A265754 -id:A265754 - cp's OEIS frontend

A266640 Reversed reduced frequency counts for A004001: a(n) = A265754(A054429(n)).

1, 2, 1, 3, 2, 1, 1, 4, 3, 2, 1, 2, 1, 1, 1, 5, 4, 3, 2, 1, 3, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 6, 5, 4, 3, 2, 1, 4, 3, 2, 1, 3, 2, 1, 2, 1, 1, 3, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 7, 6, 5, 4, 3, 2, 1, 5, 4, 3, 2, 1, 4, 3, 2, 1, 3, 2, 1, 2, 1, 1, 4, 3, 2, 1, 3, 2, 1, 2, 1, 1, 3, 2, 1, 2, 1, 1, 2, 1, 1, 1

Offset: 1

Antti Karttunen, Jan 22 2016

Deleting all 1's and decrementing the remaining terms by one gives the sequence back.

			Illustration how the sequence can be constructed by concatenating the reversed reduced frequency counts R_n of each successive level n of A004001-tree:
                              1
                             / \
                            2   1
                           /|\   \
              ____________3 2 1   1
             /    /    /  | |\ \   \
    ________4  __3    2   1 2 1 1   1
   / / / / /  / /|   /|   | |\ \ \   \
  5 4 3 2 1  3 2 1  2 1   1 2 1 1 1   1
etc.

Antti Karttunen, Table of n, a(n) for n = 1..8191
T. Kubo and R. Vakil, On Conway's recursive sequence, Discr. Math. 152 (1996), 225-252.

Cf. A004001, A054429.
Cf. A000079 (positions of records, where n appears for the first time).
Cf. A265754 (obtained from the mirror image of the same tree).

(define (A266640 n) (A265754 (A054429 n)))

a(n) = A265754(A054429(n)).
Other identities. For all n >= 0:
a(2^n) = n+1.

A265332 a(n) is the index of the column in A265901 where n appears; also the index of the row in A265903 where n appears.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 2, 4, 1, 1, 1, 2, 1, 2, 3, 5, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 6, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 7, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4

Offset: 1

Antti Karttunen, Jan 09 2016

nonn

If all 1's are deleted, the remaining terms are the sequence incremented. - after Franklin T. Adams-Watters Oct 05 2006 comment in A051135.

Ordinal transform of A162598.

			Illustration how the sequence can be constructed by concatenating the frequency counts Q_n of each successive level n of A004001-tree:
--
             1                                      Q_0 = (1)
             |
            _2__                                    Q_1 = (2)
           /    \
         _3    __4_____                             Q_2 = (1,3)
        /     /  |     \
      _5    _6  _7    __8___________                Q_3 = (1,1,2,4)
     /     /   / |   /  |  \        \
   _9    10  11 12  13  14  15___    16_________    Q_4 = (1,1,1,2,1,2,3,5)
  /     /   /  / |  /  / |   |\  \   | \  \  \  \
17    18  19 20 21 22 23 24 25 26 27 28 29 30 31 32
--
The above illustration copied from the page 229 of Kubo and Vakil paper (page 5 in PDF).

Antti Karttunen, Table of n, a(n) for n = 1..8192
T. Kubo and R. Vakil, On Conway's recursive sequence, Discr. Math. 152 (1996), 225-252.

Essentially same as A051135 apart from the initial term, which here is set as a(1)=1.
Cf. A004001, A265901, A265903.
Cf. A162598 (corresponding other index).
Cf. A265754.
Cf. also A267108, A267109, A267110.

terms = 120;
h[1] = 1; h[2] = 1;
h[n_] := h[n] = h[h[n - 1]] + h[n - h[n - 1]];
seq[nmax_] := seq[nmax] = (Length /@ Split[Sort @ Array[h, nmax, 2]])[[;; terms]];
seq[nmax = 2 terms];
seq[nmax += terms];
While[seq[nmax] != seq[nmax - terms], nmax += terms];
seq[nmax] (* Jean-François Alcover, Dec 19 2021 *)

(define (A265332 n) (if (= 1 n) 1 (A051135 n)))

a(1) = 1; for n > 1, a(n) = A051135(n).

