A072764 -id:A072764 - cp's OEIS frontend

A072771 X-projection of the tabular N X N -> N bijection A072764 and Y-projection of its transpose A072766.

0, 0, 1, 0, 0, 1, 2, 3, 0, 0, 0, 0, 0, 1, 1, 2, 4, 5, 3, 6, 7, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 4, 9, 10, 5, 11, 12, 13, 3, 3, 6, 14, 15, 7, 16, 17, 18, 8, 19, 20, 21, 22, 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

Offset: 1

Antti Karttunen, Jun 12 2002

nonn

This corresponds to Lisp/Scheme function 'car' computed with respect to the lexicographical ordering of parenthesizations/planar binary trees (A014486), i.e. with planar binary trees this is equal to extracting the left subtree (from the root), with general parenthesizations equal to taking the first sub-parenthesization of the top-level list and with general plane trees equal to taking the leftmost branch of the tree (at the root).

A. Karttunen, Gatomorphisms (with the complete Scheme source)

A072772 Y-projection of the tabular N X N -> N bijection A072764 and X-projection of its transpose A072766.

Original entry on oeis.org

0, 1, 0, 2, 3, 1, 0, 0, 4, 5, 6, 7, 8, 2, 3, 1, 0, 0, 1, 0, 0, 0, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 4, 5, 6, 7, 8, 2, 3, 1, 0, 0, 1, 0, 0, 0, 2, 3, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44

Offset: 1

Antti Karttunen, Jun 12 2002

nonn

This corresponds to Lisp/Scheme function 'cdr' computed with respect to the lexicographical ordering of parenthesizations/planar binary trees (A014486), i.e. with planar binary trees this is equal to extracting the right subtree (from the root), with general parenthesizations equal to discarding the first sub-parenthesization of the top-level list and with general plane trees equal to discarding the leftmost branch from the root.

A. Karttunen, Gatomorphisms (with the complete Scheme source)

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

A072765 Inverse permutation to A072764.

Original entry on oeis.org

0, 1, 3, 2, 6, 10, 5, 4, 7, 15, 21, 28, 36, 45, 9, 14, 8, 11, 16, 12, 22, 29, 37, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 20, 27, 35, 44, 54, 13, 19, 17, 46, 56, 23, 67, 79, 92, 18, 25, 30, 106, 121, 38, 137, 154, 172, 47, 191, 211, 232, 254, 300

Offset: 0

Antti Karttunen, Jun 12 2002

nonn

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

Cf. A072767, A001477, A072771, A072772.

A072773 The upper triangular region of A072764.

Original entry on oeis.org

1, 3, 6, 7, 16, 42, 8, 19, 51, 52, 17, 44, 126, 127, 385, 18, 47, 135, 136, 413, 414, 20, 53, 154, 155, 475, 476, 477, 21, 56, 163, 164, 503, 504, 505, 506, 22, 60, 177, 178, 551, 552, 553, 554, 555, 45, 128, 390, 391, 1243, 1244, 1245, 1246, 1247, 4089, 46

Offset: 0

Antti Karttunen, Jun 12 2002

A. Karttunen, Gatomorphisms (with the complete Scheme source)

Cf. also A014486, A003056, A002262.

A057548 A014486-indices of Catalan mountain ranges with no sea-level valleys, i.e., the rooted plane general trees with root degree = 1.

Original entry on oeis.org

1, 3, 7, 8, 17, 18, 20, 21, 22, 45, 46, 48, 49, 50, 54, 55, 57, 58, 59, 61, 62, 63, 64, 129, 130, 132, 133, 134, 138, 139, 141, 142, 143, 145, 146, 147, 148, 157, 158, 160, 161, 162, 166, 167, 169, 170, 171, 173, 174, 175, 176, 180, 181, 183, 184, 185, 187, 188

Offset: 0

Antti Karttunen, Sep 07 2000

nonn

This sequence is induced by the unary form of function 'list' (present in Lisp and Scheme) when it acts on symbolless S-expressions encoded by A014486/A063171.

Index entries for the sequences induced by list functions of Lisp

We have A057515(A057548(n)) = 1 for all n. Row 0 of A072764. Column 1 of A085203. Cf. A057517, A057549, A057551.

a(n) = A080300(A057547(n)) = A069770(A072795(n)).

A081291 Complement of A072795.

Original entry on oeis.org

0, 3, 6, 7, 8, 14, 15, 16, 17, 18, 19, 20, 21, 22, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128

Offset: 0

Antti Karttunen, Mar 17 2003

nonn

This gives positions of those terms in A063171 which begin with digits 11... Also the elements of table A072764 which do not occur in the leftmost column. See the comment at A081292.

Cf. A072795, A014137, A081288.

Cf. A080300, A081289, A081290, A081292.

A014137(n) = sum(k=0, n, binomial(2*k, k)/(k+1););
A081288(n) = my(i=0); while(binomial(2*i, i)/(i+1) <= n, i++); i;
a(n) = if (n==0, 0,  n + A014137(A081288(n)-1)); \\ Michel Marcus, Apr 28 2020

a(0)=0, a(n) = n + A014137(A081288(n)-1).

a(n) = A080300(A081292(n)) = A081289(n) + n - A081290(n).

A072795 A014486-indices of the plane binary trees AND plane general trees whose left subtree is just a stick: \. thus corresponding to the parenthesizations whose first element (of the top-level list) is an empty parenthesization: ().

