A020737 Pisot sequence L(5,9).

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, 4294967297, 8589934593

Offset: 0

Author

David W. Wilson

Keywords

• nonn
• easy

Comments

An Engel expansion of 1/2 to the base 2 as defined in A181565, with the associated series expansion 1/2 = 2/5 + 2^2/(5*9) + 2^3/(5*9*17) + 2^4/(5*9*17*33) + ... . - Peter Bala, Oct 28 2013

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..238
• Index entries for linear recurrences with constant coefficients, signature (3,-2).

Crossrefs

Subsequence of A000051. See A008776 for definitions of Pisot sequences.

Cf. A083575, A083683, A083686, A083705, A168596, A181565, A195744.

Programs

Magma

[2^(n+2)+1: n in [0..40] ]; // Vincenzo Librandi, Apr 28 2011

Mathematica

LinearRecurrence[{3,-2},{5,9},40] (* Harvey P. Dale, Jun 10 2015 *)

PARI

a(n)=2^(n+2)+1 \\ Charles R Greathouse IV, Jun 05 2013

PARI

Vec(-(6*x-5)/((x-1)*(2*x-1)) + O(x^100)) \\ Colin Barker, Jun 21 2014

Formula

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

a(n) = 3*a(n-1) - 2*a(n-2).

G.f.: -(6*x-5) / ((x-1)*(2*x-1)). - Colin Barker, Jun 21 2014

E.g.f.: exp(x)*(1 + 4*exp(x)). - Stefano Spezia, Oct 08 2022