cp's OEIS Frontend

A356875 Square array, n >= 0, k >= 0, read by descending antidiagonals. A(n,k) = A022341(n)*2^k.

%I A356875 #8 Sep 07 2022 18:58:08
%S A356875 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 A356875 136,84,66,37,128,320,288,272,168,132,74,41,256,640,576,544,336,264,
%U A356875 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 A356875 Square array, n >= 0, k >= 0, read by descending antidiagonals. A(n,k) = A022341(n)*2^k.
%C A356875 The nonzero Fibbinary numbers (A003714) arranged in rows where each successive term is twice the preceding term; a (transposed) Fibbinary equivalent of A054582.
%C A356875 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 A356875 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 A356875 A(n,0) = A022341(n), otherwise A(n,k) = 2*A(n,k-1).
%F A356875 A287870(n+1,k+1) = A356874(floor(A(n,k)/2)).
%F A356875 A035513(n+1,k+1) = A022290(A(n,k)).
%e A356875 Square array A(n,k) begins:
%e A356875    1    2    4    8    16    32    64   128 ...
%e A356875    5   10   20   40    80   160   320   640 ...
%e A356875    9   18   36   72   144   288   576  1152 ...
%e A356875   17   34   68  136   272   544  1088  2176 ...
%e A356875   21   42   84  168   336   672  1344  2688 ...
%e A356875   33   66  132  264   528  1056  2112  4224 ...
%e A356875   37   74  148  296   592  1184  2368  4736 ...
%e A356875   41   82  164  328   656  1312  2624  5248 ...
%e A356875   65  130  260  520  1040  2080  4160  8320 ...
%e A356875   69  138  276  552  1104  2208  4416  8832 ...
%e A356875   ...
%e A356875 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 A356875       1      10      100      1000 ...
%e A356875     101    1010    10100    101000 ...
%e A356875    1001   10010   100100   1001000 ...
%e A356875   10001  100010  1000100  10001000 ...
%e A356875   10101  101010  1010100  10101000 ...
%e A356875   ...
%Y A356875 See the comments for the relationship to: A000045, A003714, A035513, A054582, A287870.
%Y A356875 See the formula section for the relationship to: A022290, A022341, A356874.
%K A356875 nonn,easy,tabl
%O A356875 0,2
%A A356875 _Peter Munn_, Sep 02 2022