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 1-10 of 12 results. Next

A085204 Transpose of A085203.

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 3, 7, 5, 3, 4, 8, 12, 8, 4, 5, 17, 13, 21, 10, 5, 6, 18, 31, 22, 26, 13, 6, 7, 20, 32, 58, 27, 35, 15, 7, 8, 21, 34, 59, 73, 36, 40, 18, 8, 9, 22, 35, 62, 74, 100, 41, 49, 22, 9, 10, 45, 36, 63, 76, 101, 115, 50, 63, 24, 10, 11, 46, 87, 64, 77, 104, 116, 142, 64, 68, 27
Offset: 0

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Author

Antti Karttunen, Jun 23 2003

Keywords

A085225 Row 1 of A085203.

Original entry on oeis.org

1, 3, 5, 8, 10, 13, 15, 18, 22, 24, 27, 29, 32, 36, 38, 41, 43, 46, 50, 52, 55, 59, 64, 66, 69, 71, 74, 78, 80, 83, 85, 88, 92, 94, 97, 101, 106, 108, 111, 113, 116, 120, 122, 125, 127, 130, 134, 136, 139, 143, 148, 150, 153, 155, 158, 162, 164, 167, 171, 176, 178, 181
Offset: 0

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Formula

a(n) = A085203bi(n, 1).

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

Views

Author

Antti Karttunen, Sep 07 2000

Keywords

Comments

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.

Links

Crossrefs

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

Formula

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

A085201 Array A(x,y): Position of the concatenation of binary strings A014486(x) & A014486(y) in A014486, listed antidiagonalwise as A(0,0), A(1,0), A(0,1), A(2,0), A(1,1), A(0,2), ...

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 4, 4, 3, 4, 6, 9, 5, 4, 5, 9, 14, 10, 9, 5, 6, 11, 23, 15, 23, 10, 6, 7, 14, 28, 24, 37, 24, 11, 7, 8, 16, 37, 29, 65, 38, 25, 12, 8, 9, 19, 42, 38, 79, 66, 39, 26, 13, 9, 10, 23, 51, 43, 107, 80, 67, 40, 27, 23, 10, 11, 25, 65, 52, 121, 108, 81, 68, 41, 65, 24, 11
Offset: 0

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Comments

This table is induced by the 2-ary form of the list-function 'append' present in the programming languages like Lisp, Scheme and Prolog.

Links

Crossrefs

Transpose: A085202. Variant: A085203. Row 1: A085223, Column 1: A072795.
Cf. also A014486, A080300, A025581, A002262.

Formula

a(0, y)=y, a(x, y) = A072764bi(A072771(x), a(A072772(x), y))
a(x, y) = A080300(A085207bi(A014486(x), A014486(y))) = A085200(A085215bi(A071155(y), A071155(x)))

A085159 Permutation of natural numbers induced by the Catalan bijection gma085159 acting on symbolless S-expressions encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 5, 8, 9, 11, 14, 16, 19, 17, 20, 12, 10, 13, 21, 15, 18, 22, 23, 25, 28, 30, 33, 37, 39, 42, 44, 47, 51, 53, 56, 60, 45, 48, 54, 57, 61, 31, 34, 26, 24, 27, 35, 29, 32, 36, 58, 62, 40, 38, 41, 49, 43, 46, 50, 63, 52, 55, 59, 64, 65, 67, 70, 72, 75
Offset: 0

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Comments

This Catalan bijection rotates the interpretations (pp)-(rr) of Stanley, using the "rising slope" mapping illustrated in A085161.

Links

Crossrefs

Inverse: A085160. a(n) = A085161(A085160(A085161(n))) = A085169(A082315(A085170(n))) = A074684(A082315(A074683(n))) = A085173(A085173(n)). Occurs in A073200. Cf. also A085165-A085168, A086429. Scheme-function app-to-xrt given in A085203.
Number of cycles: A054357. Number of fixed points: A046698. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

A085163 Permutation of natural numbers induced by the Catalan bijection gma085163 acting on symbolless S-expressions encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 5, 8, 9, 11, 14, 16, 19, 17, 20, 12, 10, 18, 21, 15, 13, 22, 23, 25, 28, 30, 33, 37, 39, 42, 44, 56, 51, 53, 47, 60, 45, 48, 54, 57, 61, 31, 34, 26, 24, 46, 49, 38, 32, 59, 58, 62, 40, 29, 55, 35, 43, 27, 50, 63, 52, 41, 36, 64, 65, 67, 70, 72, 75
Offset: 0

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Links

Crossrefs

Inverse: A085164. a(n) = A085161(A057508(n)). Occurs in A073200. Cf. also A085171, A085172. Scheme-function app-to-xrt given in A085203.
Number of fixed points 1, 1, 2, 2, 3, 4, 7, 11, 21, 36, 71, ... in range [A014137(n-1)..A014138(n-1)] of this permutation appears to be A051920 after the initial two ones.

