A083318 a(0) = 1; for n>0, a(n) = 2^n + 1.

1, 3, 5, 9, 17, 33, 65, 129, 257, 513, 1025, 2049, 4097, 8193, 16385, 32769, 65537, 131073, 262145, 524289, 1048577, 2097153, 4194305, 8388609, 16777217, 33554433, 67108865, 134217729, 268435457, 536870913, 1073741825, 2147483649

Offset: 0

Author

Paul Barry, Apr 25 2003

Keywords

• nonn
• easy

Comments

Inverse binomial transform of A005056.

Also, A000533 interpreted as binary numbers, written in base 10. Numbers whose representation in base 2 is has n+1 digits and the digit "1" is the initial and final digit and if n>1 then the internal digits are "0" (see example). - Omar E. Pol, Feb 24 2008

a(n) equals the number of ternary sequences of length n such that no two consecutive terms differ by 1. - David Nacin, May 31 2017

Examples

			From _Omar E. Pol_, Feb 24 2008: (Start)
------------------------------
n .... a(n) .. a(n) in base 2
------------------------------
0 ..... 1 ..... 1
1 ..... 3 ..... 11
2 ..... 5 ..... 101
3 ..... 9 ..... 1001
4 .... 17 ..... 10001
5 .... 33 ..... 100001
6 .... 65 ..... 1000001
7 ... 129 ..... 10000001
8 ... 257 ..... 100000001
9 ... 513 ..... 1000000001
(End)
G.f. = 1 + 3*x + 5*x^2 + 9*x^3 + 17*x^4 + 33*x^5 + 65*x^6 + 129*x^7 + ... - _Michael Somos_, Jun 04 2016
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Ramon Carbó-Dorca, Boolean Hypercubes: The Origin of a Tagged Recursive Logic and the Limits of Artificial Intelligence, Universitat de Girona (Spain, 2020).
• Quynh Nguyen, Jean Pedersen, and Hien T. Vu, New Integer Sequences Arising From 3-Period Folding Numbers, Vol. 19 (2016), Article 16.3.1. Cites this sequence.
• Index entries for linear recurrences with constant coefficients, signature (3,-2).

Crossrefs

Except for the leading term, the same as A000051.

Cf. A000533, A083319, A132749.

Programs

GAP

Concatenation([1], List([1..40], n-> 2^n +1)); # G. C. Greubel, Nov 20 2019

Magma

[2^n+1-0^n : n in [0..40]]; // Vincenzo Librandi, Sep 01 2011

Maple

seq(`if`(n=0, 1, 2^n + 1), n=0..40); # G. C. Greubel, Nov 20 2019

Mathematica

Join[{1},2^Range[40]+1] (* Harvey P. Dale, May 17 2013 *)

PARI

{a(n) = if( n<1, n==0, 2^n + 1)}; /* Michael Somos, Jun 04 2016 */

Sage

[1]+[2^n +1 for n in (1..40)] # G. C. Greubel, Nov 20 2019

Formula

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

G.f.: (1-2*x^2)/((1-x)*(1-2x)).

E.g.f.: exp(2*x) + exp(x) - exp(0).

a(n) = Sum_{k=0..n} 0^(k*(n-k))*2^(n-k). - Paul Barry, Feb 09 2005

a(n) = Min{m: A008687(m) = n+1}. - Reinhard Zumkeller, Jul 25 2006

Row sums of triangle A132749; = binomial transform of [1, 2, 0, 2, 0, 2, 0, 2, ...]. - Gary W. Adamson, Aug 28 2007

A020650(a(n)) = 1. - Yosu Yurramendi, Jun 01 2016

Extensions

Edited by N. J. A. Sloane, Sep 28 2007