A072654 -id:A072654 - cp's OEIS frontend

A072655 Binary widths of the terms of A072654.

Original entry on oeis.org

0, 1, 3, 5, 8, 14, 26, 49, 94, 183, 362, 718, 1429, 2852, 5696, 11384

Offset: 0

Antti Karttunen, Jun 02 2002

Cf. A072641, A072642.

a(11)-a(15) from Sean A. Irvine, Oct 18 2024

A072639 a(0) = 0, a(n) = Sum_{i=0..n-1} 2^((2^i)-1).

Original entry on oeis.org

0, 1, 3, 11, 139, 32907, 2147516555, 9223372039002292363, 170141183460469231740910675754886398091, 57896044618658097711785492504343953926805133516280751251469702679711451218059

Offset: 0

Antti Karttunen, Jun 02 2002

nonn

Maximum position in A072644 where the value n occurs.
Also partial sums of A058891, i.e. the first differences are there. - R. J. Mathar, May 15 2007
A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Then a(n) is the minimum BII-number of a set-system with n distinct vertices. - Gus Wiseman, Jul 24 2019

Binary width of each term: A000079. Cf. A072638, A072640, A072654.
Cf. A058891.
Cf. A000120, A014221, A029931, A034797, A048793, A070939, A326031, A326702.

A072639 := proc(n) local i; add(2^((2^i)-1),i=0..(n-1)); end;

a[n_] := Sum[2^(2^i - 1), {i, 0, n - 1}]; Table[a[n], {n, 0, 9}] (* Jean-François Alcover, Mar 06 2016 *)

a(n) = if (n, sum(i=0, n-1, 2^((2^i)-1)), 0); \\ Michel Marcus, Mar 06 2016

A072638 Number of unary-binary rooted trees of height at most n.

Original entry on oeis.org

0, 1, 3, 10, 66, 2278, 2598060, 3374961778891, 5695183504492614029263278, 16217557574922386301420536972254869595782763547560

Offset: 0

Antti Karttunen, Jun 02 2002

nonn

A unary-binary tree is one in which the degree of every node is <= 3.
a(n+1) = (a(n)+1)-th triangular numbers = A000217(a(n)+1). a(n+1) = (a(n) + 1) * (a(n) + 2) / 2. a(n+1) = A006894(n+2) - 1. - Jaroslav Krizek, Sep 11 2009
a(n) is the smallest integer that is the sum of n distinct members of the complete sequence A000124. See A204009 for the binary vectors that select the terms from A000124. - Frank M Jackson, Jan 09 2012

Index entries for sequences related to rooted trees

Maximal position in A071673 where the value n occurs.

Binary width of each term: A072641. Cf. A072639, A072640, A072654.

a[ -1]:=0:a[0]:=1:for n from 1 to 50 do a[n]:=binomial(a[n-1]+2,2) od: seq(a[n], n=-1..9); # Zerinvary Lajos, Jun 08 2007

Clear[a]; a[0] = 0; a[n_] := a[n] = 1 + (a[n-1]*(a[n-1]+3))/2; Table[a[n], {n, 0, 9}] (* Jean-François Alcover, Jan 31 2013 *)

a(n+1) = 1 + (a(n)*(a(n)+3))/2.
Conjecture: a(n) = A006894(n+1) - 1. - R. J. Mathar, Apr 23 2007
a(n) := C(a(n-1) + 2, 2), n >= -1. - Zerinvary Lajos, Jun 08 2007

Edited by Christian G. Bower, Oct 23 2002

A072645 Size of the parenthesizations obtained with the global ranking/unranking scheme A072646/A072647.

Original entry on oeis.org

0, 1, 2, 3, 2, 4, 3, 3, 5, 3, 4, 4, 4, 7, 4, 4, 5, 3, 6, 5, 5, 9, 4, 5, 5, 5, 6, 4, 5, 7, 6, 6, 6, 5, 10, 5, 4, 5, 4, 5, 6, 6, 6, 4, 5, 6, 5, 8, 7, 7, 7, 6, 8, 6, 7, 11, 5, 6, 5, 6, 6, 5, 4, 6, 7, 6, 6, 6, 8, 5, 6, 7, 5, 7, 6, 6, 10, 8, 7, 7, 7, 7, 7, 8, 8, 7, 7, 8, 6, 12, 6, 5, 6, 4, 7, 7, 7, 8, 5, 6, 5, 6

Offset: 0

Antti Karttunen, Jun 02 2002

nonn

Each value v occurs A000108(v) times. The maximum position for value v to occur is A072654(v). Permutations: A071673, A072643, A072644, A072660.

A072640 Maximum position in A072660 where the value n occurs.

Original entry on oeis.org

0, 1, 3, 11, 100, 2090, 415729, 15670134831, 2990945138477441778, 28553369883190773267638351857949, 22025764139998888788032314837232820672093881927968648914

Offset: 0

Antti Karttunen, Jun 02 2002

nonn

Binary width of each term: A072642. Cf. A072638, A072639, A072654.

cp's OEIS Frontend

A072655 Binary widths of the terms of A072654.

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A072639 a(0) = 0, a(n) = Sum_{i=0..n-1} 2^((2^i)-1).

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A072638 Number of unary-binary rooted trees of height at most n.

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Maple

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A072645 Size of the parenthesizations obtained with the global ranking/unranking scheme A072646/A072647.

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A072640 Maximum position in A072660 where the value n occurs.

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