A266348 a(1) = 1; for n > 1, a(n) = A004001(n+1) - A072376(n).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 1, 2, 3, 3, 4, 4, 4, 4, 1, 2, 3, 4, 4, 5, 6, 6, 7, 7, 7, 8, 8, 8, 8, 8, 1, 2, 3, 4, 5, 5, 6, 7, 8, 8, 9, 10, 10, 11, 11, 11, 12, 13, 13, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 10, 11, 12, 13, 13, 14, 15, 15, 16, 16, 16, 17, 18, 19, 19, 20, 21, 21

Offset: 1

Antti Karttunen, Jan 22 2016

When the terms are arranged as successively larger batches of 2^n, the terms A(n,k), k = 1 .. 2^n, on row n give the cumulative number of 1's encountered since the beginning of the row n of similarly organized irregular table A265754, up to and including the k-th term on that row:
1;
1, 1;
1, 2, 2, 2;
1, 2, 3, 3, 4, 4, 4, 4;
1, 2, 3, 4, 4, 5, 6, 6, 7, 7, 7, 8, 8, 8, 8, 8;
...

Antti Karttunen, Table of n, a(n) for n = 1..8191

Cf. A000523, A004001, A072376, A265754.

lim = 100; b[1] = 1; b[2] = 1; b[n_] := b[n] = b[b[n - 1]] + b[n - b[n - 1]]; s = CoefficientList[Series[1/(2 - 2 x) (2 x - x^2 + Sum[ 2^(k - 1) x^2^k, {k, Floor@ Log2@ lim}]), {x, 0, lim}], x]; {1}~Join~Table[b[n + 1] - s[[n + 1]], {n, 2, lim}] (* Michael De Vlieger, Jan 26 2016, after Robert G. Wilson v at A004001 *)

(define (A266348 n) (if (= 1 n) 1 (- (A004001 (+ 1 n)) (A072376 n))))

a(1) = 1; for n > 1, a(n) = A004001(n+1) - A072376(n) = A004001(n+1) - 2^(A000523(n)-1).

A266349 a(n) = 1 + A053644(n) - A004001(n+1) = 1 + A072376(n) - A266348(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 3, 2, 2, 1, 1, 1, 1, 8, 7, 6, 5, 5, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 16, 15, 14, 13, 12, 12, 11, 10, 9, 9, 8, 7, 7, 6, 6, 6, 5, 4, 4, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 32, 31, 30, 29, 28, 27, 27, 26, 25, 24, 23, 23, 22, 21, 20, 20, 19, 18, 18, 17, 17, 17, 16, 15, 14, 14, 13, 12, 12, 11, 11, 11, 10

Offset: 1

Antti Karttunen, Jan 22 2016

nonn

Used in a recursive formula of A265754.

Antti Karttunen, Table of n, a(n) for n = 1..8191

Cf. A004001, A053644, A072376, A266348, A265754, A266640.

b[1] = 1; b[2] = 1; b[n_] := b[n] = b[b[n - 1]] + b[n - b[n - 1]]; Table[1 + 2^(Ceiling@ Log2[n + 1] - 1) - b[n + 1], {n, 96}] (* Michael De Vlieger, Jan 26 2016, after Robert G. Wilson v at A004001 *)

a(n) = 1 + A053644(n) - A004001(n+1).

a(n) = 1 + A072376(n) - A266348(n).

A293959 Construct a triangle T(n,k) (0 <= k <= n) of strings of integers, where T(0,0) = {0}, T(n,n) = {n}, and otherwise T(n,k) is the concatenation of T(n-1,k-1) and T(n-1,k). The sequence is obtained by reading across the rows of the triangle, concatenating the successive strings.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 0, 1, 2, 3, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 4, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 2, 0, 0, 0, 1, 0, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 4

Offset: 0

N. J. A. Sloane, Nov 05 2017

The string T(n,k) contains binomial(n,k) numbers.

			The first few rows of the triangle (where the strings T(n,k) are shown without spaces for legibility) are:
0,
0,1,
0,01,2,
0,001,012,3,
0,0001,001012,0123,4,
0,00001,0001001012,0010120123,01234,5,
...

Subtracting 1 from each term gives A265754.

Cf. A007318.

cp's OEIS Frontend

A266640 Reversed reduced frequency counts for A004001: a(n) = A265754(A054429(n)).

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A265332 a(n) is the index of the column in A265901 where n appears; also the index of the row in A265903 where n appears.

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A266348 a(1) = 1; for n > 1, a(n) = A004001(n+1) - A072376(n).

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Formula

A266349 a(n) = 1 + A053644(n) - A004001(n+1) = 1 + A072376(n) - A266348(n).

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A293959 Construct a triangle T(n,k) (0 <= k <= n) of strings of integers, where T(0,0) = {0}, T(n,n) = {n}, and otherwise T(n,k) is the concatenation of T(n-1,k-1) and T(n-1,k). The sequence is obtained by reading across the rows of the triangle, concatenating the successive strings.

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