A004119 a(0) = 1; thereafter a(n) = 3*2^(n-1) + 1.

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, 6442450945, 12884901889

Offset: 0

Author

N. J. A. Sloane, R. K. Guy

Keywords

• nonn
• easy

Comments

Also Pisot sequence L(4,7) (cf. A008776).

Alternatively, define the sequence S(a(1),a(2)) by: a(n+2) is the least integer such that a(n+2)/a(n+1) > a(n+1)/a(n) for n > 0. This is S(4,7).

a(n) = number of terms of the arithmetic progression with first term 2^(2n-1) and last term 2^(2n+1). So common difference is 2^n. E.g., a(2)=7 corresponds to (8,12,16,20,24,28,32). - Augustine O. Munagi, Feb 21 2007

Equals binomial transform of [1, 3, 0, 3, 0, 3, 0, 3, ...]. - Gary W. Adamson, Aug 27 2010

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993
• S. W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663.
• S. W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663. [Annotated scanned copy]
• A. O. Munagi and T. Shonhiwa, On the partitions of a number into arithmetic progressions, J. Integer Sequences, 11 (2008), #08.5.4.
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• Index entries for linear recurrences with constant coefficients, signature (3, -2).
• Index entries for Pisot sequences

Crossrefs

Cf. A049988, A049775, A000051, A008776.

A181565 is an essentially identical sequence.

For primes see A002253 and A039687.

Programs

Magma

[1] cat [n le 1 select 4 else 2*Self(n-1)-1: n in [1..40]]; // Vincenzo Librandi, Dec 16 2015

Maple

A004119:=-(-1-z+3*z**2)/(2*z-1)/(z-1); # Simon Plouffe in his 1992 dissertation

Mathematica

s=4;lst={1,s};Do[s=s+(s-1);AppendTo[lst,s],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 30 2009 *)
Prepend[Table[3*2^n + 1, {n, 0, 32}], 1] (* or *)
{1}~Join~LinearRecurrence[{3, -2}, {4, 7}, 33] (* Michael De Vlieger, Dec 16 2015 *)

PARI

a(n)=3<Charles R Greathouse IV, Sep 28 2015

Formula

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

For n>3, a(3)=13, a(n)= a(n-1)+2*floor(a(n-1)/2). - Benoit Cloitre, Aug 14 2002

For n>=1, a(n) = A049775(n+1)/2^(n-2). For n>=2, a(n) = 2a(n-1)-1; see also A000051. - Philippe Deléham, Feb 20 2004

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

For n>0, a(n) = 2*a(n-1)-1. - Vincenzo Librandi, Dec 16 2015

E.g.f.: exp(x)*(1 + 3*sinh(x)). - Stefano Spezia, May 06 2023

Extensions

Edited by N. J. A. Sloane, Dec 16 2015 at the suggestion of Bruno Berselli