Original entry on oeis.org

1, 2, 4, 5, 9, 10, 11, 12, 13, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 197, 198, 199

Offset: 0

Antti Karttunen Jun 12 2002

nonn

This sequence is induced by the 'flipped form' of the function 'list': (define (flippedlist x) (cons '() x)) when it acts on symbolless S-expressions encoded by A014486/A063171.

Gives in A063171 positions of the terms which begin with digits 10...

Column 0 of A072764, row 0 of A072766, column 1 of A085201. Complement: A081291. Cf. A085223.

Range[0, Length[#]-1] + CatalanNumber[#] & [Flatten[Array[Table[#, CatalanNumber[#]] &, 7, 0]]] (* Paolo Xausa, Mar 01 2024 *)

a(n) = n + A000108(A072643(n)) = A069770(A057548(n)) = A080300(A083937(n))

A218787 a(n) = A014486-index for the n-th tendril of infinite beanstalk (A213730(n)), with the "lesser numbers to the left side" construction.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 2, 0, 8, 0, 0, 1, 0, 0, 1, 2, 0, 8, 0, 0, 1, 8, 0, 0, 3, 0, 2, 1, 0, 0, 0, 1, 2, 0, 8, 0, 0, 1, 8, 0, 0, 3, 0, 2, 1, 0, 8, 0, 0, 3, 0, 60, 0, 0, 172, 0, 2, 0, 1, 0, 0, 1, 2, 0, 8, 0, 0, 1, 8, 0, 0, 3, 0, 2, 1, 0, 8, 0, 0

Offset: 1

Antti Karttunen, Nov 11 2012

nonn

"Tendrils" of the beanstalk are the finite side-trees sprouting from its infinite trunk (see A179016) at the numbers given by A213730.

			A213730(9)=22, and from that branches 24 and 25 (as both A011371(24)=A011371(25)=22) and while 24 is a leaf (in A055938) the other branch 25 further branches to two leaves (as both A011371(28)=A011371(29)=25).
When we construct a binary tree from this in such a fashion that the lesser numbers go to the left, we obtain:
...........
...28...29.
.....\./...
..24..25...
...\ /.....
....22.....
...........
and the binary tree
........
...\./..
....*...
.\./....
..*.....
........
is located as A014486(2) in the normal encoding order of binary trees, thus a(9)=2.

A. Karttunen, Table of n, a(n) for n = 1..8727
A. Karttunen, Illustration of how binary trees (the second rightmost column) are encoded by A014486

These are the mirror-images of binary trees given in A218788, i.e. a(n) = A057163(A218788(n)). A218786 gives the sizes of these trees. Cf. A072764, A218609, A218611.

A218788 a(n) = A014486-index for the n-th tendril of infinite beanstalk (A213730(n)), with the "lesser numbers to the right side" construction.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 3, 0, 0, 0, 1, 3, 0, 4, 0, 0, 1, 0, 0, 1, 3, 0, 4, 0, 0, 1, 4, 0, 0, 2, 0, 3, 1, 0, 0, 0, 1, 3, 0, 4, 0, 0, 1, 4, 0, 0, 2, 0, 3, 1, 0, 4, 0, 0, 2, 0, 37, 0, 0, 110, 0, 3, 0, 1, 0, 0, 1, 3, 0, 4, 0, 0, 1, 4, 0, 0, 2, 0, 3, 1, 0, 4, 0, 0

Offset: 1

Antti Karttunen, Nov 11 2012

nonn

"Tendrils" of the beanstalk are the finite side-trees sprouting from its infinite trunk (see A179016) at the numbers given by A213730.

			A213730(9)=22, and from that branches 24 and 25 (as both A011371(24)=A011371(25)=22) and while 24 is a leaf (in A055938) the other branch 25 further branches to two leaves (as both A011371(28)=A011371(29)=25).
When we construct a binary tree from this in such a fashion that the larger numbers go to the left, we obtain:
..........
29...28...
..\./.....
...25..24.
....\./...
.....22...
..........
and the binary tree
.......
.\./...
..*....
...\./.
....*..
.......
is located as A014486(3) in the normal encoding order of binary trees, thus a(9)=3.

A. Karttunen, Table of n, a(n) for n = 1..8727
A. Karttunen, Illustration of how binary trees (the second rightmost column) are encoded by A014486

These are the mirror-images of binary trees given in A218787, i.e. a(n) = A057163(A218787(n)). A218786 gives the sizes of these trees. Cf. A072764, A218610, A218611.

cp's OEIS Frontend

A072771 X-projection of the tabular N X N -> N bijection A072764 and Y-projection of its transpose A072766.

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A072772 Y-projection of the tabular N X N -> N bijection A072764 and X-projection of its transpose A072766.

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

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A072765 Inverse permutation to A072764.

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A072773 The upper triangular region of A072764.

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A057548 A014486-indices of Catalan mountain ranges with no sea-level valleys, i.e., the rooted plane general trees with root degree = 1.

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A081291 Complement of A072795.

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PARI

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A072795 A014486-indices of the plane binary trees AND plane general trees whose left subtree is just a stick: \. thus corresponding to the parenthesizations whose first element (of the top-level list) is an empty parenthesization: ().

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A218787 a(n) = A014486-index for the n-th tendril of infinite beanstalk (A213730(n)), with the "lesser numbers to the left side" construction.

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A218788 a(n) = A014486-index for the n-th tendril of infinite beanstalk (A213730(n)), with the "lesser numbers to the right side" construction.

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Examples

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