      {11}  {11}
            {12}
            {22}
            {11,12}
            {11,22}
            {12,22}
            {11,12,22}
(End)

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 547 (Fig. 9.7), 573.
G. Everest, A. van der Poorten, I. Shparlinski, and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; p. 178.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 517.
F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 178.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 3, Eq. (1.1.2).
J. Propp, Enumeration of matchings: problems and progress, in: New perspectives in geometric combinatorics, L. Billera et al., eds., Mathematical Sciences Research Institute series, vol. 38, Cambridge University Press, 1999.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 0..50
F. Ardila and R. P. Stanley, Tilings, arXiv:math/0501170 [math.CO], 2005.
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. - From _N. J. A. Sloane_, Oct 08 2012
Anders Björner and Richard P. Stanley, A combinatorial miscellany, 2010.
Tobias Boege and Thomas Kahle, Construction Methods for Gaussoids, arXiv:1902.11260 [math.CO], 2019.
Taylor Brysiewicz and Fulvio Gesmundo, The Degree of Stiefel Manifolds, arXiv:1909.10085 [math.AG], 2019.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Mihai Ciucu, Perfect matchings of cellular graphs, J. Algebraic Combin., 5 (1996) 87-103.
Mihai Ciucu, Enumeration of perfect matchings in graphs with reflective symmetry, J. Combin. Theory Ser. A 77 (1997), no. 1, 67-97.
Thierry de la Rue and Élise Janvresse, Pavages aléatoires par touillage de dominos, Images des Mathématiques, CNRS, 2023. In French.
Noam Elkies, Greg Kuperberg, Michael Larsen, and James Propp, Alternating sign matrices and domino tilings. Part I, Journal of Algebraic Combinatorics 1-2, 111-132 (1992).
Noam Elkies, Greg Kuperberg, Michael Larsen, and James Propp, Alternating sign matrices and domino tilings. Part II, Journal of Algebraic Combinatorics 1-3, 219-234 (1992).
Sen-Peng Eu and Tung-Shan Fu, A simple proof of the Aztec diamond theorem, arXiv:math/0412041 [math.CO], 2004.
D. Grensing, I. Carlsen, and H.-Chr. Zapp, Some exact results for the dimer problem on plane lattices with non-standard boundaries, Phil. Mag. A, 41:5 (1980), 777-781.
Harald Helfgott and Ira M. Gessel, Enumeration of tilings of diamonds and hexagons with defects, arXiv:math/9810143 [math.CO], 1998.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
Pakawut Jiradilok, Some Combinatorial Formulas Related to Diagonal Ramsey Numbers, arXiv:2404.02714 [math.CO], 2024. See p. 19.
William Jockusch, Perfect matchings and perfect squares J. Combin. Theory Ser. A 67 (1994), no. 1, 100-115.
Eric H. Kuo, Applications of graphical condensation for enumerating matchings and tilings, Theoretical Computer Science, Vol. 319, No. 1-3 (2004), pp. 29-57, arXiv preprint, arXiv:math/0304090 [math.CO], 2003.
C. L. Mallows & N. J. A. Sloane, Emails, May 1991
W. H. Mills, David P. Robbins, and Howard Rumsey, Jr., Alternating sign matrices and descending plane partitions, J. Combin. Theory Ser. A 34 (1983), 340-359.
Götz Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
James Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics
James Propp, Enumeration of matchings: problems and progress, arXiv:math/9904150 [math.CO], 1999.
James Propp, Lessons I learned from Richard Stanley, arXiv preprint [math.CO], 2015.
James Propp and R. P. Stanley, Domino tilings with barriers, arXiv:math/9801067 [math.CO], 1998.
Steven S. Skiena, Generating graphs.
David E. Speyer, Perfect matchings and the octahedron recurrence, Journal of Algebraic Combinatorics, Vol. 25, No. 3 (2007), pp. 309-348, arXiv preprint, arXiv:math/0402452 [math.CO], 2004.
Jan Tóth and Ondřej Kuželka, Complexity of Weighted First-Order Model Counting in the Two-Variable Fragment with Counting Quantifiers: A Bound to Beat, arXiv:2404.12905 [cs.LO], 2024. See p. 24.
Herman Tulleken, Polyominoes 2.2: How they fit together, (2019).
Eric Weisstein's World of Mathematics, Connected Graph.
Eric Weisstein's World of Mathematics, Labeled Graph.
Eric Weisstein's World of Mathematics, Symmetric Matrix.
Doron Zeilberger, Dave Robbins's Art of Guessing, Adv. in Appl. Math. 34 (2005), 939-954.
Index entries for sequences related to dominoes

