A048578 Pisot sequence L(3,5).

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, 4294967297, 8589934593, 17179869185

Offset: 0

Author

David W. Wilson

Keywords

• nonn
• easy

Comments

Lexicographically earliest (when ordered) minimal set of generators for A001969 (numbers with an even number of binary 1's) as a group under A003987(.,.) the XOR operation. - Peter Munn, Aug 21 2019

Odd numbers with binary weight = 2. - David James Sycamore, Feb 02 2025

References

• G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..2000
• Josef Eschgfäller and Andrea Scarpante, Dichotomic random number generators, arXiv:1603.08500 [math.CO], 2016.
• Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
• Eric Weisstein's World of Mathematics, Group.
• Wikipedia, Generating set of a group.
• Index entries for linear recurrences with constant coefficients, signature (3,-2).

Crossrefs

Subsequence of A000051.

See A008776 for definitions of Pisot sequences.

Cf. A001969, A003987.

Essentially the same as A020737 and A000051.

Programs

Magma

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

Mathematica

LinearRecurrence[{3,-2},{3,5},40] (* Harvey P. Dale, Sep 10 2017 *)

PARI

my(x='x+O('x^99)); Vec(1/(1-x)+2/(1-2*x)) \\ Altug Alkan, Mar 29 2016

Formula

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

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

O.g.f.: (3-4*x)/(1-3*x+2*x^2). - R. J. Mathar, Nov 23 2007

E.g.f.: exp(x)*(1 + 2*exp(x)). - Elmo R. Oliveira, Dec 06 2024