A142964 a(n) = 6*2^n - 2*n - 5.

1, 5, 15, 37, 83, 177, 367, 749, 1515, 3049, 6119, 12261, 24547, 49121, 98271, 196573, 393179, 786393, 1572823, 3145685, 6291411, 12582865, 25165775, 50331597, 100663243, 201326537, 402653127, 805306309, 1610612675, 3221225409, 6442450879, 12884901821

Offset: 0

Author

Wolfdieter Lang, Sep 15 2008

Keywords

• nonn
• easy

Comments

Previous name was: One half of second column (m=1) of triangle A142963.

Essentially a duplicate of A050488. - Johannes W. Meijer, Feb 20 2009

Examples

			a(3) = 6*2^3 - 2*3 - 5 = 37.
		

References

• Eric Billault, Walter Damin, Robert Ferréol, and Rodolphe Garin, MPSI Classes Prépas - Khôlles de Maths, Exercices corrigés, Ellipses, 2012, exercice 2.22 (1) pp 26, 43-44.

Links

• Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Crossrefs

Cf. A142965 (m=2 column/4).

Equals A050488(n+1).

Equals A156920(n+1,1).

Equals A156919(n+1,1)/2^n.

Cf. A156925, A339771.

Partial sums of A033484.

Programs

Maple

seq(6*2^n-2*n-5,n=0..40); # Bernard Schott, Dec 16 2020

Mathematica

a[n_]:=6*2^n-2n-5;Array[a,32,0] (* or *) CoefficientList[Series[(1+x)/((1-x)^2*(1-2*x)),{x,0,31}],x] (* or *) LinearRecurrence[{4,-5,2},{1,5,15},32] (* James C. McMahon, Aug 12 2025 *)

PARI

Vec((1+z)/((1-z)^2*(1-2*z)) + O(z^50)) \\ Michel Marcus, Jun 18 2017

Formula

a(n) = A142693(n+2,1)/2.

From Johannes W. Meijer, Feb 20 2009: (Start)

a(n) = 4a(n-1) - 5a(n-2) + 2a(n-3) for n > 2 with a(0) = 1, a(1) = 5, a(2) = 15.

G.f.: (1+z)/((1-z)^2*(1-2*z)). (End)

a(n) = Sum_{i=0..n} Sum_{j=0..n} 2^min(i,j) (Billault et al) (compare with A339771 that has max instead of min). - Bernard Schott, Dec 16 2020

a(n) = 2*A066524(n+1) - A339771(n). - Kevin Ryde, Dec 17 2020

E.g.f.: 6*exp(2*x) - exp(x)*(5 + 2*x). - Stefano Spezia, Dec 17 2020

Extensions

New name using a formula of Bernard Schott by Peter Luschny, Dec 17 2020