Cf. A000568 for the unlabeled analog, A053763, A006253, A004003.
Cf. A001187 (connected labeled graphs).
Cf. A095340, A103904, A005329, A114604, A299998.
Cf. A158474. - Gary W. Adamson, Mar 20 2009
Cf. A136652 (log). - Paul D. Hanna, Dec 04 2009
The unlabeled version is A000088, or A002494 without isolated vertices.
The directed version is A002416.
The covering case is A006129.
The version for hypergraphs is A058891, or A016031 without singletons.
Row sums of A143543.
The case of connected edge set is A287689.
Cf. A000169, A000295, A036442, A059167, A100743, A136284, A327078.

[2^(n*(n-1) `div` 2) | n <- [0..20]] -- José E. Solsona, Feb 05 2023

[2^(n*(n-1) div 2): n in [0..20]]; // Vincenzo Librandi, Jul 03 2019

Join[{1}, 2^Accumulate[Range[0, 20]]] (* Harvey P. Dale, May 09 2013 *)
Table[2^(n (n - 1) / 2), {n, 0, 20}] (* Vincenzo Librandi, Jul 03 2019 *)
Table[2^Binomial[n, 2], {n, 0, 15}] (* Eric W. Weisstein, Jan 03 2021 *)
2^Binomial[Range[0, 15], 2] (* Eric W. Weisstein, Jan 03 2021 *)
Prepend[Table[Count[Tuples[{0, 1}, {n, n}], ?SymmetricMatrixQ], {n, 5}], 1] (* _Eric W. Weisstein, Jan 03 2021 *)
Prepend[Table[Count[Tuples[{-1, 1}, {n, n}], ?PositiveDefiniteMatrixQ], 1], {n, 4}] (* _Eric W. Weisstein, Jan 03 2021 *)

A006125(n):=2^(n*(n-1)/2)$ makelist(A006125(n), n, 0, 30); /* Martin Ettl, Oct 24 2012 */

a(n)=1<Charles R Greathouse IV, Jun 15 2011

def A006125(n): return 1<<(n*(n-1)>>1) # Chai Wah Wu, Nov 09 2023

Sequence is given by the Hankel transform of A001003 (Schroeder's numbers) = 1, 1, 3, 11, 45, 197, 903, ...; example: det([1, 1, 3, 11; 1, 3, 11, 45; 3, 11, 45, 197; 11, 45, 197, 903]) = 2^6 = 64. - Philippe Deléham, Mar 02 2004
a(n) = 2^floor(n^2/2)/2^floor(n/2). - Paul Barry, Oct 04 2004
G.f. satisfies: A(x) = 1 + x*A(2x). - Paul D. Hanna, Dec 04 2009
a(n) = 2 * a(n-1)^2 / a(n-2). - Michael Somos, Dec 30 2012
G.f.: G(0)/x - 1/x, where G(k) = 1 + 2^(k-1)*x/(1 - 1/(1 + 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 26 2013
E.g.f. satisfies A'(x) = A(2x). - Geoffrey Critzer, Sep 07 2013
Sum_{n>=1} 1/a(n) = A299998. - Amiram Eldar, Oct 27 2020
a(n) = s_lambda(1,1,...,1) where s is the Schur polynomial in n variables and lambda is the partition (n,n-1,n-2,...,1). - Leonid Bedratyuk, Feb 06 2022
a(n) = Product_{1 <= j <= i <= n-1}  (i + j)/(2*i - 2*j + 1). Cf. A007685.  - Peter Bala, Oct 25 2024

More terms from Vladeta Jovovic, Apr 09 2000

A190404 Decimal expansion of (1/2)(1 + Sum_{k>=1} (1/2)^T(k)), where T=A000217 (triangular numbers); based on row 1 of the natural number array, A000027.

