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a(n) is the greatest common divisor of the first two terms of row n of the Wythoff array (A035513).
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%I #14 Jan 15 2025 16:43:10

%S 1,1,2,3,4,1,1,1,2,1,1,1,3,1,2,5,1,1,6,1,1,7,1,1,8,1,1,9,2,1,10,1,1,

%T 11,2,1,1,1,1,1,2,1,3,1,4,1,2,1,1,1,2,3,1,1,2,5,4,3,1,1,2,1,1,1,1,1,6,

%U 1,1,1,2,1,2,1,1,1,4,7,1,1,1,3,2,1,1,1,2,1,8,1,3,1,2,1,1,5,12,1,2,1,1,1,2,1,13,3,1,1

%N a(n) is the greatest common divisor of the first two terms of row n of the Wythoff array (A035513).

%C a(n) is also the gcd of every pair of consecutive terms of row n of the Wythoff array. Conjectures: the maximal number of consecutive 1's is 5, and the limiting proportion of 1's exists. See A332938.

%C If seems that for all primes p > 3, a(1+p) = 1. - _Antti Karttunen_, Jan 15 2025

%H Antti Karttunen, <a href="/A332937/b332937.txt">Table of n, a(n) for n = 1..20000</a>

%e See A332938.

%t W[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k]; (* A035513 *)

%t t = Table[GCD[W[n, 1], W[n, 2]], {n, 1, 160}] (* A332937 *)

%t Flatten[Position[t, 1]] (* A332938 *)

%o (PARI) T(n, k) = (n+sqrtint(5*n^2))\2*fibonacci(k+1) + (n-1)*fibonacci(k); \\ A035513

%o a(n) = gcd(T(n, 0), T(n, 1)); \\ _Michel Marcus_, Mar 03 2020

%Y Cf. A000045, A173027, A173028, A035513, A332938 (positions of 1's).

%K nonn,easy

%O 1,3

%A _Clark Kimberling_, Mar 03 2020

%E More terms from _Antti Karttunen_, Jan 15 2025