A007283 -id:A007283 - cp's OEIS frontend

A215366 Triangle T(n,k) read by rows in which n-th row lists in increasing order all partitions lambda of n encoded as Product_{i in lambda} prime(i); n>=0, 1<=k<=A000041(n).

1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 12, 16, 11, 14, 15, 18, 20, 24, 32, 13, 21, 22, 25, 27, 28, 30, 36, 40, 48, 64, 17, 26, 33, 35, 42, 44, 45, 50, 54, 56, 60, 72, 80, 96, 128, 19, 34, 39, 49, 52, 55, 63, 66, 70, 75, 81, 84, 88, 90, 100, 108, 112, 120, 144, 160, 192, 256

Offset: 0

Alois P. Heinz, Aug 08 2012

The concatenation of all rows (with offset 1) gives a permutation of the natural numbers A000027 with fixed points 1-6, 9, 10, 14, 15, 21, 22, 33, 49, 1095199, ... and inverse permutation A215501.
Number m is positioned in row n = A056239(m).  The number of different values m, such that both m and m+1 occur in row n is A088850(n).  A215369 lists all values m, such that both m and m+1 are in the same row.
The power prime(i)^j of the i-th prime is in row i*j for j in {0,1,2, ... }.
Column k=2 contains the even semiprimes A100484, where 10 and 22 are replaced by the odd semiprimes 9 and 21, respectively.
This triangle is related to the triangle A145518, see in both triangles the first column, the right border, the second right border and the row sums. - Omar E. Pol, May 18 2015

			The partitions of n=3 are {[3], [2,1], [1,1,1]}, encodings give {prime(3), prime(2)*prime(1), prime(1)^3} = {5, 3*2, 2^3} => row 3 = [5, 6, 8].
For n=0 the empty partition [] gives the empty product 1.
Triangle T(n,k) begins:
   1;
   2;
   3,  4;
   5,  6,  8;
   7,  9, 10, 12, 16;
  11, 14, 15, 18, 20, 24, 32;
  13, 21, 22, 25, 27, 28, 30, 36, 40, 48, 64;
  17, 26, 33, 35, 42, 44, 45, 50, 54, 56, 60, 72, 80, 96, 128;
  ...
Corresponding triangle of integer partitions begins:
  ();
  1;
  2, 11;
  3, 21, 111;
  4, 22, 31, 211, 1111;
  5, 41, 32, 221, 311, 2111, 11111;
  6, 42, 51, 33, 222, 411, 321, 2211, 3111, 21111, 111111;
  7, 61, 52, 43, 421, 511, 322, 331, 2221, 4111, 3211, 22111, 31111, 211111, 1111111;  - _Gus Wiseman_, Dec 12 2016

Column k=1 gives: A008578(n+1).
Last elements of rows give: A000079.
Second to last elements of rows give: A007283(n-2) for n>1.
Row sums give: A145519.
Row lengths are: A000041.
Cf. A129129 (with row elements using order of A080577).
LCM of terms in row n gives A138534(n).
Cf. A000027, A000040, A056239, A063008, A088850, A100484, A215501.
Cf. A112798, A246867 (the same for partitions into distinct parts).
Cf. A324939, A377852, A378175.

b:= proc(n, i) option remember; `if`(n=0 or i<2, [2^n],
       [seq(map(p->p*ithprime(i)^j, b(n-i*j, i-1))[], j=0..n/i)])
    end:
T:= n-> sort(b(n, n))[]:
seq(T(n), n=0..10);
# (2nd Maple program)
with(combinat): A := proc (n) local P, A, i: P := partition(n): A := {}; for i to nops(P) do A := `union`(A, {mul(ithprime(P[i][j]), j = 1 .. nops(P[i]))}) end do: A end proc; # the command A(m) yields row m. # Emeric Deutsch, Jan 23 2016
# (3rd Maple program)
q:= 7: S[0] := {1}: for m to q do S[m] := `union`(seq(map(proc (f) options operator, arrow: ithprime(j)*f end proc, S[m-j]), j = 1 .. m)) end do; # for a given positive integer q, the program yields rows 0, 1, 2,...,q. # Emeric Deutsch, Jan 23 2016

b[n_, i_] := b[n, i] = If[n == 0 || i<2, {2^n}, Table[Function[#*Prime[i]^j] /@ b[n - i*j, i-1], {j, 0, n/i}] // Flatten]; T[n_] := Sort[b[n, n]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Mar 12 2015, after Alois P. Heinz *)
nn=7;HeinzPartition[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]//Reverse];
Take[GatherBy[Range[2^nn],Composition[Total,HeinzPartition]],nn+1] (* Gus Wiseman, Dec 12 2016 *)
Table[Map[Times @@ Prime@ # &, IntegerPartitions[n]], {n, 0, 8}] // Flatten (* Michael De Vlieger, Jul 12 2017 *)