Original entry on oeis.org

8, 2, 0, 8, 1, 6, 2, 8, 0, 3, 2, 7, 5, 7, 6, 9, 3, 3, 1, 4, 6, 9, 2, 1, 3, 8, 5, 1, 1, 2, 7, 1, 4, 7, 1, 7, 1, 1, 3, 0, 3, 0, 7, 6, 8, 9, 7, 8, 3, 6, 9, 8, 7, 3, 9, 0, 2, 3, 2, 5, 8, 1, 1, 1, 9, 0, 0, 7, 2, 3, 0, 1, 8, 6, 6, 6, 7, 5, 8, 8, 7, 8, 0, 0, 1, 8, 2, 0, 8, 5, 8, 1, 1, 6, 7, 9, 5, 6, 6, 5, 4, 3, 0, 4, 4, 8, 6, 7, 6, 5, 8, 1, 7, 1, 8, 0, 9, 7, 3, 0

Offset: 0

Clark Kimberling, May 10 2011

Suppose that F={f(i,j): i>=1, j>=1} is an array of positive integers such that every positive integer occurs exactly once in F.
Let G=G(F) denote the array defined by g(i,j)=(1/2)^f(i,j);
R(i)=Sum_{j>=1} g(i,j); i-th row sum of G;
C(j)=Sum_{i>=1} g(i,j); j-th column sum of G;
U(j)=Sum_{i>=1} g(i,i+j-1); j-th upper diagonal sum of G;
L(i)=Sum_{j>=1} g(i+j,j); i-th lower diagonal sum of G;
R(odds)=Sum_{i>=1} R(2i-1); sum, odd numbered rows of G;
R(evens)=Sum_{i>=1} R(2i); sum, even numbered rows of G;
C(odds)=Sum_{j>=1} R(2j-1); sum, odd numbered cols of G;
C(evens)=Sum_{j>=1} R(2j); sum, even numbered cols of G;
UT=Sum_{j>=1} U(j); sum, upper triangular subarray of G;
LT=Sum_{i>=1} L(i); sum, lower triangular subarray of G.
...
Note that R(odds)+R(evens)=C(odds)+C(evens)=UT+LT=1.
...
For the natural number array F=A000027:
R(1)=0.820816280327576933146921385113... (A190404)
R(2)=0.160408140163788466573460692556...
R(3)=0.0177040700818942332867303462782...
R(4)=0.00103953504094711664336517313909...
R(5)=0.0000314862704735583216825865695447...
...
R(odds)=0.838551840434481240061632331355800... (A190408)
R(evens)=0.161448159565518759938367668644199...(A190409)
...
C(1)=0.64163256065515386629... (A190405)
C(2)=0.28326512131030773259...
C(3)=0.066530242620615465175...
C(4)=0.0080604852412309303507...
C(5)=0.00049597048246186070148...
...
C(odds)=0.7086590131172367153696485920526...(A190410)
C(evens)=0.29134098688276328463035140794... (A190411)
...
D(1)=0.53137210011527713548... (A190406)
D(2)=0.25391006493009715683...
D(3)=0.062744200230554270960...
D(4)=0.0078201298601943136650...
D(5)=0.00048840046110854191952...
...
E(1)=0.12695503246504857842... (A190407)
E(2)=0.015686050057638567740...
E(3)=0.00097751623252428920813...
E(4)=0.000030525028819283869970...
E(5)=0.00000047686626214460406264...
...
UT=0.8563503956097795739814618239914245448... (A190412)
LT=0.1436496043902204260185381760085754551... (A190415)

			0.820816280327576933146921385113...

