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.

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A329743 Number of compositions of n with runs-resistance n - 3.

Original entry on oeis.org

0, 0, 0, 1, 2, 6, 9, 16, 8
Offset: 0

Views

Author

Gus Wiseman, Nov 21 2019

Keywords

Comments

A composition of n is a finite sequence of positive integers with sum n.
For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined as the number of applications required to reach a singleton.

Examples

			The a(3) = 1 through a(8) = 8 compositions:
  (3)  (22)    (14)   (114)    (1123)    (12113)
       (1111)  (23)   (411)    (1132)    (12212)
               (32)   (1113)   (1141)    (13112)
               (41)   (1221)   (1411)    (21131)
               (131)  (2112)   (2122)    (21221)
               (212)  (3111)   (2212)    (31121)
                      (11112)  (2311)    (121112)
                      (11211)  (3211)    (211121)
                      (21111)  (11131)
                               (11212)
                               (11221)
                               (12211)
                               (13111)
                               (21211)
                               (111121)
                               (121111)
For example, repeatedly taking run-lengths starting with (1,2,1,1,3) gives (1,2,1,1,3) -> (1,1,2,1) -> (2,1,1) -> (1,2) -> (1,1) -> (2), which is 5 steps, and 5 = 8 - 3, so (1,2,1,1,3) is counted under a(8).

Crossrefs

Column k = n - 3 of A329744.
Column k = 3 of A329750.
Compositions with runs-resistance 2 are A329745.
Cf. A000740, A008965, A098504, A242882, A318928, A329746, A329747, A329767.

Programs

  • Mathematica
    runsres[q_]:=If[Length[q]==1,0,Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],runsres[#]==n-3&]],{n,10}]

A338271 a(n) is the number of compositions of n, b_1 + ... + b_t = n such that sqrt(b_1 + sqrt(b_2 + ... + sqrt(b_t)...)) is an integer.

Original entry on oeis.org

1, 0, 0, 2, 0, 2, 0, 2, 2, 4, 2, 6, 2, 8, 4, 14, 6, 20, 8, 28, 14, 44, 20, 66, 30, 96, 46, 146, 70, 220, 102, 326, 154, 490, 232, 740, 346, 1102, 520, 1652, 782, 2484, 1166, 3716, 1750, 5568, 2628, 8358, 3936, 12518, 5900, 18760, 8848, 28138, 13256, 42170
Offset: 1

Views

Author

Peter Kagey, Oct 19 2020

Keywords

Comments

a(n) <= Sum_{k=1..floor(sqrt(n)/2)} A338286(floor((n-4*k^2)/2)) when n is even.
a(n) <= Sum_{k=1..floor((sqrt(n) - 1)/2)} A338286(floor((n-4*k^2-4*k-1)/2)) when n is odd and greater than 1.

Examples

			(Let s(k) = sqrt(k) for brevity.)
For n = 14, the a(14) = 8 valid compositions are:
14 = 2+2+2+2+2+3+1 and 2 = s(2+s(2+s(2+s(2+s(2+s(3+s(1)))))))
14 = 1+7+2+3+1     and 2 = s(1+s(7+s(2+s(3+s(1)))))
14 = 2+1+7+3+1     and 2 = s(2+s(1+s(7+s(3+s(1)))))
14 = 2+2+1+8+1     and 2 = s(2+s(2+s(1+s(8+s(1)))))
14 = 2+2+2+2+2+4   and 2 = s(2+s(2+s(2+s(2+s(2+s(4))))))
14 = 1+7+2+4       and 2 = s(1+s(7+s(2+s(4))))
14 = 2+1+7+4       and 2 = s(2+s(1+s(7+s(4))))
14 = 2+2+1+9       and 2 = s(2+s(2+s(1+s(9))))

Links

Crossrefs

Cf. A000196, A338268.
Cf. A098504, A101391, A143862, A238351, A238686, A325676. A335942.

Formula

a(n) = Sum_{i=k..A000196(n)} A338268(n,k).

A329768 Number of finite sequences of positive integers whose sum minus runs-resistance is n.

Original entry on oeis.org

8, 17, 42, 104, 242, 541, 1212, 2664, 5731, 12314
Offset: 1

Views

Author

Gus Wiseman, Nov 21 2019

Keywords

Comments

A composition of n is a finite sequence of positive integers with sum n.
For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined as the number of applications required to reach a singleton.

Examples

			The a(1) = 8 and a(2) = 17 compositions whose sum minus runs-resistance is n:
  (1)        (2)
  (1,1)      (1,3)
  (1,2)      (3,1)
  (2,1)      (1,1,1)
  (1,1,2)    (1,1,3)
  (2,1,1)    (1,2,1)
  (1,1,2,1)  (1,2,2)
  (1,2,1,1)  (2,2,1)
             (3,1,1)
             (1,1,1,2)
             (1,1,3,1)
             (1,3,1,1)
             (2,1,1,1)
             (1,1,1,2,1)
             (1,2,1,1,1)
             (1,2,1,1,2)
             (2,1,1,2,1)

Links

Crossrefs

Column sums of A329750.
Cf. A000740, A008965, A098504, A242882, A318928, A329744, A329745, A329746, A329747, A329767.

A330029 Numbers whose binary expansion has cuts-resistance <= 2.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 18, 19, 20, 21, 22, 25, 26, 37, 38, 41, 42, 43, 44, 45, 50, 51, 52, 53, 74, 75, 76, 77, 82, 83, 84, 85, 86, 89, 90, 101, 102, 105, 106, 149, 150, 153, 154, 165, 166, 169, 170, 171, 172, 173, 178, 179, 180, 181, 202, 203
Offset: 1

Views

Author

Gus Wiseman, Nov 27 2019

Keywords

Comments

For the operation of shortening all runs by 1, cuts-resistance is defined to be the number of applications required to reach an empty word.
Also numbers whose binary expansion is a balanced word (see A027383 for definition).
Also numbers whose binary expansion has all run-lengths 1 or 2 and whose sequence of run-lengths has no odd-length run of 1's sandwiched between two 2's.

Examples

			The sequence of terms together with their binary expansions begins:
    0:
    1:        1
    2:       10
    3:       11
    4:      100
    5:      101
    6:      110
    9:     1001
   10:     1010
   11:     1011
   12:     1100
   13:     1101
   18:    10010
   19:    10011
   20:    10100
   21:    10101
   22:    10110
   25:    11001
   26:    11010
   37:   100101
   38:   100110

Crossrefs

Union of A000975 and A329862.
Balanced binary words are counted by A027383.
Compositions with cuts-resistance <= 2 are A330028.
Cuts-resistance of binary expansion is A319416.
Cf. A027383, A098504, A107907, A164707, A329860, A329861, A329863, A329865.

Programs

  • Mathematica
    degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
Select[Range[0,100],degdep[IntegerDigits[#,2]]<=2&]
Previous Showing 31-34 of 34 results.

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