A072644 -id:A072644 - cp's OEIS frontend

A072639 a(0) = 0, a(n) = Sum_{i=0..n-1} 2^((2^i)-1).

0, 1, 3, 11, 139, 32907, 2147516555, 9223372039002292363, 170141183460469231740910675754886398091, 57896044618658097711785492504343953926805133516280751251469702679711451218059

Offset: 0

Antti Karttunen, Jun 02 2002

nonn

Maximum position in A072644 where the value n occurs.
Also partial sums of A058891, i.e. the first differences are there. - R. J. Mathar, May 15 2007
A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Then a(n) is the minimum BII-number of a set-system with n distinct vertices. - Gus Wiseman, Jul 24 2019

Binary width of each term: A000079. Cf. A072638, A072640, A072654.
Cf. A058891.
Cf. A000120, A014221, A029931, A034797, A048793, A070939, A326031, A326702.

A072639 := proc(n) local i; add(2^((2^i)-1),i=0..(n-1)); end;

a[n_] := Sum[2^(2^i - 1), {i, 0, n - 1}]; Table[a[n], {n, 0, 9}] (* Jean-François Alcover, Mar 06 2016 *)

a(n) = if (n, sum(i=0, n-1, 2^((2^i)-1)), 0); \\ Michel Marcus, Mar 06 2016

A072643 Half of the binary width of the terms of A014486, the number of digits in A063171(n)/2.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jun 02 2002

Paolo Xausa, Table of n, a(n) for n = 0..10000

Each value v occurs A000108(v) times. The maximum position for value v to occur is A014138(v). Permutations: A071673, A072644, A072645, A072660. Cf. also A002024, A072649.

a[n_] := Module[{i, c, a}, i = c = 0; a = 1; While[n>c, a *= (4*i+2)/(i+2); i++; c += a]; i];
Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Dec 26 2017, from Sage code *)
Flatten[Array[Table[#, CatalanNumber[#]]&, 7, 0]] (* Paolo Xausa, Feb 13 2024 *)

def A072643(n) :
    i = c = 0; a = 1
    while n > c :
        a *= (4*i+2)/(2+i)
        i += 1; c += a
    return i
[A072643(n) for n in (0..100)] # Peter Luschny, Sep 07 2012

Sum_{n>=1} (-1)^(n+1)/a(n) = Sum_{n>=1} (-1)^(n+1)/(2^n-1) = 0.76449978034844420919... . - Amiram Eldar, Feb 18 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

A072660 Size of the parenthesizations obtained with the global ranking/unranking scheme A072656-A072659.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 3, 5, 4, 4, 4, 5, 4, 4, 6, 5, 5, 5, 5, 4, 6, 5, 4, 4, 5, 7, 6, 6, 6, 5, 5, 5, 6, 5, 5, 7, 6, 6, 6, 5, 4, 6, 5, 8, 7, 7, 7, 6, 7, 6, 6, 6, 6, 5, 7, 6, 5, 4, 6, 8, 7, 7, 7, 6, 6, 5, 7, 5, 5, 8, 6, 6, 9, 8, 8, 8, 7, 8, 7, 7, 9, 7, 7, 7, 6, 6, 5, 7, 5, 6, 8, 7, 7, 6, 6, 4, 7, 5, 9, 8

Offset: 0

Antti Karttunen, Jun 02 2002

nonn

Cf. A072657-A072659. Permutations: A071673, A072643, A072644, A072645.

A072634 Permutation of natural numbers induced by reranking plane binary trees given in the standard lexicographic order (A014486) with an "arithmetic global ranking algorithm", using A054238 as the pairing function N X N -> N.

Original entry on oeis.org

0, 1, 3, 2, 11, 9, 4, 6, 5, 139, 131, 33, 41, 35, 12, 10, 8, 70, 66, 7, 17, 21, 18, 32907, 32779, 2051, 2179, 2059, 161, 137, 129, 8233, 8201, 43, 515, 547, 521, 140, 132, 34, 42, 36, 16, 14, 72, 16454, 16390, 68, 1026, 1090, 1030, 15, 13, 19, 81, 69, 23, 65

Offset: 0

Antti Karttunen, Jun 02 2002

nonn

Antti Karttunen, Alternative Catalan Orderings
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A072635.
Cf. A014486, A000695, A054238, A071651, A072636, A072646, A072656, A072658, A072644.
Cf. also A296689.

;; Functions below show the essential idea:
(define A072634 (lexrank->arithrank-bijection packA054238))
(define (lexrank->arithrank-bijection packfun) (lambda (n) (rank-bintree (binexp->parenthesization (A014486 n)) packfun)))
(define (rank-bintree bt packfun) (cond ((not (pair? bt)) 0) (else (1+ (packfun (rank-bintree (car bt) packfun) (rank-bintree (cdr bt) packfun))))))
(define (packA054238 x y) (+ (A000695 x) (* 2 (A000695 y))))

A072637 Inverse permutation to A072636.

