cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A002416 a(n) = 2^(n^2).

Original entry on oeis.org

1, 2, 16, 512, 65536, 33554432, 68719476736, 562949953421312, 18446744073709551616, 2417851639229258349412352, 1267650600228229401496703205376, 2658455991569831745807614120560689152, 22300745198530623141535718272648361505980416, 748288838313422294120286634350736906063837462003712
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

For n >= 1, a(n) is the number of n X n (0, 1) matrices.
Also number of directed graphs on n labeled nodes allowing self-loops (cf. A053763).
1/2^(n^2) is the Hankel transform of C(n, n/2)*(1 + (-1)^n)/(2*2^n), or C(2n, n)/4^n with interpolated zeros. - Paul Barry, Sep 27 2007
Hankel transform of A064062. - Philippe Deléham, Nov 19 2007
a(n) is also the order of the semigroup (monoid) of all binary relations on an n-set. - Abdullahi Umar, Sep 14 2008
With offset = 1, a(n) is the number of n X n (0, 1) matrices with an even number of 1's in every row and in every column. - Geoffrey Critzer, May 23 2013
a(n) is the number of functions from an n-set to its power set (by definition of function including the empty function only when n = 0). - Rick L. Shepherd, Dec 27 2014

Examples

			G.f. = 1 + 2*x + 16*x^2 + 512*x^3 + 65536*x^4 + 33554432*x^5 + ...

References

  • John M. Howie, Fundamentals of semigroup theory. Oxford: Clarendon Press, (1995). - Abdullahi Umar, Sep 14 2008

Links

Crossrefs

Bisection of A060656.
Cf. A053763, A064062, A064231, A319015.

Programs

Formula

G.f. satisfies: A(x) = 1 + 2*x*A(4x). - Paul D. Hanna, Dec 04 2009
a(n) = 2^n * Sum_{i = 0..C(n, 2)} C(C(n, 2), i)*3^i. The summation conditions on the number of symmetric pairs (a,b) with aA027465, A013610. - Geoffrey Critzer, Nov 05 2024
G.f.: 1 / (1 - 2^1*x / (1 - 2^1*(2^2-1)*x / (1 - 2^5 * x / (1 - 2^3*(2^4-1)*x / (1 - 2^9*x / (1 - 2^5*(2^6-1)*x / ...)))))). - Michael Somos, May 12 2012
a(n) = [x^n] 1/(1 - 2^n*x). - Ilya Gutkovskiy, Oct 10 2017
Sum_{n>=0} 1/a(n) = A319015. - Amiram Eldar, Oct 14 2020

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

Views

Author

Ilya Gutkovskiy, Sep 07 2018

Keywords

Comments

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

Examples

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

Crossrefs

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

Programs

  • Mathematica
    RealDigits[EllipticTheta[2, 0, 1/2]/2^(3/4), 10, 110] [[1]]
  • PARI
    suminf(k=0, 1/2^(k*(k+1))) \\ Michel Marcus, Sep 08 2018

Formula

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
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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