A020522 -id:A020522 - cp's OEIS frontend

A032085 Number of reversible strings with n beads of 2 colors. If more than 1 bead, not palindromic.

2, 1, 2, 6, 12, 28, 56, 120, 240, 496, 992, 2016, 4032, 8128, 16256, 32640, 65280, 130816, 261632, 523776, 1047552, 2096128, 4192256, 8386560, 16773120, 33550336, 67100672, 134209536, 268419072, 536854528

Offset: 1

Christian G. Bower

a(n) is also the number of induced subgraphs with odd number of edges in the path graph P(n) if n>0. - Alessandro Cosentino (cosenal(AT)gmail.com), Feb 06 2009
A common recurrence of the bisections A020522 and A006516 means a(n+4) = 6*a(n+2) - 8*a(n), n>1. - Yosu Yurramendi, Aug 07 2008
Also, the decimal representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 566", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, Jul 05 2017

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
M. Archibald, A. Blecher, A. Knopfmacher, M. E. Mays, Inversions and Parity in Compositions of Integers, J. Int. Seq., Vol. 23 (2020), Article 20.4.1.
C. G. Bower, Transforms (2)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1022
S. J. Cyvin et al., Theory of polypentagons, J. Chem. Inf. Comput. Sci., 33 (1993), 466-474.
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (2,2,-4).

Cf. A005418, A016116. Essentially the same as A122746.
Row sums of triangle A034877.
Cf. A289404, A289405, A052551.

[2] cat [2^(n-1)-2^Floor((n-1)/2) : n in [2..40]]; // Wesley Ivan Hurt, Jul 03 2020

Join[{2}, LinearRecurrence[{2, 2, -4}, {1, 2, 6}, 29]] (* Jean-François Alcover, Oct 11 2017 *)

a(n)=([0,1,0; 0,0,1; -4,2,2]^(n-1)*[2;1;2])[1,1] \\ Charles R Greathouse IV, Oct 21 2022

"BHK" (reversible, identity, unlabeled) transform of 2, 0, 0, 0, ...
a(n) = 2^(n-1)-2^floor((n-1)/2), n > 1. - Vladeta Jovovic, Nov 11 2001
G.f.: 2*x+x^2/((1-2*x)*(1-2*x^2)). - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 25 2004
a(n) = A005418(n+1)-A016116(n+2), n>1. - Yosu Yurramendi, Aug 07 2008
a(n+1) = A077957(n) + 2*a(n), n>1. a(n+2) = A000079(n+1) + 2*a(n), n>1. - Yosu Yurramendi, Aug 10 2008
First differences: a(n+1)-a(n) = A007179(n) = A156232(n+2)/4, n>1. - Paul Curtz, Nov 16 2009
a(n) = 2*(a(n-1) bitwiseOR a(n-2)), n>3. - Pierre Charland, Dec 12 2010
a(n) = 2*a(n-1) + 2*a(n-2) - 4*a(n-3). - Wesley Ivan Hurt, Jul 03 2020

A099393 a(n) = 4^n + 2^n - 1.

Original entry on oeis.org

1, 5, 19, 71, 271, 1055, 4159, 16511, 65791, 262655, 1049599, 4196351, 16781311, 67117055, 268451839, 1073774591, 4295032831, 17180000255, 68719738879, 274878431231, 1099512676351, 4398048608255, 17592190238719

Offset: 0

Ralf Stephan, Oct 20 2004

Number of occurrences of letter 2 in the (n+1)-st Peano word.
In binary representation, a leading one followed by n zeros then by n ones. - Reinhard Zumkeller, Feb 07 2006
The number of involutions in group G_n G_{n+1} = G_n(operation) D_8. For example, Q_8->1 involution; D_8->5 involutions - Roger L. Bagula, Aug 08 2007

			n=5: a(5)=4^5+2^5-1=1024+32-1=1055 -> '10000011111'.