Danny Rorabaugh, Table of n, a(n) for n = 0..10000

Cf. A190405-A190412, A190415, A299998.

f[i_, j_] := i + (j + i - 2)(j + i - 1)/2;
TableForm[Table[f[i,j],{i,1,10},{j,1,10}]] (* A000027 *)
r[i_] := Sum[2^-f[i, j], {j,1,400}];    (* C(row i) *)
c[j_] := Sum[2^-f[i,j], {i,1,400}];     (* C(col j) *)
d[h_] := Sum[2^-f[i,i+h-1], {i,1,200}]; (* C(udiag h) *)
e[h_] := Sum[2^-f[i+h,i], {i,1,200}];   (* C(ldiag h) *)
RealDigits[r[1], 10, 120, -1]  (* A190404 *)
N[r[1], 30]
N[r[2], 30]
N[r[3], 30]
N[r[4], 30]
N[r[5], 30]
N[r[6], 30]
RealDigits[c[1], 10, 120, -1] (* A190405 *)
N[c[1], 20]
N[c[2], 20]
N[c[3], 20]
N[c[4], 20]
N[c[5], 20]
N[c[6], 20]
RealDigits[d[1], 10, 20, -1] (* A190406 *)
N[d[1], 20]
N[d[2], 20]
N[d[3], 20]
N[d[4], 20]
N[d[5], 20]
N[d[6], 20]
RealDigits[e[1], 10, 20, -1] (* A190407 *)
N[e[1], 20]
N[e[2], 20]
N[e[3], 20]
N[e[4], 20]
N[e[5], 20]
N[e[6], 20]

def A190404(b):  # Generate the constant with b bits of precision
    return N(sum([(1/2)^(1+j*(j+1)/2) for j in range(1,b)])+1/2,b)
A190404(409) # Danny Rorabaugh, Mar 25 2015

A190404: (1/2)(1 + Sum_{k>=1} (1/2)^T(k)), where T=A000217 (triangular numbers).
A190405: Sum_{k>=1} (1/2)^T(k), where T=A000217.
A190406: Sum_{k>=1} (1/2)^S(k-1), where S=A001844 (centered square numbers).
A190407: Sum_{k>=1} (1/2)^V(k), where V=A058331 (1+2*k^2).
Equals Product_{k>=1} 1 - 1/(2^(2*k + 1) - 1). - Antonio Graciá Llorente, Oct 01 2024
Equals A299998/2. - Hugo Pfoertner, Oct 01 2024

A073587 a(n) = a(n-1)*2^n + 1 where a(0)=1.

Original entry on oeis.org

1, 3, 13, 105, 1681, 53793, 3442753, 440672385, 112812130561, 57759810847233, 59146046307566593, 121131102837896382465, 496152997224023582576641, 4064485353259201188467843073, 66592528027798752271857140908033, 2182103958414909514444214793274425345

Offset: 0

Felice Russo, Aug 28 2002

Also, number of nodes in an n-ary tree with increasing fanout as the level increases. - Dhruv Matani, Apr 12 2012

Cf. A000225 (nodes in a binary tree).

Cf. A006125, A117261, A299998.

a = 1; Table[a = a*2^n + 1, {n, 14}] (* Jayanta Basu, Jul 02 2013 *)

25 A=1
30 for I=1 to 20
40 A=A*2^I+1
50 print A
60 next
70 end

a(n) = floor(D*2^(n*(n+1)/2)) where D is a constant, D=1.64163256065515386629... = Sum_{k>=0} 1/2^(k(k+1)/2) = A299998. - Benoit Cloitre, Sep 01 2002

Added a(0)=1. Added information from duplicate sequence A182104. - N. J. A. Sloane, Dec 28 2015

A319015 Decimal expansion of Sum_{k>=0} 1/2^(k^2).