\\ From M. F. Hasler, Dec 06 2016 (Start)
A215366_row(n)=vecsort([vecprod([prime(p)|p<-P])|P<-partitions(n)]) \\ bug fix & syntax update by M. F. Hasler, Oct 20 2023
A215366_vec(N)=concat(apply(A215366_row,[0..N])) \\ "flattened" rows 0..N (End)

Recurrence relation, explained for the set S(4) of entries in row 4: multiply the entries of S(3) by 2 (= 1st prime), multiply the entries of S(2) by 3 (= 2nd prime), multiply the entries of S(1) by 5 (= 3rd prime), multiply the entries of S(0) by 7 (= 4th prime); take the union of all the obtained products. The 3rd Maple program is based on this recurrence relation. - Emeric Deutsch, Jan 23 2016

A016116 a(n) = 2^floor(n/2).

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 8, 8, 16, 16, 32, 32, 64, 64, 128, 128, 256, 256, 512, 512, 1024, 1024, 2048, 2048, 4096, 4096, 8192, 8192, 16384, 16384, 32768, 32768, 65536, 65536, 131072, 131072, 262144, 262144, 524288, 524288, 1048576, 1048576, 2097152

Offset: 0

N. J. A. Sloane, Dec 11 1999

Powers of 2 doubled up. The usual OEIS policy is to omit the duplicates in such cases (when this would become A000079). This is an exception.
Number of symmetric compositions of n: e.g., 5 = 2+1+2 = 1+3+1 = 1+1+1+1+1 so a(5) = 4; 6 = 3+3 = 2+2+2 = 1+4+1 = 2+1+1+2 = 1+2+2+1 = 1+1+2+1+1 = 1+1+1+1+1+1 so a(6) = 8. - Henry Bottomley, Dec 10 2001
This sequence is the number of digits of each term of A061519. - Dmitry Kamenetsky, Jan 17 2009
Starting with offset 1 = binomial transform of [1, 1, -1, 3, -7, 17, -41, ...]; where A001333 = (1, 1, 3, 7, 17, 41, ...). - Gary W. Adamson, Mar 25 2009
a(n+1) is the number of symmetric subsets of [n]={1,2,...,n}. A subset S of [n] is symmetric if k is an element of S implies (n-k+1) is an element of S. - Dennis P. Walsh, Oct 27 2009
INVERT and inverse INVERT transforms give A006138, A039834(n-1).
The Kn21 sums, see A180662, of triangle A065941 equal the terms of this sequence. - Johannes W. Meijer, Aug 15 2011
First differences of A027383. - Jason Kimberley, Nov 01 2011
Run lengths in A079944. - Jeremy Gardiner, Nov 21 2011
Number of binary palindromes (A006995) between 2^(n-1) and 2^n (for n>1). - Hieronymus Fischer, Feb 17 2012
Pisano period lengths: 1, 1, 4, 1, 8, 4, 6, 1, 12, 8, 20, 4, 24, 6, 8, 1, 16, 12, 36, 8, ... . - R. J. Mathar, Aug 10 2012
Range of row n of the Circular Pascal array of order 4. - Shaun V. Ault, May 30 2014
a(n) is the number of permutations of length n avoiding both 213 and 312 in the classical sense which are breadth-first search reading words of increasing unary-binary trees. For more details, see the entry for permutations avoiding 231 at A245898. - Manda Riehl, Aug 05 2014
Also, the decimal representation of the diagonal from the origin to the corner (and from the corner to the origin except for the initial term) of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 190", based on the 5-celled von Neumann neighborhood when initialized with a single black (ON) cell at stage zero. - Robert Price, May 10 2017
a(n + 1) + n - 1, n > 0, is the number of maximal subsemigroups of the monoid of partial order-preserving or -reversing mappings on a set with n elements. See the East et al. link. - James Mitchell and Wilf A. Wilson, Jul 21 2017
Number of symmetric stairs with n cells. A stair is a snake polyomino allowing only two directions for adjacent cells: east and north. See A005418. - Christian Barrientos, May 11 2018
For n >= 4, a(n) is the exponent of the group of the Gaussian integers in a reduced system modulo (1+i)^(n+2). See A302254. - Jianing Song, Jun 27 2018
a(n) is the number of length-(n+1) binary sequences, denoted , with s(1)=1 and with s(i+1)=s(i) for odd i. - Dennis P. Walsh, Sep 06 2018
a(n+1) is the number of subsets of {1,2,..,n} in which all differences between successive elements of subsets are even.  For example, for n = 7, a(6) = 8 and the 8 subsets are {7}, {1,7}, {3,7}, {5,7}, {1,3,7}, {1,5,7}, {3,5,7}, {1,3,5,7}.  For odd differences between elements see Comment in A000045 (Fibonacci numbers). - Enrique Navarrete, Jul 01 2020
Also, the number of walks of length n on the graph x--y--z, starting at x. - Sean A. Irvine, May 30 2025