Vincenzo Librandi, Table of n, a(n) for n = 0..170
A. M. Cohen and D. E. Taylor, On a Certain Lie Algebra Defined By a Finite Group, American Mathematical Monthly, volume 114, number 7, August-September 2007, pages 633-638. Also preprint. a(n) = t_n in proof of theorem 6.2.
Sergey Kitaev and Toufik Mansour, The Peano curve and counting occurrences of some patterns, arXiv:math/0210268 [math.CO], 2002. Section 3 lemma 1, d_2^n = a(n-1).
Sergey Kitaev, Toufik Mansour, and Patrice Séébold, Generating the Peano curve and counting occurrences of some patterns, Journal of Automata, Languages and Combinatorics, volume 9, number 4, 2004, pages 439-455. Also at ResearchGate. Section 4, |P_n|_r = a(n-1).
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

See the formula section for the relationships with A000120, A000217, A000225, A002378, A007582, A020522, A023416, A030101, A063376, A070939, A083420, A279396.

[4^n + 2^n - 1: n in [0..60]]; // Vincenzo Librandi, Apr 26 2011

LinearRecurrence[{7,-14,8},{1,5,19},30] (* Harvey P. Dale, Sep 06 2015 *)

a(n)=4^n+2^n-1; \\ Charles R Greathouse IV, Sep 24 2015

def A099393(n): return ((1<Chai Wah Wu, Mar 10 2025

a(n) = A063376(n)-1.
a(n) = A020522(n) + A000225(n+1) = A083420(n) - A020522(n); A000120(a(n)) = n+1; A023416(a(n))=n; A070939(a(n)) = 2*n+1; 2*A020522(n)+1 = A030101(a(n)). - Reinhard Zumkeller, Feb 07 2006
a(n) = 2^(2*n-1) + 2*a(n-1) + 1. - Roger L. Bagula, Aug 08 2007
From Mohammad K. Azarian, Jan 15 2009: (Start)
G.f.: 1/(1-4*x) + 1/(1-2*x) - 1/(1-x).
E.g.f.: e^(4*x) + e^(2*x) - e^x. (End)
a(n) = A279396(n+4, 4). - Wolfdieter Lang, Jan 10 2017
a(n) = A002378(2^n) - 1 = 2*A000217(2^n) - 1 = 2*A007582(n) - 1. - Peter Munn, Nov 20 2022

A140690 A positive integer n is included if n written in binary can be subdivided into a number of runs all of equal-length, the first run from the left consisting of all 1's, the next run consisting of all 0's, the next run consisting of all 1's, the next run consisting of all 0's, etc.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 12, 15, 21, 31, 42, 51, 56, 63, 85, 127, 170, 204, 240, 255, 341, 455, 511, 682, 819, 992, 1023, 1365, 2047, 2730, 3276, 3640, 3855, 4032, 4095, 5461, 8191, 10922, 13107, 16256, 16383, 21845, 29127, 31775, 32767, 43690, 52428, 61680

Offset: 1

Leroy Quet, Jul 11 2008

Also: numbers of the form (2^s-1)*[4^{s*(k+1)}-1]/(4^s-1) or 2^s(2^s-1)*[4^{s*(k+1)}-1]/(4^s-1), s>=1, k>=0. Subsequences are, with the possible exception of terms at n=0, A002450(n), A043291(n), A015565(2n), A093134(2n+1), A000225(n), A020522(n). [R. J. Mathar, Aug 04 2008]
From Emeric Deutsch, Jan 25 2018: (Start)
Also the indices of the compositions having equal parts.
We define the index of a composition to be the positive integer whose binary form has run-lengths (i.e. runs of 1's, runs of 0's, etc., from left to right) equal to the parts of the composition. Example: the composition [1,1,3,1] has index 46 since the binary form of 46 is 101110. The integer 992 is in the sequence since its binary form is 1111100000 and the composition [5,5] has equal parts. The integer 100 is not in the sequence since its binary form is 1100100 and the composition [2,2,1,2] does not have equal parts.
The command c(n) from the Maple program yields the composition having index n. (End)