Original entry on oeis.org

1, 5, 6, 4, 4, 6, 8, 4, 1, 3, 6, 0, 5, 9, 3, 8, 5, 7, 9, 3, 3, 4, 7, 2, 9, 2, 7, 4, 2, 7, 2, 4, 7, 5, 6, 6, 2, 3, 0, 6, 2, 5, 8, 2, 6, 9, 9, 7, 0, 4, 3, 9, 0, 4, 6, 4, 4, 4, 5, 0, 5, 5, 9, 6, 0, 2, 8, 4, 8, 0, 1, 3, 3, 1, 7, 9, 5, 7, 8, 4, 0, 6, 6, 5, 9, 1, 3, 0, 6, 4, 0, 1, 6, 2, 4, 6, 9, 1, 4, 8, 4, 4, 7, 4, 0, 2, 4, 7, 1, 6

Offset: 1

Ilya Gutkovskiy, Sep 07 2018

The binary expansion is the characteristic function of the squares (A010052).

This constant is transcendental (Nesterenko, 1996). - Amiram Eldar, Apr 30 2020

			1.5644684136059385793347... = (1.1001000010000001000000001...)_2.
                               | |  |    |      |        |
                               0 1  4    9     16       25

David H. Bailey, Jonathan M. Borwein, Richard E. Crandall, and Carl Pomerance, On the binary expansions of algebraic numbers, Journal de théorie des nombres de Bordeaux, Vol. 16, No. 3 (2004), pp. 487-518. See p. 490.
Yu. V. Nesterenko, Modular functions and transcendence questions (in Russian), Sbornik: Mathematics, Vol. 187 (1996), pp. 65-96, English translation, ibid., pp. 1319-1348.
Michael Ian Shamos, Property Enumerators and a Partial Sum Theorem, 2011; alternative link.
Index entries for transcendental numbers.

Cf. A000290, A002416, A008836, A010052, A190405, A299998.

RealDigits[(1 + EllipticTheta[3, 0, 1/2])/2, 10, 110] [[1]]

suminf(k=0, 1/2^(k^2)) \\ Michel Marcus, Sep 08 2018

Equals (1 + theta_3(1/2))/2, where theta_3 is the Jacobi theta function.
Equals 1 + Sum_{k>=1} lambda(k)/(2^k - 1), where lambda is the Liouville function (A008836). - Amiram Eldar, Apr 30 2020
Equals 1 + Sum_{k>=1} floor(sqrt(k))/2^(k+1) (Shamos, 2011, p. 4). - Amiram Eldar, Mar 12 2024

A319016 Decimal expansion of Sum_{k>=0} 1/2^(k*(k+1)).

Original entry on oeis.org

1, 2, 6, 5, 8, 7, 0, 0, 9, 5, 2, 3, 0, 8, 6, 6, 3, 6, 8, 4, 1, 8, 9, 2, 1, 3, 1, 4, 5, 4, 3, 5, 4, 3, 4, 2, 7, 4, 6, 4, 2, 6, 5, 4, 4, 6, 3, 9, 9, 6, 3, 8, 7, 1, 6, 8, 2, 0, 0, 5, 3, 3, 4, 1, 8, 1, 4, 8, 9, 3, 4, 9, 2, 5, 1, 1, 2, 7, 4, 8, 9, 4, 4, 3, 7, 0, 6, 4, 5, 9, 7, 4, 8, 3, 5, 3, 0, 5, 6, 7, 3, 9, 0, 8, 4, 2, 7, 1, 1, 4

Offset: 1

Ilya Gutkovskiy, Sep 07 2018

The binary expansion is the characteristic function of the oblong numbers (A005369).

The Engel expansion of this constant are the powers of 4 (A000302). - Amiram Eldar, Dec 07 2020

			1.2658700952308663684189... = (1.010001000001000000010000000001...)_2.
                               |  |   |     |       |         |
                               0  2   6    12      20        30

Cf. A000302, A002378, A005369, A053763, A190405, A299998, A319015.

RealDigits[EllipticTheta[2, 0, 1/2]/2^(3/4), 10, 110] [[1]]

suminf(k=0, 1/2^(k*(k+1))) \\ Michel Marcus, Sep 08 2018

Equals theta_2(1/2)/2^(3/4), where theta_2 is the Jacobi theta function.

Equals Product_{k>=1} (1 - 1/4^k)^((-1)^k). - Antonio Graciá Llorente, Oct 01 2024

A333797 Total number of saturated chains in the lattices L_n(2) of subspaces (ordered by inclusion) of the vector space GF(2)^n.

