A072789 -id:A072789 - cp's OEIS frontend

A072790 Maximum position in A072789 where the value n occurs.

0, 1, 3, 13, 38, 230, 6902, 5963330, 4445170540277, 2469942641522759680444872, 762577081551553466590743598512392989564579443140, 72690475663460578305076472056787461439583062421562715705151023011944336192599271053989785936377

Offset: 0

Antti Karttunen, Jun 12 2002

nonn

Binary widths given by A072791. See the comment at A072787.

A072792 Triangle A072789 computed modulo 2.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1

Offset: 0

Antti Karttunen, Jun 12 2002

To get a cleaner looking table, the term a(0)=0 is not listed here. This is just for the sake of abstract art.

A071673 Sequence a(n) obtained by setting a(0) = 0; then reading the table T(x,y)=a(x)+a(y)+1 in antidiagonal fashion.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 3, 3, 4, 4, 3, 4, 4, 5, 4, 4, 4, 5, 5, 5, 5, 4, 4, 5, 6, 5, 6, 5, 4, 4, 5, 6, 6, 6, 6, 5, 4, 5, 5, 6, 6, 7, 6, 6, 5, 5, 5, 6, 6, 6, 7, 7, 6, 6, 6, 5, 4, 6, 7, 6, 7, 7, 7, 6, 7, 6, 4, 5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 5, 5, 5, 6, 6, 7, 8, 7, 7, 7, 8, 7, 6, 6, 5, 6, 6, 7, 6, 8, 8, 7, 7, 8, 8, 6, 7, 6, 6

Offset: 0

Antti Karttunen, May 30 2002

The fixed point of RASTxx transformation. The repeated applications of RASTxx starting from A072643 seem to converge toward this sequence. Compare to A072768 from which this differs first time at the position n=37, where A072768(37) = 4, while A071673(37) = 5.
Each term k occurs A000108(k) times, and maximal position where k occurs is A072638(k).
The size of each Catalan structure encoded by the corresponding terms in triangles A071671 & A071672 (i.e., the number of digits / 2), as obtained with the global ranking/unranking scheme presented in A071651-A071654.

			The first 15 rows of this irregular triangular table:
               0,
               1,
              2, 2,
             3, 3, 3,
            3, 4, 4, 3,
           4, 4, 5, 4, 4,
          4, 5, 5, 5, 5, 4,
         4, 5, 6, 5, 6, 5, 4,
        4, 5, 6, 6, 6, 6, 5, 4,
       5, 5, 6, 6, 7, 6, 6, 5, 5,
      5, 6, 6, 6, 7, 7, 6, 6, 6, 5,
     4, 6, 7, 6, 7, 7, 7, 6, 7, 6, 4,
    5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 5, 5,
   5, 6, 6, 7, 8, 7, 7, 7, 8, 7, 6, 6, 5,
  6, 6, 7, 6, 8, 8, 7, 7, 8, 8, 6, 7, 6, 6
etc.
E.g., we have
  a(1) = T(0,0) = a(0) + a(0) + 1 = 1,
  a(2) = T(1,0) = a(1) + a(0) + 1 = 2,
  a(3) = T(0,1) = a(0) + a(1) + 1 = 2,
  a(4) = T(2,0) = a(2) + a(0) + 1 = 3, etc.

Antti Karttunen, Table of n, a(n) for n = 0..10440 (rows 0..144 of the triangle, flattened)
N. J. A. Sloane, Transforms (Maple code for RASTxx transform)

Same triangle computed modulo 2: A071674.
Permutations of this sequence include: A072643, A072644, A072645, A072660, A072768, A072789, A075167.
Cf. also A000108, A002260, A004736, A002262, A025581, A072638.

up_to = 105;
A002260(n) = (n-binomial((sqrtint(8*n)+1)\2, 2)); \\ From A002260
A004736(n) = (1-n+(n=sqrtint(8*n)\/2)*(n+1)\2); \\ From A004736
A071673list(up_to) = { my(v=vector(1+up_to)); v[1] = 0; for(n=1,up_to,v[1+n] = 1 + v[A004736(n)] + v[A002260(n)]); (v); };
v071673 = A071673list(up_to);
A071673(n) = v071673[1+n]; \\ Antti Karttunen, Aug 17 2021

(define (A071673 n) (cond ((zero? n) n) (else (+ 1 (A071673 (A025581 (-1+ n))) (A071673 (A002262 (-1+ n)))))))

a(0) = 0, a(n) = 1 + a(A025581(n-1)) + a(A002262(n-1)) = 1 + a(A004736(n)) + a(A002260(n)).

