A079946 -id:A079946 - cp's OEIS frontend

A081292 Intersection of A014486 and A079946.

0, 12, 50, 52, 56, 202, 204, 210, 212, 216, 226, 228, 232, 240, 810, 812, 818, 820, 824, 842, 844, 850, 852, 856, 866, 868, 872, 880, 906, 908, 914, 916, 920, 930, 932, 936, 944, 962, 964, 968, 976, 992, 3242, 3244, 3250, 3252, 3256, 3274, 3276, 3282, 3284

Offset: 1

Antti Karttunen, Mar 17 2003

nonn

These are A014486-encodings of the binary trees (or equivalent general trees) whose left subtree (equivalently: leftmost branch) is not just a single edge.

Antti Karttunen, Initial terms illustrated in positions 0,3,6,7,8,14,15,16,17,18,...

Cf. A014486, A079946.

a(n) = A014486(A081291(n)).

A090324 Second in a series of triangular arrays generating the natural numbers (cf. A079946).

Original entry on oeis.org

6, 13, 10, 27, 21, 18, 55, 43, 37, 34, 111, 87, 75, 69, 66, 223, 175, 151, 139, 133, 130, 447, 351, 303, 279, 267, 261, 258, 895, 703, 607, 559, 535, 523, 517, 514

Offset: 1

Alford Arnold, Jan 26 2004

Note that for each triangular array T(row+1,col) = 2 * T(row,col) +1 and that the change from T(row,col+1) to T(row,col) can be readily discerned. In the first array that pattern is 1 2 4 8 ...while in the second, it becomes 3 6 12 24 48 ... and in general, the pattern for the k-th array will begin with 2k-1.

			The first array begins
1
3 2
7 5 4
15 11 9 8
31 23 19 17 16
so a(1) begins with six. In general each triangular array will begin with
the next term from A079946.

A090774 A permutation of the natural numbers related to sequence A079946.

Original entry on oeis.org

1, 3, 6, 2, 13, 12, 7, 10, 25, 14, 5, 27, 20, 29, 24, 4, 21, 51, 22, 49, 26, 15, 18, 41, 59, 40, 53, 28, 11, 55, 36, 45, 99, 42, 57, 30, 9, 43, 103, 38, 81, 107, 44, 61, 48, 8, 37, 83, 119, 72, 85, 115, 46, 97, 50, 31, 34, 73, 91, 199, 74, 89, 123, 80, 101, 52, 23, 111, 68, 77

Offset: 1

Alford Arnold, Feb 07 2004

When viewed as a square array the first row begin with 1 6 12 14 24 ..., cf. A079946 and the second column is A090324.

			When viewed as a square array each column can be interpreted as a triangular array. Thus column two would become as illustrated below:
6
13 10
27 21 18
55 43 37 34

Cf. A090324.

A004755 Binary expansion starts 11.

Original entry on oeis.org

3, 6, 7, 12, 13, 14, 15, 24, 25, 26, 27, 28, 29, 30, 31, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122

Offset: 1

N. J. A. Sloane

a(n) is the smallest value > a(n-1) (or > 1 for n=1) for which A001511(a(n)) = A001511(n). - Franklin T. Adams-Watters, Oct 23 2006

			12 in binary is 1100, so 12 is in the sequence.

Kenny Lau, Table of n, a(n) for n = 1..16383 (first 1023 terms from T. D. Noe)
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions

Equals union of A079946 and A080565.
Cf. A004754 (10), A004756 (100), A004757 (101), A004758 (110), A004759 (111).
Cf. A004760, A053644, A062050, A076877.

import Data.List (transpose)
a004755 n = a004755_list !! (n-1)
a004755_list = 3 : concat (transpose [zs, map (+ 1) zs])
                   where zs = map (* 2) a004755_list
-- Reinhard Zumkeller, Dec 04 2015

a:= proc(n) n+2*2^floor(log(n)/log(2)) end: seq(a(n),n=1..60); # Muniru A Asiru, Oct 16 2018

Flatten[Table[FromDigits[#,2]&/@(Join[{1,1},#]&/@Tuples[{0,1},n]),{n,0,5}]] (* Harvey P. Dale, Feb 05 2015 *)

PARI
```
a(n)=n+2*2^floor(log(n)/log(2))
```

is(n)=n>2 && binary(n)[2] \\ Charles R Greathouse IV, Sep 23 2012

f = open('b004755.txt', 'w')
lo = 3
hi = 4
i = 1
while i<16384:
    for x in range(lo,hi):
        f.write(str(i)+" "+str(x)+"\n")
        i += 1
    lo <<= 1
    hi <<= 1
# Kenny Lau, Jul 05 2016

def A004755(n): return n+(1<Chai Wah Wu, Jul 13 2022

a(2n) = 2*a(n), a(2n+1) = 2*a(n) + 1 + 2*[n==0].
a(n) = n + 2 * 2^floor(log_2(n)) = A004754(n) + A053644(n).
a(n) = 2n + A080079(n). - Benoit Cloitre, Feb 22 2003
G.f.: (1/(1+x)) * (1 + Sum_{k>=0, t=x^2^k} 2^k*(2t+t^2)/(1+t)).
a(n) = n + 2^(floor(log_2(n)) + 1) = n + A062383(n). - Franklin T. Adams-Watters, Oct 23 2006
a(2^m+k) = 2^(m+1) + 2^m + k, m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Aug 08 2016

Edited by Ralf Stephan, Oct 12 2003

A079905 a(1)=1; then a(n) is smallest positive integer which is consistent with sequence being monotonically increasing and satisfying a(a(n)) = 2n+1 for n>1.

