A126646 a(n) = 2^(n+1) - 1.

1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, 16383, 32767, 65535, 131071, 262143, 524287, 1048575, 2097151, 4194303, 8388607, 16777215, 33554431, 67108863, 134217727, 268435455, 536870911, 1073741823, 2147483647

Offset

0,2

Comments

a(n) is the number of integers k less than 10^n such that the decimal representation of k lacks the digits 1,2,3,4,5,6 and 7 and at least one of the digits 8,9.

Partial sums of the powers of 2 (A000079).

a(n) is the number of elements (all m-dimensional faces) in an n-dimensional simplex (0 <= m <= n). - Sergey Pavlov, Aug 15 2015

A261461(a(n)) != A261922(a(n)). - Reinhard Zumkeller, Sep 17 2015

a(n) is the total number of matches in a knockout tournament with 2^n players. - Paul Duckett, Dec 12 2022

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Jerry Metzger and Thomas Richards, A Prisoner Problem Variation, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.7.

Wikipedia, Simplex Elements (see last column of table).

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

Formula

a(n-1)^2 + a(n) = a(2n) + 1, a square. - Vincenzo Librandi and Ralf Stephan, Nov 23 2010

G.f.: 1/ ( (1-2*x)*(1-x) ). - R. J. Mathar, Dec 02 2013

a(n) = 3*a(n-1) - 2*a(n-2), n > 1. - Wesley Ivan Hurt, Aug 21 2015

E.g.f.: 2*exp(2*x) - exp(x). - G. C. Greubel, Mar 31 2021

Example

a(8) = 2^9 - 1 = 511.

Maple

A126646:=n->2*2^n-1; seq(A126646(n), n=0..50); # Wesley Ivan Hurt, Dec 02 2013

Mathematica

Table[2^(n+1) - 1, {n, 0, 50}] (* Wesley Ivan Hurt, Dec 02 2013 *)
LinearRecurrence[{3, -2}, {1, 3}, 40] (* Harvey P. Dale, Mar 23 2018 *)

Prog

(PARI) first(m)=vector(m, i, i--; 2^(i+1)-1) /* Anders Hellström, Aug 19 2015 */
(Magma) [2^(n+1)-1: n in [0.. 35]]; // Vincenzo Librandi, Aug 20 2015
(Haskell)
a126646 = (subtract 1) . (2 ^) . (+ 1)
a126646_list = iterate ((+ 1) . (* 2)) 1
-- Reinhard Zumkeller, Sep 17 2015
(Sage) [2^(n+1) -1 for n in (0..50)] # G. C. Greubel, Mar 31 2021
(Python)
def A126646(n): return (1<<n+1)-1 # Chai Wah Wu, Mar 18 2024

Crossrefs

Essentially the same as A000225.

Cf. A125630, A125945, A125947, A125948, A125940, A125909, A125908, A125880, A125897, A125904, A125858.

Cf. A000079, A168604.

Cf. A261461, A261922.

Sequence in context: A336700 A097002 A060152 * A225883 A255047 A000225

Adjacent sequences: A126643 A126644 A126645 * A126647 A126648 A126649

Keyword

nonn,easy

Author

Aleksandar M. Janjic and Milan Janjic, Feb 08 2007, Feb 13 2007

Status

approved