A181565 a(n) = 3*2^n + 1.

4, 7, 13, 25, 49, 97, 193, 385, 769, 1537, 3073, 6145, 12289, 24577, 49153, 98305, 196609, 393217, 786433, 1572865, 3145729, 6291457, 12582913, 25165825, 50331649, 100663297, 201326593, 402653185, 805306369, 1610612737, 3221225473

Offset: 0

Author

M. F. Hasler, Oct 30 2010

Keywords

• nonn
• easy

Comments

From Peter Bala, Oct 28 2013: (Start)

Let x and b be positive real numbers. We define an Engel expansion of x to the base b to be a (possibly infinite) nondecreasing sequence of positive integers [a(1), a(2), a(3), ...] such that we have the series representation x = b/a(1) + b^2/(a(1)*a(2)) + b^3/(a(1)*a(2)*a(3)) + .... Depending on the values of x and b such an expansion may not exist, and if it does exist it may not be unique. When b = 1 we recover the ordinary Engel expansion of x.

This sequence gives an Engel expansion of 2/3 to the base 2, with the associated series expansion 2/3 = 2/4 + 2^2/(4*7) + 2^3/(4*7*13) + 2^4/(4*7*13*25) + ....

More generally, for n and m positive integers, the sequence [m + 1, n*m + 1, n^2*m + 1, ...] gives an Engel expansion of the rational number n/m to the base n. See the cross references for several examples. (End)

The only squares in this sequence are 4, 25, 49. - Antti Karttunen, Sep 24 2023

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Solomon W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663.
• Solomon W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663. [Annotated scanned copy]
• Hunar Sherzad Taher and Saroj Kumar Dash, On sums of k-generalized Fibonacci and k-generalized Lucas numbers as first and second kinds of Thabit numbers, Notes Num. Theor. Disc. Math. (2025) Vol. 31, No. 3, 448-459. See p. 2.
• Wikipedia, Engel Expansion
• Index entries for linear recurrences with constant coefficients, signature (3,-2)

Crossrefs

Cf. A007283, A020737, A083575, A083683, A083686, A168596, A083705, A195744.

Essentially a duplicate of A004119.

A002253 and A039687 give the primes in this sequence, and A181492 is the subsequence of twin primes.

Programs

Magma

[3*2^n + 1: n in [0..30]]; // Vincenzo Librandi, May 19 2011

Mathematica

3*2^Range[0,50]+1 (* Vladimir Joseph Stephan Orlovsky, Mar 24 2011 *)
LinearRecurrence[{3,-2},{4,7},40] (* Harvey P. Dale, Sep 19 2024 *)

PARI

A181565(n)=3<
				

Formula

a(n) = A004119(n+1) = A103204(n+1) for all n >= 0.

From Ilya Gutkovskiy, Jun 01 2016: (Start)

O.g.f.: (4 - 5*x)/((1 - x)*(1 - 2*x)).

E.g.f.: (1 + 3*exp(x))*exp(x).

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

a(n) = 2*a(n-1) - 1. - Miquel Cerda, Aug 16 2016

For n >= 0, A005940(a(n)) = A001248(1+n). - Antti Karttunen, Sep 24 2023