A072768 -id:A072768 - cp's OEIS frontend

A072769 Differences between A071673 and A072768.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 1, 0, 0, 0, 0, 0, 1, 0, -1, 0, -1, 0, 1, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, 1, 0, 0, 0, 1, 0, -1, 0, 0, 1, 0, 0, -1, 0, 1, 0, 0, 1, 0, -1, 0, 0, 1, 0, 1, 0, 0, -1, 0, 1, 0, 1, 0, -1, 0, 0, 1, 0

Offset: 1

Antti Karttunen, Jun 12 2002

The term a(0) = A071673(0)-A072768(0) = 0-0 = 0 is not explicitly listed here as to get a better looking triangle.

(define (A072769 n) (- (A071673 n) (A072768 n)))

A072770 Triangle A072768 computed modulo 2.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 12 2002

Differs from A071674 first time at the position n=37, where A072770(37) = 0, while A071674(37) = 1.

The term a(0)=0 is not explicitly listed here as to get a better looking triangle.

a(n) = (A072769(n) + A071674(n)) mod 2.

(define (A072770 n) (modulo (A072768 n) 2))

A072764 Tabular N X N -> N bijection induced by Lisp/Scheme function 'cons' combining the two planar binary trees/general trees/parenthesizations encoded by A014486(X) and A014486(Y).

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 4, 8, 16, 14, 5, 17, 19, 42, 15, 9, 18, 44, 51, 43, 37, 10, 20, 47, 126, 52, 121, 38, 11, 21, 53, 135, 127, 149, 122, 39, 12, 22, 56, 154, 136, 385, 150, 123, 40, 13, 45, 60, 163, 155, 413, 386, 151, 124, 41, 23, 46, 128, 177, 164, 475, 414, 387, 152

Offset: 0

Antti Karttunen Jun 12 2002

Index entries for the sequences induced by list functions of Lisp
Index entries for sequences that are permutations of the natural numbers
A. Karttunen, Gatomorphisms (with the complete Scheme source)

Inverse permutation: A072765. a(n) = A069770(A072766(n)). Also transpose of A072766, i.e. a(n) = A072766(A038722(n)). The upper triangular region: A072773. Projection functions are A072771 ('car') & A072772 ('cdr'). The sizes of the corresponding Catalan structures: A072768. The first row: A057548, the first column: A072795, diagonal: A083938. Cf. also A080300, A025581, A002262.

a(0)=0 prepended by Sean A. Irvine, Oct 25 2024

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

A072766 Transpose of A072764, 'cons' with arguments swapped.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 5, 14, 16, 8, 9, 15, 42, 19, 17, 10, 37, 43, 51, 44, 18, 11, 38, 121, 52, 126, 47, 20, 12, 39, 122, 149, 127, 135, 53, 21, 13, 40, 123, 150, 385, 136, 154, 56, 22, 23, 41, 124, 151, 386, 413, 155, 163, 60, 45, 24, 107, 125, 152, 387, 414, 475, 164

Offset: 0

Antti Karttunen, Jun 12 2002

A. Karttunen, Gatomorphisms (with the complete Scheme source)
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A072767. a(n) = A069770(A072764(n)). Also transpose of A072764, i.e. a(n) = A072764(A038722(n)). Projection functions are A072772 & A072771. The sizes of the corresponding Catalan structures: A072768. The first column: A057548, the first row: A072795. Cf. also A025581, A002262.

a(0)=0 prepended by Sean A. Irvine, Oct 25 2024

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A072769 Differences between A071673 and A072768.

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A072770 Triangle A072768 computed modulo 2.

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A072764 Tabular N X N -> N bijection induced by Lisp/Scheme function 'cons' combining the two planar binary trees/general trees/parenthesizations encoded by A014486(X) and A014486(Y).

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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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A072766 Transpose of A072764, 'cons' with arguments swapped.

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