A182780 Twice A024537.

2, 4, 8, 18, 42, 100, 240, 578, 1394, 3364, 8120, 19602, 47322, 114244, 275808, 665858, 1607522, 3880900, 9369320, 22619538, 54608394, 131836324, 318281040, 768398402, 1855077842, 4478554084, 10812186008, 26102926098, 63018038202, 152139002500, 367296043200, 886731088898, 2140758220994, 5168247530884

Offset: 0

Author

N. J. A. Sloane, Dec 23 2012

Keywords

• nonn

Comments

a(n) = A078057(n) + 1 (see A288213). - Michel Dekking, Sep 29 2019

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• J. V. Leyendekkers and A. G. Shannon, Pellian sequence relationships among pi, e, sqrt(2), Notes on Number Theory and Discrete Mathematics, Vol. 18, 2012, No. 2, 58-62. See Table 2.
• Index entries for linear recurrences with constant coefficients, signature (3,-1,-1).

Crossrefs

Cf. A024537.

Programs

Magma

a:=[2,4,8]; [n le 3 select a[n] else 3*Self(n-1) - Self(n-2) - Self(n-3):n in [1..35]]; // Marius A. Burtea, Sep 29 2019

PARI

Vec(2*(1 - x - x^2) / ((1 - x)*(1 - 2*x - x^2)) + O(x^40)) \\ Colin Barker, May 26 2018

Formula

From Colin Barker, May 26 2018: (Start)

G.f.: 2*(1 - x - x^2) / ((1 - x)*(1 - 2*x - x^2)).

a(n) = (2 + (1-sqrt(2))^(1+n) + (1+sqrt(2))^(1+n)) / 2.

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

(End)