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

A073294 Permutation A069767 applied four times or permutation A073290 applied twice.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 9, 11, 13, 12, 14, 15, 16, 18, 17, 19, 20, 22, 21, 26, 27, 25, 24, 23, 29, 28, 33, 35, 36, 30, 34, 32, 31, 38, 37, 39, 41, 40, 42, 43, 47, 49, 50, 44, 48, 46, 45, 51, 52, 53, 55, 54, 60, 61, 63, 64, 56, 57, 62, 59, 58, 73, 74, 76, 77, 78, 72, 75
Offset: 0

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Author

Antti Karttunen, Jun 25 2002

Keywords

Links

Crossrefs

Inverse permutation: A073295. Cf. A073290-A073299. Occurs in A073200 as row 142647371.

Formula

a(n) = A069767(A069767(A069767(A069767(n)))) = A073290(A073290(n)).

A073296 Permutation A069767 applied five times or composition of the permutations A073290 & A073292.

Original entry on oeis.org

0, 1, 3, 2, 7, 8, 6, 5, 4, 18, 17, 20, 22, 21, 16, 19, 15, 13, 12, 14, 11, 9, 10, 49, 50, 48, 46, 45, 55, 54, 61, 63, 64, 57, 62, 59, 58, 47, 44, 53, 60, 56, 43, 52, 41, 35, 36, 40, 34, 32, 31, 42, 51, 39, 33, 30, 37, 28, 24, 23, 38, 29, 25, 27, 26, 142, 143, 146, 147, 148
Offset: 0

Views

Author

Antti Karttunen, Jun 25 2002

Keywords

Links

Crossrefs

Inverse permutation: A073297. Cf. A073290-A073299. Occurs in A073200 as row 36169536789029067.

Formula

a(n) = A069767^5(n) = A073290(A073292(n)).

A073298 Permutation A069767 applied six times or permutation A073290 "cubed" or permutation A073292 "squared".

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 6, 8, 7, 13, 12, 11, 9, 10, 15, 14, 19, 22, 21, 16, 20, 17, 18, 35, 36, 34, 32, 31, 33, 30, 28, 24, 23, 29, 25, 27, 26, 41, 40, 39, 37, 38, 52, 51, 60, 63, 64, 56, 62, 59, 58, 43, 42, 53, 61, 57, 44, 54, 46, 45, 47, 55, 48, 50, 49, 100, 101, 104, 105, 106
Offset: 0

Views

Author

Antti Karttunen, Jun 25 2002

Keywords

Links

Crossrefs

Inverse permutation: A073299. Cf. A073290-A073297. Occurs in A073200 as row 2596188043348680205834878944583883.

Formula

a(n) = A069767^6(n) = A073290(A073290(A073290(n))) = A073292(A073292(n))

A073200 Number of simple Catalan bijections of type B.

Original entry on oeis.org

0, 1, 0, 3, 1, 0, 2, 2, 1, 0, 7, 3, 3, 1, 0, 8, 4, 2, 3, 1, 0, 6, 6, 8, 2, 3, 1, 0, 4, 5, 7, 7, 2, 3, 1, 0, 5, 7, 6, 6, 8, 2, 3, 1, 0, 17, 8, 5, 8, 7, 7, 2, 2, 1, 0, 18, 9, 4, 4, 6, 8, 7, 3, 3, 1, 0, 20, 10, 22, 5, 5, 5, 8, 4, 2, 2, 1, 0, 21, 14, 21, 17, 4, 4, 6, 5, 8, 3, 3, 1, 0
Offset: 0