			819 in binary is 1100110011. The runs of 0's and 1's are (11)(00)(11)(00)(11). Each run (alternating 1's and 0's) is the same length. So 819 is in the sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to binary expansion of n

Cf. A101211, A298644.

import Data.Set (singleton, deleteFindMin, insert)
a140690 n = a140690_list !! (n-1)
a140690_list = f $ singleton (1, 1, 2) where
   f s | k == 1 = m : f (insert (2*b-1, 1, 2*b) $ insert (b*m, k+1, b) s')
       | even k    = m : f (insert (b*m+b-1, k+1, b) s')
       | otherwise = m : f (insert (b*m, k+1, b) s')
       where ((m, k, b), s') = deleteFindMin s
-- Reinhard Zumkeller, Feb 21 2014

Runs := proc (L) local j, r, i, k: j := 1: r[j] := L[1]: for i from 2 to nops(L) do if L[i] = L[i-1] then r[j] := r[j], L[i] else j := j+1: r[j] := L[i] end if end do: [seq([r[k]], k = 1 .. j)] end proc: RunLengths := proc (L) map(nops, Runs(L)) end proc: c := proc (n) ListTools:-Reverse(convert(n, base, 2)): RunLengths(%) end proc: A := {}: for n to 62000 do if nops(convert(c(n), set)) = 1 then A := `union`(A, {n}) else  end if end do: A; # most of the Maple program is due to W. Edwin Clark. - Emeric Deutsch, Jan 25 2018

Select[Range[62000],Length[Union[Length/@Split[IntegerDigits[#,2]]]]==1&] (* Harvey P. Dale, Mar 22 2012 *)

Terms beyond 42 from R. J. Mathar, Aug 04 2008

A171476 a(n) = 6a(n-1) - 8a(n-2) for n > 1, a(0)=1, a(1)=6.

Original entry on oeis.org

1, 6, 28, 120, 496, 2016, 8128, 32640, 130816, 523776, 2096128, 8386560, 33550336, 134209536, 536854528, 2147450880, 8589869056, 34359607296, 137438691328, 549755289600, 2199022206976, 8796090925056, 35184367894528

Offset: 0

Klaus Brockhaus, Dec 09 2009

Binomial transform of A048473; second binomial transform of A151821; third binomial transform of A010684; fourth binomial transform of A084633 without second term 0; fifth binomial transform of A168589.
Inverse binomial transform of A081625; second inverse binomial transform of A081626; third inverse binomial transform of A081627.
Partial sums of A010036.
Essentially first differences of A006095.
a(n) = A109241(n) converted from binary to decimal. - Robert Price, Jan 19 2016
a(n) is the area enclosed by a Hilbert curve with unit length segments after n iterations, when the start and end points are joined. - Jennifer Buckley, Apr 23 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jennifer Buckley, The Area Enclosed by a Hilbert Curve with Unit Length Segments
Gordon Hamilton, Cookie Monster Counts Cookies, Youtube video, 2010.
Index entries for linear recurrences with constant coefficients, signature (6,-8).

Cf. A006516 (2^(n-1)*(2^n-1)), A020522 (4^n-2^n), A048473 (2*3^n-1), A151821 (powers of 2, omitting 2 itself), A010684 (repeat 1, 3), A084633 (inverse binomial transform of repeated odd numbers), A168589 ((2-3^n)*(-1)^n), A081625 (2*5^n-3^n), A081626 (2*6^n-4^n), A081627 (2*7^n-5^n), A010036 (sum of 2^n, ..., 2^(n+1)-1), A006095 (Gaussian binomial coefficient [n, 2] for q=2), A171472, A171473.

