A057547 -id:A057547 - cp's OEIS frontend

A057549 The local ranks of each term of A057547.

0, 1, 3, 4, 8, 9, 11, 12, 13, 22, 23, 25, 26, 27, 31, 32, 34, 35, 36, 38, 39, 40, 41, 64, 65, 67, 68, 69, 73, 74, 76, 77, 78, 80, 81, 82, 83, 92, 93, 95, 96, 97, 101, 102, 104, 105, 106, 108, 109, 110, 111, 115, 116, 118, 119, 120, 122, 123, 124, 125, 127, 128, 129

Offset: 0

Antti Karttunen, Sep 07 2000

nonn

Cf. A057519, A057548, A057550.

a(n) = CatalanRank(floor(binwidth(A057547[n])/2), A057547[n])

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

Antti Karttunen, Sep 07 2000

nonn

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.

Index entries for the sequences induced by list functions of Lisp

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

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

A079946 Numbers k whose binary expansion begins with two or more 1's and ends with at least one 0.

Original entry on oeis.org

6, 12, 14, 24, 26, 28, 30, 48, 50, 52, 54, 56, 58, 60, 62, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 192, 194, 196, 198, 200, 202, 204, 206, 208, 210, 212, 214, 216, 218, 220, 222, 224, 226, 228, 230, 232, 234, 236, 238, 240, 242, 244, 246

Offset: 1

N. J. A. Sloane, Feb 21 2003

a(n) = b(n+1), with b(2n) = 2b(n), b(2n+1) = 2b(n)+2+4[n==0]. - Ralf Stephan, Oct 11 2003

Yifan Xie, Table of n, a(n) for n = 1..10001 (first 1000 terms from Harvey P. Dale)
Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, arXiv:math/0305308 [math.NT], 2003.
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions

A004755 = union of this and A080565. A057547(n) = a(A014486(n)) for n >= 1.

Maple
```
A079946 := n -> 2*(2^(1+A000523(n))+n);
```

Table[Union[FromDigits[Join[{1,1},#,{0}],2]&/@Tuples[{1,0},n]],{n,0,5}]//Flatten (* Harvey P. Dale, Jan 16 2018 *)

for(n=0,6, for(k=2^(n-1),2^n-1,print1((2^n+k)*2,",")))

for(n=1,59,print1((2^(floor(log(n)/log(2))+1)+n)*2,","))

a(n) = n*2 + 4<Ruud H.G. van Tol, May 10 2024

def A079946(n): return n+(1<Chai Wah Wu, Jul 13 2022

a(n) = 2^floor(log_2(4*n))+2*n. - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 22 2003
a(n) = (2^(floor(log_2(n))+1)+n)*2. - Klaus Brockhaus, Feb 23 2003
a(2n) = 2a(n), a(2n+1) = 2a(n) + 2 + 4[n==0]. Twice A004755. - Ralf Stephan, Oct 12 2003

Definition clarified by N. J. A. Sloane, May 10 2024

A057517 Binary encodings of the Catalan mountain ranges with exactly one sea-level valley, i.e., the rooted plane trees with root degree = 2.

Original entry on oeis.org

10, 44, 50, 180, 184, 204, 210, 226, 724, 728, 740, 744, 752, 820, 824, 844, 850, 866, 908, 914, 930, 962, 2900, 2904, 2916, 2920, 2928, 2964, 2968, 2980, 2984, 2992, 3012, 3016, 3024, 3040, 3284, 3288, 3300, 3304, 3312, 3380, 3384, 3404, 3410, 3426

Offset: 1

Antti Karttunen, Sep 03 2000

nonn

This bijective mapping from all rooted plane trees to one node larger, root degree = 2 trees illustrates the fact that CONV(A000108, A000108) = LEFT(A000108). (Catalan numbers shift left under convolution).

Cf. A057501 (for binexp2pars, pars2binexp, car, cdr), A057518, A057519, A057122. Single-trunked trees: A057547.

alltrees2doubletrunked := n -> pars2binexp(alltrees2doubletrunkedP(binexp2pars(n)));
alltrees2doubletrunkedP := h -> [car(h),cdr(h)];

a(n) = alltrees2doubletrunked(A014486(n)) (Starting from n=1).

A216648 Triangle T(n,k) in which n-th row lists in increasing order all positive integers with a representation as totally balanced 2n digit binary string without totally balanced proper prefixes such that all consecutive totally balanced substrings are in nondecreasing order; n>=1, 1<=k<=A000081(n).

Original entry on oeis.org

2, 12, 52, 56, 212, 216, 232, 240, 852, 856, 872, 880, 920, 936, 944, 976, 992, 3412, 3416, 3432, 3440, 3480, 3496, 3504, 3536, 3552, 3688, 3696, 3752, 3760, 3792, 3808, 3888, 3920, 3936, 4000, 4032, 13652, 13656, 13672, 13680, 13720, 13736, 13744, 13776

Offset: 1

Alois P. Heinz, Sep 12 2012

There is a simple bijection between the elements of row n and the rooted trees with n nodes. Each matching pair (1,0) in the binary string representation encodes a node, each totally balanced substring encodes a list of subtrees.

			856 is element of row 5, the binary string representation (with totally balanced substrings enclosed in parentheses) is (1(10)(10)(1(10)0)0).  The encoded rooted tree is:
.    o
.   /|\
.  o o o
.      |
.      o
Triangle T(n,k) begins:
2;
12;
52,     56;
212,   216,  232,  240;
852,   856,  872,  880,  920,  936,  944,  976,  992;
3412, 3416, 3432, 3440, 3480, 3496, 3504, 3536, 3552, 3688, 3696, ...
Triangle T(n,k) in binary:
10;
1100;
110100,       111000;
11010100,     11011000,     11101000,     11110000;
1101010100,   1101011000,   1101101000,   1101110000,   1110011000, ...
110101010100, 110101011000, 110101101000, 110101110000, 110110011000, ...

Alois P. Heinz, Rows n = 1..12, flattened

First column gives: A080675.
Last elements of rows give: A020522.
Row lengths are: A000081.
Subsequence of A057547, A081292.
Cf. A061773, A216349, A216350, A216649.

F:= proc(n) option remember; `if`(n=1, [10], sort(map(h->
      parse(cat(1, sort(h)[], 0)), g(n-1, n-1)))) end:
g:= proc(n, i) option remember; `if`(i=1, [[10$n]], [seq(seq(seq(
      [seq (F(i)[w[t]-t+1], t=1..j),v[]], w=combinat[choose](
      [$1..nops(F(i))+j-1], j)), v=g(n-i*j, i-1)), j=0..n/i)])
    end:
b:= proc(n) local h, i, r; h, r:= n, 0; for i from 0
      while h>0 do r:= r+2^i*irem(h, 10, 'h') od; r
    end:
T:= proc(n) option remember; map(b, F(n))[] end:
seq(T(n), n=1..7);

T(n,k) = A216649(n-1,k)*2 + 2^(2*n-1) for n>1.

cp's OEIS Frontend

A057549 The local ranks of each term of A057547.

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A057548 A014486-indices of Catalan mountain ranges with no sea-level valleys, i.e., the rooted plane general trees with root degree = 1.

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A079946 Numbers k whose binary expansion begins with two or more 1's and ends with at least one 0.

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A057517 Binary encodings of the Catalan mountain ranges with exactly one sea-level valley, i.e., the rooted plane trees with root degree = 2.

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