A377291 - OEIS

%I #7 Nov 15 2024 23:32:00

%S 6,14,7,98,16,34,1442,398,194,119,30,62,4354,1154,115598,322,23,

%T 155234,48,98,10402,2702,64514,727,482,3040,1154,2114,70,142,21314,

%U 5474,2498,1442,16793602,674,48497294,158402,47,48670,96,194,39202,9998,1684802,2599

%N For each row n in array A374602(n, k), the asymptotic geometric growth factor of every A377290(n) terms, represented by its nearest integer.

%C (a(n)^2-4)/A000037(n) is a square, and as such, a(n) is a member of row x of A298675(x, k), where x is the smallest value >= 3 such that (x^2-4)/A000037(n) is a square. E.g. for n=38: A000037(38)=44, x=20 ((20^2-4)/44 = 3^2), and a(38) = 158402 = A298675(20, 4).

%C The same row x of A298675(x, k) also results as integer solutions to g+(1/g) where g=(w*sqrt(d) + ceiling(w*sqrt(d)))/2 and d=A000037(n) for integers w >= 1. As such, it follows that g(n) can be expressed as a simple integer arithmetic transformation of sqrt(A000037(n)), e.g. g(1) = 2*sqrt(2)+3 (A156035), g(2) = 4*sqrt(3)+7 (A354129), g(3) = (3*sqrt(5)+7)/2 (A374883), g(4) = 20*sqrt(6)+49, and g(5) = 3*sqrt(7)+8 (A010516+8).

%H Charles L. Hohn, <a href="/A377291/b377291.txt">Table of n, a(n) for n = 1..90</a>

%F Growth factor g(n) = Lim_{k->oo}(A374602(n, k+A377290(n))/A374602(n, k)).

%F a(n) = round(g(n)) = ceiling(g(n)) = g(n)+(1/g(n)).

%F Inverse: g(n) = (sqrt(a(n)^2-4)+a(n))/2.

%F For d = A000037(n) and x in {1, 2, 4}, when d+x is a square (unless x==4 and d+x is even): a(n) = 4/x*d+2.

%F For d = A000037(n) and x in {-4, 1, 2, 4}, when n > 3 and d-x is a square (unless x==-4 and d-x is odd): a(n) = (4/abs(x))^2*d^2-16/x*d+2.

%e For n = 5, the first few terms of A374602(5, k) are {4, 5, 11, 28, 62, 79, 175, 446, 988} and the period size is A377290(5) = 4, giving A374602(5, 1+4)/A374602(5, 1) = 62/4 = 15.5, 79/5 = 15.8, 175/11 = 15.909..., 446/28 = 15.928..., 988/62 = 15.935..., ..., to limit 15.937... -> g(5), from which g(5)+(1/g(5)) = 16 -> a(5).

%Y Cf. A374602, A377290, A000037, A298675.

%Y Cf. A156035, A354129, A374883, A010516.

%K nonn

%O 1,1

%A _Charles L. Hohn_, Oct 23 2024