[2*4^n-2^n: n in [0..30]]; // Vincenzo Librandi, Jul 17 2011

LinearRecurrence[{6,-8},{1,6},30] (* Harvey P. Dale, Aug 02 2020 *)

m=23; v=concat([1, 6], vector(m-2)); for(n=3, m, v[n]=6*v[n-1]-8*v[n-2]); v

a(n) = Sum_{k=1..2^n-1} k.
a(n) = 2*4^n - 2^n.
G.f.: 1/((1-2*x)*(1-4*x)).
a(n) = A006516(n+1).
a(n) = 4*a(n-1) + 2^n for n > 0, a(0)=1. - Vincenzo Librandi, Jul 17 2011
a(n) = Sum_{k=0..n} 2^(n+k). - Bruno Berselli, Aug 07 2013
a(n) = A020522(n+1)/2. - Hussam al-Homsi, Jun 06 2021
E.g.f.: exp(2*x)*(2*exp(2*x) - 1). - Stefano Spezia, Dec 10 2021

A059153 a(n) = 2^(n+2)*(2^(n+1)-1).

Original entry on oeis.org

4, 24, 112, 480, 1984, 8064, 32512, 130560, 523264, 2095104, 8384512, 33546240, 134201344, 536838144, 2147418112, 8589803520, 34359476224, 137438429184, 549754765312, 2199021158400, 8796088827904, 35184363700224, 140737471578112, 562949919866880

Offset: 0

Jonas Wallgren, Feb 02 2001

A hierarchical sequence (S(W'2{2}c) - see A059126).
a(n) written in base 2: 100, 11000, 1110000, ..., i.e., (n+1) times 1 and (n+2) times 0 (see A163663). - Jaroslav Krizek, Aug 12 2009
Also, the number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 513", based on the 5-celled von Neumann neighborhood. - Robert Price, May 04 2016

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Harry J. Smith, Table of n, a(n) for n = 0..200
J. Wallgren, Hierarchical sequences
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (6,-8).

Cf. A006516, A020522, A059126, A163663, A173787, A272702.

Table[2^(n + 2)*(2^(n + 1) - 1), {n, 0, 23}] (* and *) LinearRecurrence[{6, -8}, {4, 24}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2012 *)

a(n) = { 2^(n + 2)*(2^(n + 1) - 1) } \\ Harry J. Smith, Jun 25 2009

a(n) = A173787(2*n+3,n+2) = 4*A006516(n+1). - Reinhard Zumkeller, Feb 28 2010
From Colin Barker, Apr 28 2013: (Start)
a(n) = 6*a(n-1) - 8*a(n-2).
G.f.: 4 / ((2*x-1)*(4*x-1)). (End)
a(n) = 2*A020522(n+1). - Hussam al-Homsi, Jun 06 2021
E.g.f.: 4*exp(2*x)*(2*exp(2*x) - 1). - Elmo R. Oliveira, Dec 10 2023

Revised by Henry Bottomley, Jun 27 2005

A093069 a(n) = (2^n + 1)^2 - 2.

Original entry on oeis.org

7, 23, 79, 287, 1087, 4223, 16639, 66047, 263167, 1050623, 4198399, 16785407, 67125247, 268468223, 1073807359, 4295098367, 17180131327, 68720001023, 274878955519, 1099513724927, 4398050705407, 17592194433023, 70368760954879, 281475010265087, 1125899973951487

Offset: 1

Eric W. Weisstein, Mar 17 2004

Cletus Emmanuel calls these "Kynea numbers".

Difference between the smallest digitally balanced number with 2n+4 binary digits and the largest digitally balanced number with 2n+2 binary digits (see A031443): 7 = 9-2 = 1001-10, 23 = 35-12 = 100011-1100, 79 = 135-56 = 10000111-111000 etc. - Juri-Stepan Gerasimov, Jun 01 2011

			G.f. = 7*x + 23*x^2 + 79*x^3 + 287*x^4 + 1087*x^5 + 4223*x^6 + 16639*x^7 + ...

Michael De Vlieger, Table of n, a(n) for n = 1..1660
Amelia Carolina Sparavigna, Binary Operators of the Groupoids of OEIS A093112 and A093069 Numbers(Carol and Kynea Numbers), Department of Applied Science and Technology, Politecnico di Torino (Italy, 2019).
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.
Eric Weisstein's World of Mathematics, Near-Square Prime
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A091514 (primes of the form (2^n + 1)^2 - 2).

Cf. A244663.

