cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A222601 Conjectured number of Fibonacci numbers with exactly n 0-bits in their binary representation.

Original entry on oeis.org

3, 5, 2, 4, 3, 1, 2, 3, 5, 1, 3, 5, 3, 3, 2, 1, 6, 1, 2, 3, 3, 2, 0, 3, 5, 4, 4, 3, 4, 2, 1, 2, 6, 1, 2, 2, 5, 4, 3, 5, 2, 2, 2, 1, 2, 2, 2, 5, 6, 3, 2, 2, 3, 1, 5, 1, 1, 0, 8, 4, 3, 3, 3, 3, 5, 4, 4, 2, 2, 2, 2, 3, 2, 6, 3, 0, 0, 2, 5, 5, 1, 6, 5, 0, 3, 5, 1
Offset: 0

Views

Author

T. D. Noe, Mar 08 2013

Keywords

Links

Crossrefs

Cf. A004685 (Fibonacci numbers in binary), A214853 (one 0-bit), A222602.
Cf. A222295, A222296.

Programs

  • Mathematica
    f = Fibonacci[Range[0,100]]; Table[Length[Select[f, Count[IntegerDigits[#, 2], 0] == n &]], {n, 0, 20}]

A222296 Irregular triangle read by rows: row n lists the Fibonacci numbers with exactly n 1's in their binary representation.

Original entry on oeis.org

0, 1, 1, 2, 8, 3, 5, 34, 144, 13, 21
Offset: 0

Views

Author

T. D. Noe, Feb 22 2013

Keywords

Comments

Besides those listed in Example section, there are no additional terms with small number of 1's in the first 10^12 Fibonacci numbers. In particular, if A000120(Fibonacci(n)) < 100, then n <= 319 or n > 10^12. - Charles R Greathouse IV, Mar 06 2014
For the theorem about S-units that Noam Elkies quotes (in the MathOverflow link), see Chapter 1 of Storey-Tijdemann, 1986. - N. J. A. Sloane, Jan 28 2017

Examples

			The irregular table begins
{0},
{1, 1, 2, 8},
{3, 5, 34, 144},
{13, 21, ...}.
It is conjectured that the previous (n=3) row is complete, and that the subsequent rows are:
{89, 610, 2584},
{55, 233, 4181},
{377, 10946, 46368, 75025},
{1597},
{987, 6765, 17711, 832040},
{121393, 2178309},
{39088169},
{28657, 196418, 317811, 1346269, 9227465},
{514229, 5702887, 14930352, 63245986, 4807526976},
{3524578, 2971215073}
...

References

  • T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge Tracts in Mathematics, 1986.

Links

Crossrefs

Cf. A004685 (Fibonacci numbers in binary), A221158 (weight 2), A222295, A222601, A222602, A222757, A222758.

Programs

  • Mathematica
    f = Fibonacci[Range[0,100]]; Table[Select[f, Total[IntegerDigits[#, 2]] == n &], {n, 0, 20}]
  • PARI
    row(n)=my(k=-1,t); while(1,t=fibonacci(k++); if(hammingweight(t)==n, print1(t", "))) \\ Charles R Greathouse IV, Mar 04 2014

Extensions

a(9)-a(10) from Noam D. Elkies, via Charles R Greathouse IV, Mar 04 2014
Truncated to established terms by Max Alekseyev, May 13 2014
Edited by Max Alekseyev, Sep 08 2016

A222757 Irregular table of conjectured indices of Fibonacci numbers with exactly n 0-bits in their binary representation.

Original entry on oeis.org

1, 2, 4, 0, 3, 5, 7, 10, 8, 16, 6, 11, 13, 14, 9, 17, 23, 20, 12, 15, 22, 27, 29, 18, 19, 21, 26, 28, 33, 24, 31, 46, 25, 34, 37, 40, 41, 30, 36, 44, 32, 35, 43, 39, 42, 49, 38, 45, 50, 55, 58, 62, 56, 51, 52, 47, 53, 65, 59, 63, 67, 48, 54, 57, 61, 73
Offset: 0

Views

Author

T. D. Noe, Mar 11 2013

Keywords

Comments

Every nonnegative integer appears.