Original entry on oeis.org

1, 3, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17, 19, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 35, 37, 39, 41, 43, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 96, 97, 98, 99, 100, 101, 102

Offset: 1

N. J. A. Sloane, Feb 21 2003

Alternate definition: a(n) is taken to be smallest positive integer greater than a(n-1) such that the condition "a(a(n)) is always odd" can be satisfied. - Matthew Vandermast, Mar 03 2003

Also: a(n)=smallest positive integer > a(n-1) such that the condition "n is in the sequence if and only if a(n) is even" is false; that is, the condition "either n is not in the sequence and a(n) is odd or n is in the sequence and a(n) is even" is satisfied. - Matthew Vandermast, Mar 05 2003

Yifan Xie, Table of n, a(n) for n = 1..10000
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, arXiv:math/0305308 [math.NT], 2003.
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint 2016.
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585.
Index entries for sequences of the a(a(n)) = 2n family

See A080637 for a nicer version. Cf. A079000.
Equals A007378(n+1)-1, n>1.
A007378, A079905, A080637, A080653 are all essentially the same sequence.
Cf. A169956, A169957.
Union of A079946 and A005408 (the odd numbers).

b[n_] := b[n] = If[n<4, n+1, If[OddQ[n], b[(n-1)/2+1]+b[(n-1)/2], 2b[n/2]]];
a[n_] := If[n == 1, 1, b[n+1] - 1];
a /@ Range[70] (* Jean-François Alcover, Oct 18 2019, after Vladeta Jovovic in A007378 *)

a(1)=1, a(2)=3, then a(3*2^k - 1 + j) = 4*2^k - 1 + 3j/2 + |j|/2 for k >= 1, -2^k <= j < 2^k.

a(n) = 1+A079945(n-1)-A079944(n-1) for n>1, a(1)=1. - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 23 2003

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 23 2003

A080675 a(n) = (5*4^n - 8)/6.

Original entry on oeis.org

2, 12, 52, 212, 852, 3412, 13652, 54612, 218452, 873812, 3495252, 13981012, 55924052, 223696212, 894784852, 3579139412, 14316557652, 57266230612, 229064922452, 916259689812, 3665038759252, 14660155037012, 58640620148052, 234562480592212, 938249922368852

Offset: 1

N. J. A. Sloane, Mar 02 2003

These numbers have a simple binary pattern: 10,1100,110100,11010100,1101010100, ... i.e., the n-th term has a binary expansion 1(10){n-1}0, that is, there are n-1 10's between the most significant 1 and the least significant 0.

Vincenzo Librandi, Table of n, a(n) for n = 1..170
Andrei Asinowski, Cyril Banderier, Benjamin Hackl, On extremal cases of pop-stack sorting, Permutation Patterns (Zürich, Switzerland, 2019).
Index entries for linear recurrences with constant coefficients, signature (5,-4).

a(n) = A072197(n-1) - 1 = A014486(|A106191(n)|). a(n) = A079946(A020988(n-2)) for n>=2. Cf. also A122229.

[(5*4^n-8)/6: n in [1..40] ]; // Vincenzo Librandi, Apr 28 2011

(5*4^Range[30]-8)/6 (* or *) LinearRecurrence[{5,-4},{2,12},30] (* Harvey P. Dale, Oct 16 2012 *)

a(n)=(5*4^n-8)/6 \\ Charles R Greathouse IV, Oct 07 2015

a(1)=2, a(2)=12, a(n)=5*a(n-1)-4*a(n-2). - Harvey P. Dale, Oct 16 2012

Further comments added by Antti Karttunen, Sep 14 2006

A057547 A014486-encodings 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

2, 12, 52, 56, 212, 216, 228, 232, 240, 852, 856, 868, 872, 880, 916, 920, 932, 936, 944, 964, 968, 976, 992, 3412, 3416, 3428, 3432, 3440, 3476, 3480, 3492, 3496, 3504, 3524, 3528, 3536, 3552, 3668, 3672, 3684, 3688, 3696, 3732, 3736, 3748, 3752, 3760

Offset: 0

Antti Karttunen Sep 07 2000

This one-to-one correspondence between all rooted plane trees and one node larger, root degree = 1 trees illustrates the fact that INVERT(A000108) = LEFT(A000108). (Catalan numbers shift left under Cameron's A transformation.)
From Ruud H.G. van Tol, May 13 2024: (Start)
Sequence on a lattice:
Tree         Paths                  Decimal  Count
|_           10                           2      1
|.         1100                        12      1
||._       110100    -111000        52,56      2
|||_._     11010100  -11110000    212-240      5
|||_|.   1101010100-1111100000  852-992     14
... (End)

