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Triangular array: T(n,k) = |round((r^n)*(s^k))|, where r = golden ratio = (1+sqrt(5))/2, s = (1-sqrt(5))/2, 1 <= k <= n, n >= 1.
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%I #11 Nov 17 2024 21:38:51

%S 1,2,1,3,2,1,4,3,2,1,7,4,3,2,1,11,7,4,3,2,1,18,11,7,4,3,2,1,29,18,11,

%T 7,4,3,2,1,47,29,18,11,7,4,3,2,1,76,47,29,18,11,7,4,3,2,1,123,76,47,

%U 29,18,11,7,4,3,2,1,199,123,76,47,29,18,11,7,4,3

%N Triangular array: T(n,k) = |round((r^n)*(s^k))|, where r = golden ratio = (1+sqrt(5))/2, s = (1-sqrt(5))/2, 1 <= k <= n, n >= 1.

%C Row n consists of the first n numbers of A169985 = (1,2,3,4,7,... ) in reverse order; these are the Lucas numbers, A000032, with order of initial two terms reversed. Every column of the triangle is A169985.

%H Clark Kimberling, <a href="/A271355/b271355.txt">Table of n, a(n) for n = 1..10000</a>

%F T(n,k) = |round((r^n)*(s^k))|, where r = golden ratio = (1+sqrt(5))/2, s = (1-sqrt(5))/2, 1 <= k <= n, n >= 1.

%F T(k+j-1,j) = A000032(k) = k-th Lucas number, for k >= 2.

%e First six rows:

%e 1

%e 2 1

%e 3 2 1

%e 4 3 2 1

%e 7 4 3 2 1

%e 11 7 4 3 2 1

%t r = N[(1 + Sqrt[5])/2, 100]; s = N[(1 - Sqrt[5])/2, 100];

%t t = Table[Abs[Round[(r^n)*(s^k)]], {n, 0, 15}, {k, 1, n}];

%t Flatten[t] (* A271355, sequence *)

%t TableForm[t] (* A271355, array *)

%Y Cf. A169985, A000032, A000045, A104762

%K nonn,easy,tabl,changed

%O 1,2

%A _Clark Kimberling_, May 01 2016