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

0,1

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

Table of n, a(n) for n=0..34.

Twitter / MathQuizzes, Puzzle relating to this sequence

Index entries for linear recurrences with constant coefficients, signature (0,3).

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)

Prog

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

Crossrefs

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

Sequence in context: A339891 A310793 A117949 * A010901 A187211 A023624

Adjacent sequences: A228876 A228877 A228878 * A228880 A228881 A228882

Keyword

nonn,easy

Author

Richard R. Forberg, Sep 06 2013

Extensions

More terms from Colin Barker, Jun 09 2014

Status

approved