Views

Author

Antti Karttunen, Jun 25 2002

Keywords

Comments

Each row is a permutation of nonnegative integers induced by a Catalan bijection (constructed as explained below) acting on the parenthesizations/plane binary trees as encoded and ordered by A014486/A063171.
The construction process is akin to the constructive mapping of primitive recursive functions to N: we have two basic primitives, A069770 (row 0) and A072796 (row 1), of which the former swaps the left and the right subtree of a binary tree and the latter exchanges the positions of the two leftmost subtrees of plane general trees, unless the tree's degree is less than 2, in which case it just fixes it. From then on, the even rows are constructed recursively from any other Catalan bijection in this table, using one of the five allowed recursion types:
0 - Apply the given Catalan bijection and then recurse down to both subtrees of the new binary tree obtained. (last decimal digit of row number = 2)
1 - First recurse down to both subtrees of the old binary tree and only after that apply the given Catalan bijection. (last digit = 4)
2 - Apply the given Catalan bijection and then recurse down to the right subtree of the new binary tree obtained. (last digit = 6)
3 - First recurse down to the right subtree of old binary tree and only after that apply the given Catalan bijection. (last digit = 8)
4 - First recurse down to the left subtree of old binary tree, after that apply the given Catalan bijection and then recurse down to the right subtree of the new binary tree. (last digit = 0)
The odd rows > 2 are compositions of the rows 0, 1, 2, 4, 6, 8, ... (i.e. either one of the primitives A069770 or A072796, or one of the recursive compositions) at the left hand side and any Catalan bijection from the same array at the right hand side. See the scheme-functions index-for-recursive-sgtb and index-for-composed-sgtb how to compute the positions of the recursive and ordinary compositions in this table.

Links

  • A. Karttunen, Gatomorphisms (With the complete source and explanation)

Crossrefs

Four other tables giving the corresponding cycle-counts: A073201, counts of the fixed elements: A073202, the lengths of the largest cycles: A073203, the LCM's of all the cycles: A073204. The ordinary compositions are encoded using the N X N -> N bijection A054238 (which in turn uses the bit-interleaving function A000695).
The first 21 rows of this table:.
Row 0: A069770. Row 1: A072796. Row 2: A057163. Row 3: A073269, Row 4: A057163 (duplicate), Row 5: A073270, Row 6: A069767, Row 7: A001477 (identity perm.), Row 8: A069768, Row 9: A073280.
Row 10: A069770 (dupl.), Row 11: A072796 (dupl.), Row 12: A057511, Row 13: A073282, Row 14: A057512, Row 15: A073281, Row 16: A057509, Row 17: A073280 (dupl.), Row 18: A057510, Row 19: A073283, Row 20: A073284.
Other Catalan bijection-induced EIS-permutations which occur in this table. Only the first known occurrence is given. Involutions are marked with *, others paired with their inverse:.
Row 164: A057164*, Row 168: A057508*, Row 179: A072797*.
Row 41: A073286 - Row 69: A073287. Row 105: A073290 - Row 197: A073291. Row 416: A073288 - Row 696: A073289.
Row 261: A057501 - Row 521: A057502. Row 2618: A057503 - Row 5216: A057504. Row 2614: A057505 - Row 5212: A057506.
Row 10435: A073292 - Row ...: A073293. Row 17517: A057161 - Row ...: A057162.
For a more practical enumeration system of (some) Catalan automorphisms see table A089840 and its various "recursive derivations".

A069767 Signature-permutation of Catalan bijection "Knick".

Original entry on oeis.org

0, 1, 3, 2, 7, 8, 6, 5, 4, 17, 18, 20, 21, 22, 16, 19, 15, 12, 13, 14, 11, 10, 9, 45, 46, 48, 49, 50, 54, 55, 57, 58, 59, 61, 62, 63, 64, 44, 47, 53, 56, 60, 43, 52, 40, 31, 32, 41, 34, 35, 36, 42, 51, 39, 30, 33, 38, 29, 26, 27, 37, 28, 25, 24, 23, 129, 130, 132, 133, 134
Offset: 0

