A037093 "Sloping binary representation" of Fibonacci numbers, slope = +1.
0, 1, 3, 14, 57, 229, 916, 7761, 29567, 117474, 469113, 3973641, 15138352, 60146777, 240187355, 2070207870, 7733090689, 30791909229, 260408711716, 991495872825, 3942106110215, 15739612088946, 133333733918417
Offset: 0
Examples
When Fibonacci numbers are written in binary (see A004685), under each other as: 0000000 (0) 0000001 (1) 0000001 (1) 0000010 (2) 0000011 (3) 0000101 (5) 0001000 (8) 0001101 (13) 0010101 (21) 0100010 (34) 0110111 (55) 1011001 (89) and one starts collecting their bits from column-0 to SW-direction (from the least to the most significant end), one gets 000... (0), ...00001 (1), ...00011 (3), ...001110 (14), etc. (See A102370 for similar transformation done on nonnegative integers).
Formula
a(n) := Sum(bit_n(A000045(n+i), i)*(2^i), i=0..inf) [ bit_n := (x, n) -> `mod`(floor(x/(2^n)), 2); ]
In practice, n can be used as an upper limit instead of infinity.
Extensions
Entry revised Dec 29 2007