[(2^n+1)^2-2 : n in [1..30]]; // Wesley Ivan Hurt, Jul 08 2014

A093069:=n->(2^n+1)^2-2: seq(A093069(n), n=1..30);

a[ n_] := If[ n < 1, 0, 4^n + 2^(n + 1) - 1]; (* Michael Somos, Jul 08 2014 *)
CoefficientList[Series[(7 - 26*x + 16*x^2)/((1 - x)*(2*x - 1)*(4*x - 1)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Jul 08 2014 *)
LinearRecurrence[{7,-14,8},{7,23,79},30] (* Harvey P. Dale, Aug 25 2025 *)

vector(100, n, (2^n+1)^2-2) \\ Colin Barker, Jul 08 2014

Vec(-(16*x^2-26*x+7)/((x-1)*(2*x-1)*(4*x-1)) + O(x^100)) \\ Colin Barker, Jul 08 2014

a(n) = 4^n+2^(n+1)-1.
G.f.: -x*(7-26*x+16*x^2) / ( (x-1)*(2*x-1)*(4*x-1) ). - R. J. Mathar, Jun 01 2011
a(n) = A092431(n+2) - A020522(n+1). - R. J. Mathar, Jun 01 2011
E.g.f.: -exp(x) + 2*exp(2*x) + exp(4*x) - 2. - Stefano Spezia, Dec 09 2019

More terms from Colin Barker, Jul 08 2014

A138147 Concatenation of n digits 1 and n digits 0.

Original entry on oeis.org

10, 1100, 111000, 11110000, 1111100000, 111111000000, 11111110000000, 1111111100000000, 111111111000000000, 11111111110000000000, 1111111111100000000000, 111111111111000000000000, 11111111111110000000000000, 1111111111111100000000000000, 111111111111111000000000000000

Offset: 1

Omar E. Pol, Mar 29 2008

Also, a(n) = binary representation of A020522(n), for n>0 (see example).

			n ... A020522(n) ..... a(n)
1 ....... 2 ........... 10
2 ...... 12 .......... 1100
3 ...... 56 ......... 111000
4 ..... 240 ........ 11110000
5 ..... 992 ....... 1111100000
6 .... 4032 ...... 111111000000
7 ... 16256 ..... 11111110000000

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 136, Ex. 4.2.2. - N. J. A. Sloane, Jul 27 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..95
Index entries for linear recurrences with constant coefficients, signature (110,-1000).

Cf. A002275, A020522, A138144, A138145, A138720, A276352.

Cf. A109241, A138118, A138119, A138120, A138146, A138721, A138826. - Omar E. Pol, Nov 08 2008

[(10^(2*n) - 10^n)/9: n in [1..30]]; // Vincenzo Librandi, Apr 26 2011

Table[FromDigits[Join[PadRight[{},n,1],PadRight[{},n,0]]],{n,15}] (* Harvey P. Dale, Nov 20 2011 *)

Vec(10*x/((10*x-1)*(100*x-1)) + O(x^100)) \\ Colin Barker, Sep 16 2013

a(n) = (10^(2*n) - 10^n)/9 = A002275(n)*10^n. - Omar E. Pol, Apr 16 2008
a(n) = 10*A109241(n-1). - Omar E. Pol, Nov 08 2008
From Colin Barker, Sep 16 2013: (Start)
a(n) = 110*a(n-1) - 1000*a(n-2).
G.f.: 10*x/((10*x-1)*(100*x-1)). (End)
From Elmo R. Oliveira, Jun 13 2025: (Start)
E.g.f.: exp(10*x)*(exp(90*x) - 1)/9.
a(n) = A276352(n)/9. (End)

A161168 a(n) = 2^n + 4^n.