Views

Author

Antti Karttunen, Apr 16 2002; entry revised Dec 20 2008

Keywords

Comments

This automorphism of binary trees first swaps the left and right subtree of the root and then proceeds recursively to the (new) right subtree, to do the same operation there. This is one of those Catalan bijections which extend to a unique automorphism of the infinite binary tree, which in this case is A153141. See further comments there.
This bijection, Knick, is a SPINE-transformation of the simple swap: SPINE(*A069770) (i.e., row 1 of A122203). Furthermore, Knick and Knack (the inverse, *A069768) have a special property, that FORK and KROF transforms (explained in A122201 and A122202) transform them to their own inverses, i.e., to each other: FORK(Knick) = KROF(Knick) = Knack and FORK(Knack) = KROF(Knack) = Knick, thus this occurs also as a row 1 in A122287 and naturally, the double-fork fixes both, e.g., FORK(FORK(Knick)) = Knick. There are also other peculiar properties.
Note: the name in Finnish is "Niks".

References

  • A. Karttunen, paper in preparation.

Links

Crossrefs

Inverse permutation: "Knack", A069768. "n-th powers" (i.e. n-fold applications), from n=2 to 6: A073290, A073292, A073294, A073296, A073298.
In range [A014137(n-1)..A014138(n-1)] of this permutation, the number of cycles is A073431, number of fixed points: A036987 (Fixed points themselves: A084108), Max. cycle size & LCM of all cycle sizes: A011782. See also: A074080.
A127302(a(n)) = A127302(n) for all n. a(n) = A057508(A057161(n)) = A057161(A069769(n)).
Row 1 of A122203 and A122287, row 15 of A122286 and A130403, row 6 of A073200.
See also bijections A073286, A082345, A082348, A082349, A130341.

A073267 Number of compositions (ordered partitions) of n into exactly two powers of 2.

Original entry on oeis.org

0, 0, 1, 2, 1, 2, 2, 0, 1, 2, 2, 0, 2, 0, 0, 0, 1, 2, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Antti Karttunen, Jun 25 2002

Keywords

Comments

Starting with 1 = self-convolution of A036987, the characteristic function of the powers of 2. [Gary W. Adamson, Feb 23 2010]

Examples

			For 2 there is only composition {1+1}, for 3 there is {1+2, 2+1}, for 4 {2+2}, for 5 {1+4, 4+1}, for 6 {2+4,4+2}, for 7 none, thus a(2)=1, a(3)=2, a(4)=1, a(5)=2, a(6)=2 and a(7)=0.

Links

Crossrefs

The second row of the table A073265. The essentially same sequence 1, 1, 2, 1, 2, 2, 0, 1, ... occurs for first time in A073202 as row 105 (the fix count sequence of A073290). The positions of 1's for n > 1 is given by the characteristic function of A000079, i.e. A036987 with offset 1 instead of 0 and the positions of 2's is given by A018900. Cf. also A023359.
Cf. A036987. [Gary W. Adamson, Feb 23 2010]

Programs

  • Haskell
    a073267 n = sum $ zipWith (*) a209229_list $ reverse $ take n a036987_list
-- Reinhard Zumkeller, Mar 07 2012
  • Maple
    f:= proc(n) local d;
d:= convert(convert(n,base,2),`+`);
if d=2 then 2 elif d=1 then 1 else 0 fi
end proc:
0, 0, seq(f(n),n=2..100); # Robert Israel, Jul 07 2016
  • Mathematica
    Table[Count[Map[{#, n - #} &, Range[0, n]], k_ /; Times @@ Boole@ Map[IntegerQ@ Log2@ # &, k] == 1], {n, 0, 88}] (* Michael De Vlieger, Jul 08 2016 *)
  • PARI
    N=166; x='x+O('x^N);
v=Vec( 'a0 + sum(k=0,ceil(log(N)/log(2)), x^(2^k) )^2 );
v[1] -= 'a0;  v
/* Joerg Arndt, Oct 21 2012 */
  • Python
    def A073267(n): return m if n>1 and (m:=n.bit_count())<3 else 0 # Chai Wah Wu, Oct 30 2024

Formula

G.f.: (Sum_{k>=0} x^(2^k) )^2. - Vladeta Jovovic, Mar 28 2005
a(n+1) = A000108(n) mod 4, n>=1 [Theorem 2.3 of Eu et al.]. - R. J. Mathar, Feb 27 2008
a(n) = sum (A209229(k)*A036987(n-k): k = 0..n), convolution of characteristic functions of 2^n and 2^n-1. [Reinhard Zumkeller, Mar 07 2012]
a(n+2) = A000168(n) mod 4. - John M. Campbell, Jul 07 2016

A073299 Permutation A069768 applied six times or permutation A073291 "cubed" or permutation A073293 "squared".