Original entry on oeis.org

0, 1, 2, 3, 6, 4, 5, 14, 15, 7, 16, 8, 19, 42, 43, 51, 52, 11, 9, 39, 37, 10, 28, 38, 112, 123, 121, 151, 149, 122, 376, 150, 466, 20, 53, 17, 44, 154, 155, 126, 127, 18, 47, 54, 156, 135, 136, 480, 481, 477, 475, 387, 385, 476, 1531, 386, 1234, 415, 413, 1542, 1540

Offset: 0

Antti Karttunen, Jun 02 2002

nonn

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

A072644 gives the size of the corresponding parenthesizations, i.e. A072644(n) = A029837(A014486(A072637(n))+1)/2 [A029837(n+1) gives the binary width of n].

A072637(n) = A057163(A072635(n)). Cf. also A014486, A059905, A059906, A071654, A072646.

A072645 Size of the parenthesizations obtained with the global ranking/unranking scheme A072646/A072647.

Original entry on oeis.org

0, 1, 2, 3, 2, 4, 3, 3, 5, 3, 4, 4, 4, 7, 4, 4, 5, 3, 6, 5, 5, 9, 4, 5, 5, 5, 6, 4, 5, 7, 6, 6, 6, 5, 10, 5, 4, 5, 4, 5, 6, 6, 6, 4, 5, 6, 5, 8, 7, 7, 7, 6, 8, 6, 7, 11, 5, 6, 5, 6, 6, 5, 4, 6, 7, 6, 6, 6, 8, 5, 6, 7, 5, 7, 6, 6, 10, 8, 7, 7, 7, 7, 7, 8, 8, 7, 7, 8, 6, 12, 6, 5, 6, 4, 7, 7, 7, 8, 5, 6, 5, 6

Offset: 0

Antti Karttunen, Jun 02 2002

nonn

Each value v occurs A000108(v) times. The maximum position for value v to occur is A072654(v). Permutations: A071673, A072643, A072644, A072660.

A072635 Inverse permutation to A072634.

Original entry on oeis.org

0, 1, 3, 2, 6, 8, 7, 19, 16, 5, 15, 4, 14, 52, 43, 51, 42, 20, 22, 53, 60, 21, 61, 56, 179, 155, 178, 154, 177, 164, 557, 163, 556, 11, 39, 13, 41, 151, 123, 153, 125, 12, 40, 33, 117, 152, 124, 471, 381, 477, 553, 479, 555, 505, 1797, 507, 1799, 478, 554, 1536

Offset: 0

Antti Karttunen, Jun 02 2002

nonn

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

A072644 gives the size of the corresponding parenthesizations, i.e. A072644(n) = A029837(A014486(A072635(n))+1)/2 [A029837(n+1) gives the binary width of n].

A072635(n) = A057163(A072637(n)). Cf. also A014486, A059905, A059906, A071652, A072647, A072637.

A072636 Permutation of natural numbers induced by reranking plane binary trees given in the standard lexicographic order (A014486) with an "arithmetic global ranking algorithm", using packA054238tr as the packing bijection N X N -> N.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 4, 9, 11, 18, 21, 17, 66, 70, 7, 8, 10, 35, 41, 12, 33, 131, 139, 261, 274, 258, 4101, 4117, 22, 65, 69, 1030, 1090, 81, 1026, 16390, 16454, 20, 23, 19, 68, 72, 13, 14, 36, 521, 547, 42, 515, 8201, 8233, 15, 16, 34, 43, 129, 132, 137, 2059, 2179

Offset: 0

Antti Karttunen, Jun 02 2002

nonn

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

Inverse permutation: A072637. Cf. also A014486, A000695, A054238, A071651, A072634, A072646, A072656, A072658, A072644.

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)))))))

cp's OEIS Frontend

A072639 a(0) = 0, a(n) = Sum_{i=0..n-1} 2^((2^i)-1).

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A072643 Half of the binary width of the terms of A014486, the number of digits in A063171(n)/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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A072660 Size of the parenthesizations obtained with the global ranking/unranking scheme A072656-A072659.

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A072634 Permutation of natural numbers induced by reranking plane binary trees given in the standard lexicographic order (A014486) with an "arithmetic global ranking algorithm", using A054238 as the pairing function N X N -> N.

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A072637 Inverse permutation to A072636.

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A072645 Size of the parenthesizations obtained with the global ranking/unranking scheme A072646/A072647.

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A072635 Inverse permutation to A072634.

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A072636 Permutation of natural numbers induced by reranking plane binary trees given in the standard lexicographic order (A014486) with an "arithmetic global ranking algorithm", using packA054238tr as the packing bijection N X N -> N.

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A072768 The RASTxx transformation of the sequence A072643.

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