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A083318 a(0) = 1; for n>0, a(n) = 2^n+1. 50
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

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

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

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

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

STATUS

approved

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Last modified August 17 03:51 EDT 2017. Contains 290631 sequences.