A303655 Bit column sums in the binary expansions of Fibonacci(n)/2^n for n >= 1.
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, 29, 32, 34, 32, 37, 34, 33, 43, 30, 37, 41, 46, 43, 44, 42, 52, 45, 51, 50, 53, 50, 51, 49, 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
Offset: 1
Keywords
Examples
The binary expansions of Fibonacci(n)/2^n for n >= 1 begin: .1 .01 .010 .0011 .00101 .001000 .0001101 .00010101 .000100010 .0000110111 .00001011001 .000010010000 .0000011101001 .00000101111001 .000001001100010 .0000001111011011 .00000011000111101 .000000101000011000 .0000001000001010101 .00000001101001101101 .000000010101011000010 .0000000100010100101111 .00000000110111111110001 .000000001011010100100000 .0000000010010010100010001 .00000000011101101000110001 .000000000101111111101000010 .0000000001001101100101110011 .00000000001111101100010110101 .000000000011001011001000101000 .0000000000101001000101011011101 .00000000001000010011110100000101 .000000000001101011100011111100010 .0000000000010101110000010011100111 .00000000000100011001100110011001001 .000000000000111000111101000110110000 .0000000000001011100001001111001111001 .00000000000010010101000111000000101001 .000000000000011110001010000111010100010 .0000000000000110000110010111111011001011 .00000000000001001110111101000110101101101 .000000000000001111111110000000110000111000 .0000000000000011001110101101001100110100101 .00000000000000101001110011101010010111011101 .000000000000001000011101001010011111110000010 .0000000000000001101101011100111110010101011111 .00000000000000010110001000110010010010011100001 .000000000000000100011110100011010000101001000000 .0000000000000000111001111101001100010111100100001 .00000000000000001011101110001100110011100101100001 ... the column sums of which form this sequence. Thus, a(n) equals the number of 1-bits in column n in the binary expansions of Fibonacci(n)/2^n for n >= 1.
Links
- Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..800 from Paul D. Hanna)
Formula
Sum_{n>=1} a(n) / 2^n = 2.