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

1,2

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.

Example

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.

Crossrefs

Cf. A000045, A037093.

Sequence in context: A094017 A092762 A017844 * A011156 A213478 A332782

Adjacent sequences: A303652 A303653 A303654 * A303656 A303657 A303658

Keyword

nonn

Author

Paul D. Hanna, Apr 27 2018

Status

approved