Original entry on oeis.org

1, 3, 14, 114, 1777, 55461, 3496868, 444131448, 113253936439, 57872769803787, 59203843739029706, 121190268142727296926, 496274148044956457612893, 4064981546636275903297015089, 66596592678542112197488335080432, 2182170552297789390998576752287351492

Offset: 0

Geoffrey Critzer, Apr 05 2020

nonn

These are the chains counted in A293844 that are saturated. A chain C in poset P is saturated if there is no z in P - C such that x < z < y for some x,y in C and such that C union {z} is a chain.

Cf. A289545, A293844.

nn = 15; eq[z_] :=Sum[z^n/FunctionExpand[QFactorial[n, q]], {n, 0, nn}];
Table[FunctionExpand[QFactorial[n, q]] /. q -> 2, {n, 0, nn}] CoefficientList[Series[eq[z]^2/(1 - z) /. q -> 2, {z, 0, nn}], z]

a(n)/A005329(n) is the coefficient of x^n in eq(x)^2/(1 - x) where eq(x) is the q-exponential function.

a(n) ~ A299998 * 2^(n*(n+1)/2). - Vaclav Kotesovec, Apr 07 2020

A332678 Decimal expansion of (1/2) * (1 + 2/1 + 4/(21) + 8/(42*1) + ... ).

Original entry on oeis.org

3, 1, 4, 1, 6, 3, 2, 5, 6, 0, 6, 5, 5, 1, 5, 3, 8, 6, 6, 2, 9, 3, 8, 4, 2, 7, 7, 0, 2, 2, 5, 4, 2, 9, 4, 3, 4, 2, 2, 6, 0, 6, 1, 5, 3, 7, 9, 5, 6, 7, 3, 9, 7, 4, 7, 8, 0, 4, 6, 5, 1, 6, 2, 2, 3, 8, 0, 1, 4, 4, 6, 0, 3, 7, 3, 3, 3, 5, 1, 7, 7, 5, 6, 0, 0, 3, 6, 4, 1, 7, 1, 6, 2, 3, 3, 5, 9, 1, 3, 3, 0, 8, 6

Offset: 1

Drew Edgette, Feb 19 2020

An approximation to Pi.

			3.1416325606551538662938427702254294342260615379567...

Cf. A000796 (Pi), A013705.

c:= sum(2^(j*(3-j)/2-1), j=0..infinity):
evalf(c, 125);  # Alois P. Heinz, Mar 03 2020

suminf(k=0, 2^(k-binomial(k,2)-1)) \\ Andrew Howroyd, Feb 21 2020

Equals (1/2)*Sum_{k>=0} 2^(k-binomial(k,2)). - Andrew Howroyd, Feb 21 2020

Equals A190405 +2.5 = A299998 +1.5. All digits the same but the first one or two. - R. J. Mathar, Mar 10 2020

cp's OEIS Frontend

A006125 a(n) = 2^(n*(n-1)/2).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

Maxima

PARI

Python

Formula

Extensions

A190404 Decimal expansion of (1/2)(1 + Sum_{k>=1} (1/2)^T(k)), where T=A000217 (triangular numbers); based on row 1 of the natural number array, A000027.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Formula

A073587 a(n) = a(n-1)*2^n + 1 where a(0)=1.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

UBASIC

Formula

Extensions

A319015 Decimal expansion of Sum_{k>=0} 1/2^(k^2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A319016 Decimal expansion of Sum_{k>=0} 1/2^(k*(k+1)).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A333797 Total number of saturated chains in the lattices L_n(2) of subspaces (ordered by inclusion) of the vector space GF(2)^n.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A332678 Decimal expansion of (1/2) * (1 + 2/1 + 4/(2*1) + 8/(4*2*1) + ... ).

Views

Author

Keywords

Comments

Examples

A332678 Decimal expansion of (1/2) * (1 + 2/1 + 4/(21) + 8/(42*1) + ... ).