			For n=5 the a(5)=4 symmetric subsets of [4] are {1,4}, {2,3}, {1,2,3,4} and the empty set. - _Dennis P. Walsh_, Oct 27 2009
For n=5 the a(5)=4 length-6 binary sequences are <1,1,0,0,0,0>, <1,1,0,0,1,1>, <1,1,1,1,0,0> and <1,1,1,1,1,1>. - _Dennis P. Walsh_, Sep 06 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Shaun V. Ault and Charles Kicey, Counting paths in corridors using circular Pascal arrays, Discrete Mathematics, Volume 332, Oct 06 2014, Pages 45-54.
Shaun V. Ault and Charles Kicey, Counting paths in corridors using circular Pascal arrays, arXiv:1407.2197 [math.CO], 2014.
Arvind Ayyer, Amritanshu Prasad, and Steven Spallone, Odd partitions in Young's lattice, arXiv:1601.01776 [math.CO], 2016. See Theorem 6 p. 12.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, J. Integer Sequ., Vol. 9 (2006), Article 06.2.4.
Francesco Battistoni and Giuseppe Molteni, An elementary proof for a generalization of a Pohst's inequality, arXiv:2101.06163 [math.NT], 2021.
Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015.
Johann Cigler, Recurrences for certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials, arXiv:2212.02118 [math.NT], 2022.
Emeric Deutsch, Problem 1633, Math. Mag., 74 #5 (2001), p. 403.
James East, Jitender Kumar, James D. Mitchell, and Wilf A. Wilson, Maximal subsemigroups of finite transformation and partition monoids, arXiv:1706.04967 [math.GR], 2017.
A. Goupil, H. Cloutier, and F. Nouboud, Enumeration of polyominoes inscribed in a rectangle Discrete Applied Mathematics 158(2010), pp. 2014-2023.
S. Heubach and T. Mansour, Counting rises, levels and drops in compositions, arXiv:math/0310197 [math.CO], 2003.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1067
D. Levin, L. Pudwell, M. Riehl, and A. Sandberg, Pattern Avoidance on k-ary Heaps, Slides of Talk, 2014.
D. Merlini, F. Uncini and M. C. Verri, A unified approach to the study of general and palindromic compositions, Integers 4 (2004), A23, 26 pp.
Agustín Moreno Cañadas, Hernán Giraldo, and Robinson Julian Serna Vanegas, Some integer partitions induced by orbits of Dynkin type, Far East Journal of Mathematical Sciences (FJMS), Vol. 101, No. 12 (2017), pp. 2745-2766.
Laurent Noé, Spaced seed design on profile HMMs for precise HTS read-mapping efficient sliding window product on the matrix semi-group, in Rapide Bilan 2012-2013, Laurent, LIFL, Université Lille 1 - INRIA Journées au vert 11 et 12 juin 2013, Laurent, Année 2012-2013.
Valentin Ovsienko, Villes paires et impaires (Oddtown and Eventown) I, Images des Mathématiques, CNRS, 2013 (in French).
Dennis Walsh, Notes on symmetric subsets of {1, 2, ..., n}
A. Yajima, How to calculate the number of stereoisomers of inositol-homologs, Bull. Chem. Soc. Jpn. 2014, 87, 1260-1264 | doi:10.1246/bcsj.20140204. See Tables 1 and 2 (and text). - _N. J. A. Sloane_, Mar 26 2015
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (0,2).

Cf. A006995, A057148, A079944, A112030, A112033.
a(n) = A094718(3, n).
Cf. A001333.
See A052955 for partial sums (without the initial term).
A000079 gives the odd-indexed terms of a(n).
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent: A029744 (s(n)); A052955 (s(n)-1), A027383 (s(n)-2), A354788 (s(n)-3), A347789 (s(n)-4), A209721 (s(n)+1), A209722 (s(n)+2), A343177 (s(n)+3), A209723 (s(n)+4); A060482, A136252 (minor differences from A354788 at the start); A354785 (3*s(n)), A354789 (3*s(n)-7). The first differences of A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences: A016116, A060546, A117575, A131572, A152166, A158780, A163403, A320770. The bisections of A029744 are A000079 and A007283. - N. J. A. Sloane, Jul 14 2022

List([0..45],n->2^Int(n/2)); # Muniru A Asiru, Apr 03 2018

[2^Floor(n/2): n in [0..50]]; // Vincenzo Librandi, Aug 16 2011

A016116:= proc(n): 2^floor(n/2) end: seq(A016116(n), n=0..42); # Dennis P. Walsh, Oct 27 2009