Examples

			The irregular table begins
{1, 2, 4},
{0, 3, 5, 7, 10},
{8, 16},
{6, 11, 13, 14},
{9, 17, 23},
{20},
{12, 15},
{22, 27, 29},
{18, 19, 21, 26, 28},
{33},
{24, 31, 46},
{25, 34, 37, 40, 41}

Links

Crossrefs

Cf. A004685 (Fibonacci numbers in binary), A222601, A222602, A222758.

Programs

  • Mathematica
    nn = 100; f = Fibonacci[Range[0, nn]]; t2 = Transpose[{Range[0, nn], f}]; Table[Select[Range[nn + 1], Count[IntegerDigits[t2[[#, 2]], 2], 0] == n &] - 1, {n, 0, nn/5}]
Insert[Flatten[Module[{nn=100,dc},dc=DigitCount[Fibonacci[Range[nn]],2,0];Table[Position[dc,n],{n,0,30}]]],0,4] (* Harvey P. Dale, Mar 17 2024 *)

A222758 Irregular table of conjectured indices of Fibonacci numbers with exactly n 1-bits in their binary representation.

Original entry on oeis.org

0, 1, 2, 3, 6, 4, 5, 9, 12, 7, 8, 11, 15, 18, 10, 13, 19, 14, 21, 24, 25, 17, 16, 20, 22, 30, 26, 32, 38, 23, 27, 28, 31, 35, 29, 34, 36, 39, 48, 33, 47, 37, 42, 45, 68, 40, 43, 54, 57, 60, 41, 51, 52, 53, 44, 49, 50, 61, 66, 59, 64, 46, 56, 55, 71, 77
Offset: 0

Views

Author

T. D. Noe, Mar 11 2013

Keywords

Comments

Every nonnegative integer appears.

Examples

			The irregular table begins
{0},
{1, 2, 3, 6},
{4, 5, 9, 12},
{7, 8},
{11, 15, 18},
{10, 13, 19},
{14, 21, 24, 25},
{17},
{16, 20, 22, 30},
{26, 32},
{38},
{23, 27, 28, 31, 35}

Links

Crossrefs

Cf. A004685 (Fibonacci numbers in binary), A222601, A222602, A222757.

Programs

  • Mathematica
    nn = 100; f = Fibonacci[Range[0, nn]]; t2 = Transpose[{Range[0, nn], f}]; Table[Select[Range[nn + 1], Count[IntegerDigits[t2[[#, 2]], 2], 1] == n &] - 1, {n, 0, nn/5}]

A222295 Conjectured number of Fibonacci numbers with exactly n bits set in their binary representation.

Original entry on oeis.org

1, 4, 4, 2, 3, 3, 4, 1, 4, 2, 1, 5, 5, 2, 2, 2, 5, 4, 3, 2, 2, 2, 3, 5, 3, 3, 2, 4, 2, 1, 4, 3, 2, 3, 3, 1, 6, 3, 2, 3, 3, 4, 4, 5, 0, 0, 3, 3, 2, 2, 5, 4, 3, 1, 5, 2, 2, 2, 5, 7, 3, 0, 0, 1, 2, 7, 3, 3, 2, 4, 3, 1, 2, 4, 4, 2, 0, 3, 1, 3, 7, 3, 4, 1, 3, 4, 3
Offset: 0

Views

Author

T. D. Noe, Feb 22 2013

Keywords

Examples

			We set a(1) = 4 because Fib(1) = 1, Fib(2) = 1, Fib(3) = 2, and Fib(6) = 8.

Links

Crossrefs

Cf. A004685 (Fibonacci numbers in binary), A221158 (two bits set), A222296.
Cf. A011373 (number of bits set in each Fibonacci number).
Cf. A222601, A222602, A222757, A222758.

Programs

  • Mathematica
    f = Fibonacci[Range[0,500]]; Table[Length[Select[f, Total[IntegerDigits[#, 2]] == n &]], {n, 0, 87}]
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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