Ruud H.G. van Tol, Table of n, a(n) for n = 0..2055
P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102.
P. J. Cameron, Some sequences of integers, in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
Index entries for encodings of plane rooted trees

Double-trunked trees: A057517. Cf. also A057548, A057549.

alltrees2singletrunked := n -> pars2binexp([binexp2pars(n)]); # Just surround with extra parentheses.

a_rows(N) = my(a=Vec([[2]], N)); for(r=1, N-1, my(b=a[r], c=List()); foreach(b, t, for(i=1, valuation(t, 2), listput(~c, (t<<2)+(2<Ruud H.G. van Tol, May 25 2024

a(n) = A014486(A057548(n)) and also from n > 0 onward = A079946(A014486(n)).

a(n) = alltrees2singletrunked(A014486[n]) (see Maple code below and in A057501).

A162932 a(n) = A053445(n-2) - A053445(n-4).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 3, 1, 3, 4, 4, 4, 8, 6, 10, 11, 13, 15, 22, 20, 28, 33, 39, 43, 58, 60, 77, 88, 104, 119, 148, 160, 197, 226, 265, 300, 363, 404, 481, 549, 638, 727, 858, 961, 1126, 1283, 1480, 1680, 1953, 2201, 2544, 2887, 3309, 3750, 4312, 4857, 5566, 6301, 7175

Offset: 6

Alford Arnold, Jul 17 2009

nonn

a(n) counts partitions of n such that all parts are >=2 and the largest part occurs at least three times, see example.

			For n = 19 the a(19) = 6 partitions are 5554, 44443, 55522, 444322, 3333322 and 33322222.

Andrew van den Hoeven, Table of n, a(n) for n = 6..10000

Cf. A162931, A079946.

a:=func; [a(n): n in [6..100]]; // Vincenzo Librandi, Dec 09 2014

Table[PartitionsP[n] - 2 PartitionsP[n - 1] + 2 PartitionsP[n - 3] - PartitionsP[n - 4], {n, 6, 70}] (* Vincenzo Librandi, Dec 09 2014 *)

From Mircea Merca, Jun 11 2012: (Start)
a(n) = p(n) - 2*p(n-1) + 2*p(n-3) - p(n-4) for n >= 6, where p(n) = A000041(n).
G.f.: -1 + x - x^3 + (1 - x)*Product_{k > 2} 1/(1 - x^k). (End)
a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi^3 / (36*sqrt(2)*n^(5/2)). - Vaclav Kotesovec, Jun 02 2018
G.f.: Sum_{n >= 1} q^(3*n+3)/Product_{k = 1..n} 1 - q^(k+1). - Peter Bala, Dec 01 2024

Keyword:tabf removed, indexing corrected, sequence extended by R. J. Mathar, Sep 17 2009

A080565 Binary expansion of n has form 11**...*1.

Original entry on oeis.org

3, 7, 13, 15, 25, 27, 29, 31, 49, 51, 53, 55, 57, 59, 61, 63, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 193, 195, 197, 199, 201, 203, 205, 207, 209, 211, 213, 215, 217, 219, 221, 223, 225, 227, 229, 231, 233

Offset: 1

Benoit Cloitre, Feb 22 2003

nonn

If n>3 is in the sequence so are 2n-1 and 2n+1.

A004755 = union of A079946 and this sequence.

A diagonal of A246830.

a(n) = 2^floor(log[2](4*(n-1)))+2*n-1 for n>1, a(1)=3. - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 22 2003

Equals 2 * A004760(n) + 1. - Ralf Stephan, Sep 16 2003

A346305 Positions of words in A076478 that start with 1 and end with 1.

Original entry on oeis.org

2, 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

Offset: 1

Clark Kimberling, Aug 16 2021

The sequences A346303, A171757, A346304, and this sequence partition the positive integers. See A076478 for a guide to related sequences.

Is this A079946 with a 2 added in front? - R. J. Mathar, Sep 07 2021

			The first fourteen words w(n) are 0, 1, 00, 01, 10, 11, 000, 001, 010, 011, 100, 101, 110, 111, so that a(1) = 2.

Cf. A007931, A076478, A171757, A346303, A346304.

Mathematica
```
(See A076478.)
```

cp's OEIS Frontend

A081292 Intersection of A014486 and A079946.

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A090324 Second in a series of triangular arrays generating the natural numbers (cf. A079946).

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A090774 A permutation of the natural numbers related to sequence A079946.

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A004755 Binary expansion starts 11.

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PARI

PARI

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Python

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A079905 a(1)=1; then a(n) is smallest positive integer which is consistent with sequence being monotonically increasing and satisfying a(a(n)) = 2n+1 for n>1.

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Mathematica

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A080675 a(n) = (5*4^n - 8)/6.

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Magma

Mathematica

PARI

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A057547 A014486-encodings of Catalan mountain ranges with no sea-level valleys, i.e., the rooted plane general trees with root degree = 1.

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Programs

Maple

PARI

Formula

A162932 a(n) = A053445(n-2) - A053445(n-4).

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