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%I #8 Sep 07 2022 18:58:08
%S 1,2,5,4,10,9,8,20,18,17,16,40,36,34,21,32,80,72,68,42,33,64,160,144,
%T 136,84,66,37,128,320,288,272,168,132,74,41,256,640,576,544,336,264,
%U 148,82,65,512,1280,1152,1088,672,528,296,164,130,69,1024,2560,2304,2176,1344,1056,592,328,260,138,73
%N Square array, n >= 0, k >= 0, read by descending antidiagonals. A(n,k) = A022341(n)*2^k.
%C The nonzero Fibbinary numbers (A003714) arranged in rows where each successive term is twice the preceding term; a (transposed) Fibbinary equivalent of A054582.
%C Write the first term in each row as Sum_{i in S} 2^i, where S is a set of nonnegative integers, then n = Sum_{i in S} F_i, where F_i is the i-th Fibonacci number, A000045(i).
%C More generally, if the terms are represented in binary, and the binary weighting of the digits (2^0, 2^1, 2^2, ...) is replaced with Fibonacci weighting (F_0, F_1, F_2, ...), we get the extended Wythoff array (A287870). If the weighting of the Zeckendorf representation is used (F_2, F_3, F_4, ...), we get the (unextended) Wythoff array (A035513).
%F A(n,0) = A022341(n), otherwise A(n,k) = 2*A(n,k-1).
%F A287870(n+1,k+1) = A356874(floor(A(n,k)/2)).
%F A035513(n+1,k+1) = A022290(A(n,k)).
%e Square array A(n,k) begins:
%e 1 2 4 8 16 32 64 128 ...
%e 5 10 20 40 80 160 320 640 ...
%e 9 18 36 72 144 288 576 1152 ...
%e 17 34 68 136 272 544 1088 2176 ...
%e 21 42 84 168 336 672 1344 2688 ...
%e 33 66 132 264 528 1056 2112 4224 ...
%e 37 74 148 296 592 1184 2368 4736 ...
%e 41 82 164 328 656 1312 2624 5248 ...
%e 65 130 260 520 1040 2080 4160 8320 ...
%e 69 138 276 552 1104 2208 4416 8832 ...
%e ...
%e The defining characteristic of a Fibbinary number is that its binary representation does not have a 1 followed by another 1. Shown in binary the array begins:
%e 1 10 100 1000 ...
%e 101 1010 10100 101000 ...
%e 1001 10010 100100 1001000 ...
%e 10001 100010 1000100 10001000 ...
%e 10101 101010 1010100 10101000 ...
%e ...
%Y See the comments for the relationship to: A000045, A003714, A035513, A054582, A287870.
%Y See the formula section for the relationship to: A022290, A022341, A356874.
%K nonn,easy,tabl
%O 0,2
%A _Peter Munn_, Sep 02 2022