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A185737
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Accumulation array of the Wythoff array, by antidiagonals.
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1
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1, 3, 5, 6, 14, 11, 11, 28, 30, 20, 19, 51, 60, 54, 32, 32, 88, 109, 108, 86, 46, 53, 148, 188, 196, 172, 123, 63, 87, 245, 316, 338, 312, 246, 168, 82, 142, 402, 523, 568, 538, 446, 336, 218, 104, 231, 656, 858, 940, 904, 769, 609, 436, 276, 129, 375, 1067, 1400, 1542, 1496, 1292, 1050, 790, 552, 342, 156, 608, 1732, 2277, 2516, 2454, 2138, 1764, 1362, 1000, 684, 413, 186
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OFFSET
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1,2
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COMMENTS
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For the definition of accumulation array, see A144112.
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LINKS
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EXAMPLE
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Northwest corner:
5 14 28 51 88
11 30 60 109 188
20 54 108 196 338
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MATHEMATICA
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(* This program creates the Wythoff array W={f(n, k)}=A035513, then the accumulation array A185736 of W, then the weight array A185736 of W *)
f[n_, 0]:=0; f[0, k_]:=0; (* Needed for the weight array *)
f[n_, k_]:=Fibonacci[k+1]Floor[n*GoldenRatio]+(n-1)Fibonacci[k];
TableForm[Table[f[n, k], {n, 1, 10}, {k, 1, 15}]] (* Wythoff array *)
Table[f[n-k+1, k], {n, 14}, {k, n, 1, -1}]//Flatten
s[n_, k_]:=Sum[f[i, j], {i, 1, n}, {j, 1, k}];
TableForm[Table[s[n, k], {n, 1, 10}, {k, 1, 15}]] (* A185736 *)
Table[s[n-k+1, k], {n, 14}, {k, n, 1, -1}]//Flatten
(* In general, the weight array W of an arbitrary rectangular array S={s(i, j):i<=1, j<=1} is defined in two steps:(1) define s(i, j)=0 if i=0 or j=0; (2) then w(m, n)=s(m, n)+s(m-1, n-1)-s(m, n-1)-s(m-1, n) for m<1, n<1. *)
w[m_, n_]:=f[m, n]+f[m-1, n-1]-f[m, n-1]-f[m-1, n]/; Or[m>0, n>0];
TableForm[Table[w[n, k], {n, 1, 10}, {k, 1, 15}]] (* A185736 *)
Table[w[n-k+1, k], {n, 20}, {k, n, 1, -1}]//Flatten
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PROG
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(PARI) W(n, k) = (n+sqrtint(5*n^2))\2*fibonacci(k+1) + (n-1)*fibonacci(k); \\ A035513
T(n, k) = sum(i=1, n, sum(j=1, k, W(i, j))); \\ Michel Marcus, Feb 25 2023
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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