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A083318 a(0) = 1; for n>0, a(n) = 2^n+1. 19
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

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).

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

EXAMPLE

------------------------------

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

G.f. = 1 + 3*x + 5*x^2 + 9*x^3 + 17*x^4 + 33*x^5 + 65*x^6 + 129*x^7 + ...

MATHEMATICA

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

PROG

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

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

CROSSREFS

Except for the leading term, the same as A000051.

Cf. A083319, A132749, A000533.

Sequence in context: A205537 A135728 * A127904 A048578 A087312 A099170

Adjacent sequences:  A083315 A083316 A083317 * A083319 A083320 A083321

KEYWORD

nonn,easy

AUTHOR

Paul Barry, Apr 25 2003

EXTENSIONS

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

Examples provided by Omar E. Pol, Feb 24 2008

STATUS

approved

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Last modified March 27 08:27 EDT 2017. Contains 284146 sequences.