A228879 a(n+2) = 3*a(n), starting 4,7.

4, 7, 12, 21, 36, 63, 108, 189, 324, 567, 972, 1701, 2916, 5103, 8748, 15309, 26244, 45927, 78732, 137781, 236196, 413343, 708588, 1240029, 2125764, 3720087, 6377292, 11160261, 19131876, 33480783, 57395628, 100442349, 172186884, 301327047, 516560652

Offset: 0

Author

Richard R. Forberg, Sep 06 2013

Keywords

• nonn
• easy

Comments

Successive terms are the square roots of expressions of prior terms. The same recursive formula (see below) can be applied using any term of A001353 as the initializing value to produce the family of sequences, as given in the array A227418. This sequence uses A001353(2) = 4, and is the third row of that array.

a(4n) are the squares of A008776(n).

Binomial transform of a(n) is A021006.

First differences of a(n) = A083658 (without initial two terms).

2nd differences of a(n) = A068911 (with initial term).

a(n-1) is the number of n-digit base 10 numbers where all the digits are even numbers, and each digit differs by 2 from the previous and the next digit. - Graeme McRae, Jun 09 2014

Links

• Paolo Xausa, Table of n, a(n) for n = 0..1000
• Twitter / MathQuizzes, Puzzle relating to this sequence
• Index entries for linear recurrences with constant coefficients, signature (0,3).

Crossrefs

Cf. A001353, A008776, A021006, A068911, A083658, A227418.

Row n = 5 of A377000.

Programs

Mathematica

LinearRecurrence[{0, 3}, {4, 7}, 50] (* Paolo Xausa, Oct 14 2024 *)

PARI

Vec(-(7*x+4)/(3*x^2-1) + O(x^100)) \\ Colin Barker, Jun 09 2014

Formula

a(n) = sqrt(3*a(n-1)^2 + (-3)^(n-1)), a(0) = 4.

This divisibility relation also applies: a(n+3) = a(n+2)*a(n+1)/a(n).

G.f.: -(7*x+4) / (3*x^2-1). - Colin Barker, Jun 09 2014

From Stefano Spezia, Mar 20 2022: (Start)

a(n) = 3^((n-1)/2)*(4*sqrt(3) + 7 + (-1)^n*(4*sqrt(3) - 7))/2.

E.g.f.: 4*cosh(sqrt(3)*x) + 7*sinh(sqrt(3)*x)/sqrt(3). (End)

Extensions

More terms from Colin Barker, Jun 09 2014