Self-referential definition added Jun 03 2002

Term a(0) = 0 prepended and the Example-section amended by Antti Karttunen, Aug 17 2021

A072768 The RASTxx transformation of the sequence A072643.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 3, 4, 4, 3, 4, 4, 5, 4, 4, 4, 5, 5, 5, 5, 4, 4, 5, 6, 5, 6, 5, 4, 4, 5, 6, 6, 6, 6, 5, 4, 4, 5, 6, 6, 7, 6, 6, 5, 4, 5, 5, 6, 6, 7, 7, 6, 6, 5, 5, 5, 6, 6, 6, 7, 7, 7, 6, 6, 6, 5, 5, 6, 7, 6, 7, 7, 7, 7, 6, 7, 6, 5, 5, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 5, 5, 6, 7, 7, 8, 7, 7, 7, 7, 8, 7, 7, 6, 5

Offset: 1

Antti Karttunen, Jun 12 2002

Also, the sizes of the parenthesizations produced by 'cons' combination A072764 and its transpose A072766.
Differs from A071673 first time at the position n=37, where A072768(37) = 4, while A071673(37) = 5. RASTxx(A072768) differs from A071673 first time at the position n=704, which leads to conjecture that the repeated applications of RASTxx starting from A072643 converge towards A071673, the fixed point of RASTxx transformation.
Each value v occurs A000108(v) times. (The term a(0)=0 is not explicitly listed here as to get a better looking triangle).

A. Karttunen, Gatomorphisms (Includes the complete Scheme source for computing this sequence)
N. J. A. Sloane, Transforms (Maple code for RASTxx transform)

Same triangle computed modulo 2: A072770. Permutations: A072643, A071673, A072644, A072645, A072660, A072789. Cf. also A072769, A025581, A002262.

(define (A072768 n) (cond ((zero? n) n) (else (+ 1 (A072643 (A025581 (-1+ n))) (A072643 (A002262 (-1+ n)))))))

A072787 Permutation of natural numbers induced by reranking plane binary trees given in the standard lexicographic order (A014486) with an "arithmetic global ranking algorithm", using A072734 as the packing bijection N X N -> N.

Original entry on oeis.org

0, 1, 3, 2, 6, 5, 13, 8, 4, 14, 10, 36, 20, 9, 25, 19, 24, 11, 12, 18, 38, 16, 7, 44, 27, 209, 77, 21, 105, 66, 104, 28, 35, 65, 230, 54, 15, 34, 33, 75, 43, 26, 85, 50, 40, 37, 22, 31, 191, 67, 23, 51, 41, 69, 107, 68, 49, 92, 30, 29, 32, 56, 211, 46, 17, 299, 120, 5671

Offset: 0

Antti Karttunen, Jun 12 2002

nonn

This ranking scheme condenses the structures of the same size (cf. A072789) somewhat better than scheme presented in A072656 (which uses the N X N -> N bijection A072793). Compare the sequences A072790 and A072640 giving the max positions where the last structure with size n will occur in these orderings and the respective binary widths A072791 & A072642. However, by using the second or third power of the bijection A072734 one gets even better results in a certain range.

A. Karttunen, Gatomorphisms
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A072788. Cf. also A014486, A072734, A072789.

A072788 Inverse permutation to A072787.

Original entry on oeis.org

0, 1, 3, 2, 8, 5, 4, 22, 7, 13, 10, 17, 18, 6, 9, 36, 21, 64, 19, 15, 12, 27, 46, 50, 16, 14, 41, 24, 31, 59, 58, 47, 60, 38, 37, 32, 11, 45, 20, 191, 44, 52, 120, 40, 23, 78, 63, 148, 144, 56, 43, 51, 111, 108, 35, 106, 61, 196, 168, 131, 178, 153, 150, 115, 116, 33, 29

Offset: 0

Antti Karttunen, Jun 12 2002

A. Karttunen, Gatomorphisms
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A072787. Cf. also A072740, A072741, A072789.

cp's OEIS Frontend

A072790 Maximum position in A072789 where the value n occurs.

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A072792 Triangle A072789 computed modulo 2.

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A071673 Sequence a(n) obtained by setting a(0) = 0; then reading the table T(x,y)=a(x)+a(y)+1 in antidiagonal fashion.

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A072768 The RASTxx transformation of the sequence A072643.

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A072787 Permutation of natural numbers induced by reranking plane binary trees given in the standard lexicographic order (A014486) with an "arithmetic global ranking algorithm", using A072734 as the packing bijection N X N -> N.

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A072788 Inverse permutation to A072787.

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