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Bit column sums in the binary expansions of Fibonacci(n)/2^n for n >= 1.
1

%I #16 Apr 28 2018 00:49:18

%S 1,2,3,4,5,5,7,12,9,10,9,14,13,18,21,17,23,16,20,24,23,23,26,26,30,29,

%T 29,32,34,32,37,34,33,43,30,37,41,46,43,44,42,52,45,51,50,53,50,51,49,

%U 55,64,48,60,53,65,73,67,58,69,62,75,65,74,71,69,68,88,89,85,67,76,82,83,76,81,89,91,98,93,92,83,104,87,95,90,85,101,91,101,105,105,114,84,104,108,116,121,104,126,104,110,131,107,111,137,109,126,124,119,127,136,127,120,122,145,132,132,127,131,122,129,130,136,144,146

%N Bit column sums in the binary expansions of Fibonacci(n)/2^n for n >= 1.

%H Chai Wah Wu, <a href="/A303655/b303655.txt">Table of n, a(n) for n = 1..10000</a> (n = 1..800 from Paul D. Hanna)

%F Sum_{n>=1} a(n) / 2^n = 2.

%e The binary expansions of Fibonacci(n)/2^n for n >= 1 begin:

%e .1

%e .01

%e .010

%e .0011

%e .00101

%e .001000

%e .0001101

%e .00010101

%e .000100010

%e .0000110111

%e .00001011001

%e .000010010000

%e .0000011101001

%e .00000101111001

%e .000001001100010

%e .0000001111011011

%e .00000011000111101

%e .000000101000011000

%e .0000001000001010101

%e .00000001101001101101

%e .000000010101011000010

%e .0000000100010100101111

%e .00000000110111111110001

%e .000000001011010100100000

%e .0000000010010010100010001

%e .00000000011101101000110001

%e .000000000101111111101000010

%e .0000000001001101100101110011

%e .00000000001111101100010110101

%e .000000000011001011001000101000

%e .0000000000101001000101011011101

%e .00000000001000010011110100000101

%e .000000000001101011100011111100010

%e .0000000000010101110000010011100111

%e .00000000000100011001100110011001001

%e .000000000000111000111101000110110000

%e .0000000000001011100001001111001111001

%e .00000000000010010101000111000000101001

%e .000000000000011110001010000111010100010

%e .0000000000000110000110010111111011001011

%e .00000000000001001110111101000110101101101

%e .000000000000001111111110000000110000111000

%e .0000000000000011001110101101001100110100101

%e .00000000000000101001110011101010010111011101

%e .000000000000001000011101001010011111110000010

%e .0000000000000001101101011100111110010101011111

%e .00000000000000010110001000110010010010011100001

%e .000000000000000100011110100011010000101001000000

%e .0000000000000000111001111101001100010111100100001

%e .00000000000000001011101110001100110011100101100001

%e ...

%e the column sums of which form this sequence.

%e Thus, a(n) equals the number of 1-bits in column n in the binary expansions of Fibonacci(n)/2^n for n >= 1.

%Y Cf. A000045, A037093.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Apr 27 2018