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.

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A083923 Characteristic function for A057548.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, May 13 2003

Keywords

Crossrefs

a(n) = A083924(A069770(n)). Used to compute A083925.

Formula

a(n) = 1 if A057515(n)=1 (equivalently: if A072772(n)=0), otherwise 0.

A083926 Inverse function of N -> N injection A072795.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, May 13 2003

Keywords

Comments

a(1)=0 because A072795(0)=1, but a(n) = 0 also for those n which do not occur as values of A072795. All positive natural numbers occur here once.

Crossrefs

a(A072795(n)) = n for all n. Cf. A083925, A083927-A083929, A083935.

Formula

a(n) = A083924(n)*A072772(n).

A127307 Positions of Dyck words beginning as UUD (110) in A014486/A063171.

Original entry on oeis.org

3, 6, 7, 14, 15, 16, 17, 18, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141
Offset: 1

Views

Author

Antti Karttunen, Jan 16 2007

Keywords

Programs

Formula

a(n) = A072764bi(A072795(A072771(n)),A072772(n)).

A367433 Number of successive Patcail predecessors of n-th binary tree.

Original entry on oeis.org

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

Views

Author

John Tromp, Nov 18 2023

Keywords

Comments

A binary tree is either 0 or a pair [s,t] of binary trees. Binary trees are counted by Catalan numbers A000108 and ordered by their binary code as given by A014486. Subtrees s and t correspond to A072771 and A072772.
Patcail defined the predecessor of [0,t] as t, and of [s,t], where s has predecessor s', as the result of replacing with [s',t] each occurrence of t within [s',t].
a(7), corresponding to [[0,[0,0]],0], is too large to show, exceeding an exponential tower of 2^63 2's. a(8), corresponding to [[[0,0],0],0], is much larger still, starting to approach Graham's Number. The next 3 terms are modest again, at a(9)=4, a(10)=7, a(11)=5.
The (A014138 indexed) subsequence for left skewed binary trees 0, [0,0], [[0,0],0], [[[0,0],0],0] ... forms an extremely fast growing sequence, at Buchholz's Ordinal in the Fast Growing Hierarchy.
Initial predecessors of these left skewed trees have sizes a(n) satisfying
a(n+1) = (a(n)+1)*(a(n)+3), which is A056207 counting the number of binary trees of height <= n.

Examples

			a(3)=5, since the 3rd binary tree is [[0,0],0] and its 5 successive Patcail predecessors are [[0,0],[0,0]], [0,[0,[0,0]]], [0,[0,0]], [0,0], and 0:
Index   n         3              6              4          2      1   0
A014486(n)       12             50             42         10      2   0
A063171(n)     1100         110010         101010       1010     10   0
Tree       [[0,0],0]  [[0,0],[0,0]]  [0,[0,[0,0]]]  [0,[0,0]]  [0,0]  0
A367433(n)        5              4              3          2      1   0
		

Crossrefs

Programs

  • Haskell
    data T = N | C T T deriving (Eq,Show)
    a014486 = [0..] >>= at where
      at 0 = [N]
      at n = [C s t | (ns,s) <- to$n-1, t <- at$n-1-ns]
      to n = (0,N):[(1+ns+nt,C s t)|n>0,(ns,s)<-to$n-1,(nt,t)<-to$n-1-ns]
    predT (C N t) = t
    predT (C s t) = go u where
      u = [predT s) t
      go v = if v==t then u else case v of
        N     -> N
        C s t -> [go s) (go t)
    a367433 = map nPred a014486 where
      nPred N = 0
      nPred t = 1 + nPred (predT t)
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