cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A080090 a(n) is the index of the first occurrence of n in A080071, or 0 for those n>0 which never occur in A080071.

Original entry on oeis.org

0, 1, 2, 4, 10, 58, 43, 191, 246, 320, 2000, 1602, 4172, 7598, 21843, 36520, 27737
Offset: 0

Views

Author

Antti Karttunen, Jan 27 2003

Keywords

Comments

17 does not occur among the first 404631 terms of A080071.

Crossrefs

Cf. A080071.

A080070 Decimal encoding of parenthesizations produced by simple iteration starting from empty parentheses and where each successive parenthesization is obtained from the previous by reflecting it as a general tree/parenthesization, then adding an extra stem below the root and then reflecting the underlying binary tree.

Original entry on oeis.org

0, 10, 1010, 101100, 10110010, 1011100100, 101100110100, 10111001001100, 1011100110100010, 101110011010011000, 10110011101001100010, 1011110010011011000100, 101100111011010001100100
Offset: 0

Views

Author

Antti Karttunen, Jan 27 2003

Keywords

Comments

Corresponding Lisp/Scheme S-expressions are (), (()), (()()), (()(())), (()(())()), (()((())())), (()(())(()())), ...
Conjecture: only the terms in positions 0,1,2 and 4 are symmetric, i.e., A057164(A080068(n)) = A080068(n) (equivalently: A036044(A080069(n)) = A080069(n)) only when n is one of {0,1,2,4}. If this is true, then the formula given in A079438 is exact. I (AK) have checked this up to n=404631 with no other occurrence of a symmetric (general) tree.

Examples

			This demonstrates how to get the fourth term 10110010 from the 3rd term 101100. The corresponding binary and general trees plus parenthesizations are shown. The first operation reflects the general tree, the second adds a new stem under the root and the third reflects the underlying binary tree, which induces changes on the corresponding general tree:
..............................................
.....\/................\/\/..........\/\/.....
......\/......\/\/......\/............\/......
.....\/........\/........\/..........\/.......
......(A057164).(A057548)..(A057163)..........
........................o.....................
........................|.....................
........o.....o.........o...o.........o.......
........|.....|..........\./..........|.......
....o...o.....o...o.......o.........o.o.o.....
.....\./.......\./........|..........\|/......
......*.........*.........*...........*.......
..[()(())]..[(())()]..[((())())]..[()(())()]..
...101100....110010....11100100....10110010...

Links

Crossrefs

Compare to similar Wolframesque plots given in A122229, A122232, A122235, A122239, A122242, A122245. See also A079438, A080067, A080071, A057119.

Formula

a(n) = A007088(A080069(n)) = A063171(A080068(n)).

A179844 a(n) = A179752(A080068(n)).

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 3, 3, 3, 3, 4, 4, 4, 4, 3, 4, 6, 6, 5, 4, 5, 6, 7, 7, 6, 6, 6, 7, 8, 6, 6, 6, 6, 5, 7, 8, 8, 7, 7, 8, 9, 8, 5, 6, 9, 11, 10, 8, 6, 7, 9, 10, 9, 9, 10, 12, 9, 9, 9, 10, 9, 10, 12, 12, 13, 10, 8, 10, 10, 11, 11, 11, 12, 14, 14, 9, 7, 8, 10, 11, 13, 13, 14, 15, 14, 13, 11, 10
Offset: 1

Views

Author

Antti Karttunen, Aug 03 2010

Keywords

Comments

a(n) is also the local maximum of A179759 in range [(2*A000217(n-1))+1,2*A000217(n)].

Links

Crossrefs

Cf. also A080071, A154477, A179840 & A179845-A179847.

A179840 a(n) = A179751(A080068(n)).

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 4, 6, 6, 6, 7, 8, 8, 7, 7, 7, 10, 11, 10, 8, 8, 10, 12, 13, 12, 11, 11, 13, 14, 14, 11, 11, 11, 10, 11, 14, 15, 14, 13, 15, 17, 17, 13, 10, 15, 20, 21, 18, 14, 13, 15, 19, 19, 18, 16, 21, 21, 18, 16, 18, 17, 19, 21, 23, 25, 23, 17, 18, 20, 20, 21, 20, 23, 25, 28
Offset: 1

Views

Author

Antti Karttunen, Aug 03 2010

Keywords

Links

Crossrefs

Cf. also A080071, A154477, A179844 & A179841-A179843.

A154477 a(n) = A153240(A080068(n)).

Original entry on oeis.org

0, 0, 0, 1, 0, 3, 2, 0, 1, 7, 4, 9, 8, 6, 7, 5, -6, -10, 16, 15, 10, -11, -15, 16, 22, 19, 24, 23, 18, 14, 28, 25, 21, 23, 11, -7, -26, 35, 34, 29, -18, 39, 38, 9, -8, 38, 33, -31, -35, 42, 37, 31, 32, 51, 48, -46, 54, 51, 40, -43, 58, 55, 43, 61, 60, 58, 52, 65, 62, -2, 68
Offset: 0

Views

Author

Antti Karttunen, Jan 11 2009

Keywords

Comments

This sequence gives some indication of how well the terms of A080068 are balanced as general trees, which has some implications as to the correctness of A123050 (see comments at A080070).

Links

Crossrefs

See also A080071.
Showing 1-5 of 5 results.

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