A336683 Sum of 2^k for all residues k found in the Fibonacci sequence mod n.

1, 3, 7, 15, 31, 63, 127, 175, 511, 1023, 1327, 4031, 7471, 16383, 32767, 43951, 127807, 238895, 502063, 1048575, 1319215, 2719023, 7798639, 10692015, 33554431, 61209903, 134217727, 259173375, 337649967, 1073741823, 1571892655, 2880154543, 5417565487, 15638470959

Offset

1,2

Comments

Row n of A079002 compactified as a binary number.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..3322

Michael De Vlieger, Plot of bits of a(n) beginning with 2^0 at left for 1 <= n <= 5000.

Formula

a(n) = Sum(2^k) for all k in row n of A189768.

a(n) = 2^(n+1) - 1 for n in A079002.

Example

a(1) = 1 by convention.
a(2) = 3 = 2^0 + 2^1, since the Fibonacci sequence mod 2 is {0,1,1} repeated, and 0 and 1 appear in the sequence.
a(8) = 175 = 2^0 + 2^1 + 2^2 + 2^3 + 2^5 + 2^7, since the Fibonacci sequence mod 8 is {0,1,1,2,3,5,0,5,5,2,7,1} repeated, and we are missing 4 and 6, leaving the exponents of 2 as shown.
Binary equivalents of first terms:
   n    a(n)   a(n) in binary
   --------------------------
   1      1                 1
   2      3                11
   3      7               111
   4     15              1111
   5     31             11111
   6     63            111111
   7    127           1111111
   8    175          10101111
   9    511         111111111
  10   1023        1111111111
  11   1327       10100101111
  12   4031      111110111111
  13   7471     1110100101111
  14  16383    11111111111111
  15  32767   111111111111111
  16  43951  1010101110101111
  ...

Mathematica

{1}~Join~Array[Block[{w = {0, 1}}, Do[If[SequenceCount[w, {0, 1}] == 1, AppendTo[w, Mod[Total@ w[[-2 ;; -1]], #]], Break[]], {i, 2, Infinity}]; Total[2^Union@ w]] &, 33, 2]
(* Second program: generate the first n terms using the plot in Links *)
With[{n = 34, img = ImageData@ ColorNegate@ Import["https://oeis.org/A336683/a336683.png"]}, Map[FromDigits[#, 2] &@ Drop[#, LengthWhile[#, # == 0 &]] &@ Reverse[IntegerPart[#]] &, img[[1 ;; n]]]] (* Michael De Vlieger, Oct 05 2020 *)

Crossrefs

Cf. A000045, A001175, A066853, A079002, A128924, A189768.

Sequence in context: A275715 A273672 A043740 * A283091 A277716 A116082

Adjacent sequences: A336680 A336681 A336682 * A336684 A336685 A336686

Keyword

nonn,easy

Author

Michael De Vlieger, Oct 04 2020

Status

approved