A085211 Array A(x,y): concatenation of the binary expansions of x & y in such a way that 'y' is inserted after the least significant 1-bit of 'x', followed by the remaining 0-bits, if any. Listed antidiagonalwise as A(0,0), A(1,0), A(0,1), A(2,0), A(1,1), A(0,2), ... Zero is expanded as an empty string.

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 3, 6, 6, 3, 4, 7, 12, 7, 4, 5, 12, 14, 14, 12, 5, 6, 11, 24, 15, 24, 13, 6, 7, 14, 22, 28, 28, 26, 14, 7, 8, 15, 28, 23, 48, 29, 28, 15, 8, 9, 24, 30, 30, 44, 52, 30, 30, 24, 9, 10, 19, 48, 31, 56, 45, 56, 31, 48, 25, 10, 11, 22, 38, 56, 60, 58, 46, 60, 56, 50, 26, 11
Offset: 0

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Examples

			A(8,3) = 56 = '1110000' in binary, is produced when '11' (binary expansion of 3) is inserted after the least significant (and only) 1-bit of '1000' (binary expansion of 8).

Crossrefs

Same array in binary: A085213. Transpose: A085212. Variant: A085207 (normal concatenation). Can be used to compute A085203.

Formula

a(0, y) = y, a(x, y) = A006519(x) * (y + ((2^A029837(y+1))*A000265(x))).

A085219 Array A(x,y): "rised concatenation" of factorial expansions of x & y, listed antidiagonalwise as A(0,0), A(1,0), A(0,1), A(2,0), A(1,1), A(0,2), ... Zero is expanded as an empty string.

Original entry on oeis.org

0, 1, 1, 2, 5, 2, 3, 15, 14, 3, 4, 17, 56, 15, 4, 5, 21, 62, 57, 22, 5, 6, 23, 80, 63, 88, 23, 6, 7, 57, 86, 81, 94, 89, 54, 7, 8, 59, 272, 87, 112, 95, 270, 55, 8, 9, 63, 278, 273, 118, 113, 294, 271, 56, 9, 10, 65, 296, 279, 424, 119, 390, 295, 272, 57, 10, 11, 69, 302, 297, 430
Offset: 0

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Comments

This is otherwise like A085215, except that to each digit in the factorial expansion of 'x' is added the most significant digit in the factorial expansion of 'y'.

Examples

			To get A(4,3) = 81 we take the factorial expansions of 4 (= '20') and 3 (= '11') and then we add 1 to each digit of the former to get '31', before concatenating them as '3111' (3*24+1*6+1*2+1*1 = 81). Similarly, for A(3,4) = 94 we add 2 to 3's expansion '11' to get '33' and then the concatenation yields '3320' (3*24+3*6+2*2=94). See A085221 for the corresponding factorial expansions.

Crossrefs

Transpose: A085220. Can be used to compute A085203. Cf. A000142, A007623, A084558, A025581, A002262.

A085168 Inverse permutation to A085167.

Original entry on oeis.org

0, 1, 3, 2, 7, 4, 8, 6, 5, 17, 9, 18, 14, 10, 20, 11, 21, 16, 12, 22, 19, 15, 13, 45, 23, 46, 37, 24, 48, 25, 49, 42, 26, 50, 51, 38, 27, 54, 28, 55, 39, 29, 57, 30, 58, 44, 31, 59, 47, 40, 32, 61, 33, 62, 53, 34, 63, 56, 43, 35, 64, 60, 52, 41, 36, 129, 65, 130, 107, 66
Offset: 0

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Links

Crossrefs

Inverse: A085167. a(n) = A069770(A085160(n)). Occurs in A073200. Cf. also A074679, A074680, A085203.

A085171 Permutation of natural numbers induced by the Catalan bijection gma085171 acting on symbolless S-expressions encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 5, 8, 9, 11, 14, 16, 19, 17, 20, 12, 10, 15, 21, 18, 13, 22, 23, 25, 28, 30, 33, 37, 39, 42, 44, 53, 51, 56, 47, 60, 45, 48, 54, 57, 61, 31, 34, 26, 24, 29, 40, 38, 43, 52, 58, 62, 49, 46, 55, 35, 32, 27, 41, 63, 59, 50, 36, 64, 65, 67, 70, 72, 75
Offset: 0

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Links

Crossrefs

Inverse: A085172. Cf. also A085163, A085164, A085203.
Showing 1-10 of 12 results. Next

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