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%I A144148 #36 Feb 16 2025 20:26:07
%S A144148 1,1,3,1,2,2,2,3,1,3,3,5,2,2,3,5,8,3,3,2,2,8,13,5,5,3,1,3,13,21,8,8,5,
%T A144148 2,2,2,21,34,13,13,8,3,3,1,3,34,55,21,21,13,5,5,2,2,3,55,89,34,34,21,
%U A144148 8,8,3,3,2,2,89,144,55,55,34,13,13,5,5,3,1,3,144,233,89,89,55,21,21,8,8,5,2,2,3
%N A144148 Weight array W={w(i,j)} of the Wythoff array A035513.
%C A144148 In general, let w(i,j) be the weight of the unit square labeled by its northeast vertex (i,j) and for each (m,n), define S(m,n) = Sum_{i=1..m} Sum_{j=1..n} w(i,j).
%C A144148 Then S(m,n) is the weight of the rectangle [0,m]x[0,n]. As in A144112, we call W the weight array of S, and S the accumulation array of W, which can be derived from S as follows:
%C A144148 (1) extend S by defining S(i,j)=0 if i=0 or j=0; and
%C A144148 (2) then w(m,n) = s(m,n) + s(m-n,n-1) - s(m,n-1) - s(n,m-1) for m>=1, n>=1.
%C A144148 For the case at hand, S is the Wythoff array, A035513. These arrays form a chain:
%C A144148 ... ->A144148->A035513->A185737-> ... Every term of this array is a Fibonacci number.
%F A144148 For m>3, if the row number is m of form floor(h*r+1), where r=(1+sqrt(5))/2, then
%F A144148 (row m)=(row 2); otherwise, (row m)=(row 3).
%F A144148 row n: (3,2,3,5,8,13,21,...) if n>1 is in the lower Wythoff sequence, A000201.
%F A144148 row n: (2,1,2,3,5,8,13,21,...) if n is in the upper Wythoff sequence, A001950.
%e A144148 Corner:
%e A144148 1 1 1 2 3 5 8 13 21 34 55 89
%e A144148 3 2 3 5 8 13 21 34 55 89 144 233
%e A144148 2 1 2 3 5 8 13 21 34 55 89 144
%e A144148 3 2 3 5 8 13 21 34 55 89 144 233
%e A144148 3 2 3 5 8 13 21 34 55 89 144 233
%e A144148 2 1 2 3 5 8 13 21 34 55 89 144
%e A144148 3 2 3 5 8 13 21 34 55 89 144 233
%e A144148 2 1 2 3 5 8 13 21 34 55 89 144
%e A144148 3 2 3 5 8 13 21 34 55 89 144 233
%e A144148 3 2 3 5 8 13 21 34 55 89 144 233
%e A144148 2 1 2 3 5 8 13 21 34 55 89 144
%e A144148 3 2 3 5 8 13 21 34 55 89 144 233
%t A144148 s[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k];
%t A144148 Grid[Table[s[n, k], {n, 1, 12}, {k, 1, 12}]] (* A035513 *)
%t A144148 s[0, k_] := 0; s[n_, 0] = 0;
%t A144148 w[m_, n_] := s[m, n] + s[m - 1, n - 1] - s[m, n - 1] - s[m - 1, n];
%t A144148 Grid[Table[w[n, k], {n, 1, 12}, {k, 1, 12}]] (* array *)
%t A144148 Table[w[k, m - k], {m, 2, 14}, {k, 1, m - 1}] // Flatten (* sequence *)
%o A144148 (PARI) s(n, k) = if ((n<=0) || (k<=0), 0, (n+sqrtint(5*n^2))\2*fibonacci(k+1) + (n-1)*fibonacci(k)); \\ A035513
%o A144148 w(n, k) = s(n,k)+s(n-1,k-1)-s(n,k-1)-s(n-1,k); \\ _Michel Marcus_, Feb 02 2025
%Y A144148 Cf. A000045, A000201, A001950, A035513, A144112, A185737, A185778.
%K A144148 nonn,tabl
%O A144148 1,3
%A A144148 _Clark Kimberling_, Sep 11 2008
%E A144148 Corrected and extended by _Michel Marcus_, Feb 02 2025
%E A144148 Some of the content of the duplicate (and now dead) sequence A185736 has been merged into this entry. - _N. J. A. Sloane_, Feb 15 2025
%E A144148 Edited by _Clark Kimberling_, Feb 16 2025