Original entry on oeis.org

2, 6, 20, 72, 272, 1056, 4160, 16512, 65792, 262656, 1049600, 4196352, 16781312, 67117056, 268451840, 1073774592, 4295032832, 17180000256, 68719738880, 274878431232, 1099512676352, 4398048608256, 17592190238720, 70368752566272

Offset: 0

Zerinvary Lajos, Jun 04 2009

Essentially a duplicate of A063376 and A028402.
a(n) written in base 2: a(0) = 10, a(n) for n >= 1: 110, 10100, 1001000, 100010000, ..., i.e., number 1, (n-1) times 0, number 1, n times 0 (see A163664). a(n) is a bisection of A005418. - Jaroslav Krizek, Aug 14 2009
Central terms of the triangle in A173786. - Reinhard Zumkeller, Feb 28 2010
For n > 0 let 2^(n+1) be the length of the even leg of a primitive Pythagorean triangle (PPT); then the odd leg is constrained to have a length of 4^n-1 and the hypotenuse to have a length of 4^n+1. The resulting triangle has a semiperimeter of 4^n + 2^n. - Frank M Jackson, Dec 28 2017
a(n) is also the number of distinct planar embeddings of the (2n+7)-triangular snake graph. - Eric W. Weisstein, May 21 2024

Iain Fox, Table of n, a(n) for n = 0..1660
Eric Weisstein's World of Mathematics, Planar Embedding.
Eric Weisstein's World of Mathematics, Triangular Snake Graph.
Index entries for linear recurrences with constant coefficients, signature (6,-8).

Cf. A005418, A007582, A020522, A028403, A063376, A163664, A173786.

Magma
```
[ 2^n+4^n: n in [0..25] ];
```

A161168:=n->2^n+4^n: seq(A161168(n), n=0..40); # Wesley Ivan Hurt, Jul 24 2017

a[n_]:=4^n+2^n; Array[a,24] (* Frank M Jackson, Dec 28 2017 *)

a(n)=2^n+4^n \\ Charles R Greathouse IV, Oct 07 2015

first(n) = Vec(2*(1 - 3*x)/((1 - 2*x)*(1 - 4*x)) + O(x^n)) \\ Iain Fox, Dec 28 2017

Sage
```
[2^n + 4^n for n in range(0,25)]
```
Sage
```
[sigma(4,n)-1for n in range(0,25)]
```

a(n) = 6*a(n-1) - 8*a(n-2); a(0)=2, a(1)=6. - Vincenzo Librandi, Dec 27 2010
G.f.: -2*(3*x-1) / ((2*x-1)*(4*x-1)). - Colin Barker, Mar 19 2013
E.g.f.: e^(2*x) + e^(4*x). - Iain Fox, Dec 28 2017
a(n) = 2*A007582(n). - R. J. Mathar, Feb 26 2018

A265736 Row sums of triangle A265705.

Original entry on oeis.org

0, 2, 8, 12, 29, 38, 46, 56, 107, 126, 144, 164, 177, 198, 218, 240, 407, 446, 484, 524, 557, 598, 638, 680, 691, 734, 776, 820, 857, 902, 946, 992, 1583, 1662, 1740, 1820, 1893, 1974, 2054, 2136, 2187, 2270, 2352, 2436, 2513, 2598, 2682, 2768, 2727, 2814

Offset: 0

Reinhard Zumkeller, Dec 15 2015

nonn

a(A000225(n)) = A020522(n).

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A265705, A000225, A020522.

Haskell
```
a265736 = sum . a265705_row
```

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

A032085 Number of reversible strings with n beads of 2 colors. If more than 1 bead, not palindromic.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A099393 a(n) = 4^n + 2^n - 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Formula

A140690 A positive integer n is included if n written in binary can be subdivided into a number of runs all of equal-length, the first run from the left consisting of all 1's, the next run consisting of all 0's, the next run consisting of all 1's, the next run consisting of all 0's, etc.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Extensions

A171476 a(n) = 6*a(n-1) - 8*a(n-2) for n > 1, a(0)=1, a(1)=6.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A059153 a(n) = 2^(n+2)*(2^(n+1)-1).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A093069 a(n) = (2^n + 1)^2 - 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

A171476 a(n) = 6a(n-1) - 8a(n-2) for n > 1, a(0)=1, a(1)=6.