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%I #30 Mar 20 2022 18:28:14
%S 4,7,12,21,36,63,108,189,324,567,972,1701,2916,5103,8748,15309,26244,
%T 45927,78732,137781,236196,413343,708588,1240029,2125764,3720087,
%U 6377292,11160261,19131876,33480783,57395628,100442349,172186884,301327047,516560652
%N a(n+2) = 3*a(n), starting 4,7.
%C 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.
%C a(4n) are the squares of A008776(n).
%C Binomial transform of a(n) is A021006.
%C First differences of a(n) = A083658 (without initial two terms).
%C 2nd differences of a(n) = A068911 (with initial term).
%C 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
%H Twitter / MathQuizzes, <a href="https://twitter.com/MathQuizzes/status/476017057138757633">Puzzle relating to this sequence</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,3).
%F a(n) = sqrt(3*a(n-1)^2 + (-3)^(n-1)), a(0) = 4.
%F This divisibility relation also applies: a(n+3) = a(n+2)*a(n+1)/a(n).
%F G.f.: -(7*x+4) / (3*x^2-1). - _Colin Barker_, Jun 09 2014
%F From _Stefano Spezia_, Mar 20 2022: (Start)
%F a(n) = 3^((n-1)/2)*(4*sqrt(3) + 7 + (-1)^n*(4*sqrt(3) - 7))/2.
%F E.g.f.: 4*cosh(sqrt(3)*x) + 7*sinh(sqrt(3)*x)/sqrt(3). (End)
%o (PARI) Vec(-(7*x+4)/(3*x^2-1) + O(x^100)) \\ _Colin Barker_, Jun 09 2014
%Y Cf. A001353, A008776, A021006, A068911, A083658, A227418.
%K nonn,easy
%O 0,1
%A _Richard R. Forberg_, Sep 06 2013
%E More terms from _Colin Barker_, Jun 09 2014
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