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 6, 8, 7, 12, 13, 11, 10, 9, 15, 14, 19, 21, 22, 16, 20, 18, 17, 32, 31, 34, 36, 35, 30, 33, 29, 27, 26, 28, 25, 23, 24, 40, 41, 39, 38, 37, 52, 51, 56, 59, 58, 60, 62, 64, 63, 43, 42, 53, 57, 61, 47, 55, 50, 49, 44, 54, 48, 45, 46, 92, 91, 90, 87, 88, 97, 96
Offset: 0

Views

Author

Antti Karttunen, Jun 25 2002

Keywords

Links

Crossrefs

Inverse permutation: A073298. Cf. A073290-A073297.

Formula

a(n) = A069768^6(n) = A073291(A073291(A073291(n))) = A073293(A073293(n))

A073292 Permutation A069767 applied three times ("cubed").

Original entry on oeis.org

0, 1, 3, 2, 8, 7, 6, 4, 5, 21, 22, 20, 18, 17, 19, 16, 14, 10, 9, 15, 11, 13, 12, 58, 59, 62, 63, 64, 57, 61, 55, 49, 50, 54, 48, 46, 45, 56, 60, 53, 47, 44, 51, 42, 38, 26, 27, 37, 25, 24, 23, 52, 43, 39, 29, 28, 41, 33, 35, 36, 40, 30, 34, 32, 31, 170, 171, 174, 175, 176
Offset: 0

Views

Author

Antti Karttunen, Jun 25 2002

Keywords

Links

Crossrefs

Inverse permutation: A073293. Occurs for first time in A073200 as row 10435. Cf. A073290-A073299.

Formula

a(n) = A069767(A069767(A069767(n))).

A073291 Permutation A069768 applied twice ("squared").

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 6, 8, 7, 13, 12, 11, 9, 10, 15, 14, 19, 22, 21, 16, 20, 17, 18, 36, 35, 34, 31, 32, 33, 30, 28, 23, 24, 29, 25, 26, 27, 41, 40, 39, 37, 38, 52, 51, 60, 64, 63, 56, 62, 58, 59, 43, 42, 53, 61, 57, 44, 54, 45, 46, 47, 55, 48, 49, 50, 106, 105, 104, 100, 101
Offset: 0

Views

Author

Antti Karttunen, Jun 25 2002

Keywords

Links

Crossrefs

Inverse permutation: A073290. Occurs for first time in A073200 as row 197. Cf. A073292-A073299.

Formula

a(n) = A069768(A069768(n)).

A073293 Permutation A069768 applied three times ("cubed").

Original entry on oeis.org

0, 1, 3, 2, 7, 8, 6, 5, 4, 18, 17, 20, 22, 21, 16, 19, 15, 13, 12, 14, 11, 9, 10, 50, 49, 48, 45, 46, 55, 54, 61, 64, 63, 57, 62, 58, 59, 47, 44, 53, 60, 56, 43, 52, 41, 36, 35, 40, 34, 31, 32, 42, 51, 39, 33, 30, 37, 28, 23, 24, 38, 29, 25, 26, 27, 148, 147, 146, 142, 143
Offset: 0

Views

Author

Antti Karttunen, Jun 25 2002

Keywords

Links

Crossrefs

Inverse permutation: A073292. Cf. A073290-A073299.

Formula

a(n) = A069768(A069768(A069768(n))).
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