Table[ 2^Floor[n/2], {n, 0, 42}] (* Robert G. Wilson v, Jun 05 2004 *)
With[{c=2^Range[0,30]},Riffle[c,c]] (* Harvey P. Dale, Jan 23 2015 *)
CoefficientList[Series[(1+x)/(1-2*x^2), {x, 0, 50}], x] (* Stefano Spezia, Sep 07 2018 *)

makelist(2^floor(n/2), n, 0, 50); /* Martin Ettl, Oct 17 2012 */

PARI
```
a(n)=if(n<0,0,2^(n\2))
```

def A016116(n): return 1 << n//2 # Chai Wah Wu, Jun 07 2022

def A016116():
    x, y = -1, 0
    while True:
        yield -x
        x, y = x + y, x - y
a = A016116(); [next(a) for i in range(40)]  # Peter Luschny, Jul 11 2013

a(n) = a(n-1)*a(n-2)/a(n-3) = 2*a(n-2) = 2^A004526(n).
G.f.: (1+x)/(1-2*x^2).
a(n) = (1/2 + sqrt(1/8))*sqrt(2)^n + (1/2 - sqrt(1/8))*(-sqrt(2))^n. - Ralf Stephan, Mar 11 2003
E.g.f.: cosh(sqrt(2)*x) + sinh(sqrt(2)*x)/sqrt(2). - Paul Barry, Jul 16 2003
The signed sequence (-1)^n*2^floor(n/2) has a(n) = (sqrt(2))^n(1/2 - sqrt(2)/4) + (-sqrt(2))^n(1/2 + sqrt(2)/4). It is the inverse binomial transform of A000129(n-1). - Paul Barry, Apr 21 2004
Diagonal sums of A046854. a(n) = Sum_{k=0..n} binomial(floor(n/2), k). - Paul Barry, Jul 07 2004
a(n) = a(n-2) + 2^floor((n-2)/2). - Paul Barry, Jul 14 2004
a(n) = Sum_{k=0..floor(n/2)} binomial(floor(n/2), floor(k/2)). - Paul Barry, Jul 15 2004
E.g.f.: cosh(asinh(1) + sqrt(2)*x)/sqrt(2). - Michael Somos, Feb 28 2005
a(n) = Sum_{k=0..n} A103633(n,k). - Philippe Deléham, Dec 03 2006
a(n) = 2^(n/2)*((1 + (-1)^n)/2 + (1-(-1)^n)/(2*sqrt(2))). - Paul Barry, Nov 12 2009
a(n) = 2^((2*n - 1 + (-1)^n)/4). - Luce ETIENNE, Sep 20 2014

A027383 a(2n) = 32^n - 2; a(2*n+1) = 2^(n+2) - 2.

Original entry on oeis.org

1, 2, 4, 6, 10, 14, 22, 30, 46, 62, 94, 126, 190, 254, 382, 510, 766, 1022, 1534, 2046, 3070, 4094, 6142, 8190, 12286, 16382, 24574, 32766, 49150, 65534, 98302, 131070, 196606, 262142, 393214, 524286, 786430, 1048574, 1572862, 2097150, 3145726, 4194302, 6291454

Offset: 0

R. K. Guy

Number of balanced strings of length n: let d(S) = #(1's) - #(0's), # == count in S, then S is balanced if every substring T of S has -2 <= d(T) <= 2.
Number of "fold lines" seen when a rectangular piece of paper is folded n+1 times along alternate orthogonal directions and then unfolded. - Quim Castellsaguer (qcastell(AT)pie.xtec.es), Dec 30 1999
Also the number of binary strings with the property that, when scanning from left to right, once the first 1 is seen in position j, there must be a 1 in positions j+2, j+4, ... until the end of the string. (Positions j+1, j+3, ... can be occupied by 0 or 1.) - Jeffrey Shallit, Sep 02 2002
a(n-1) is also the Moore lower bound on the order of a (3,n)-cage. - Eric W. Weisstein, May 20 2003 and Jason Kimberley, Oct 30 2011
Partial sums of A016116. - Hieronymus Fischer, Sep 15 2007
Equals row sums of triangle A152201. - Gary W. Adamson, Nov 29 2008
From John P. McSorley, Sep 28 2010: (Start)
a(n) = DPE(n+1) is the total number of k-double-palindromes of n up to cyclic equivalence. See sequence A180918 for the definitions of a k-double-palindrome of n and of cyclic equivalence. Sequence A180918 is the 'DPE(n,k)' triangle read by rows where DPE(n,k) is the number of k-double-palindromes of n up to cyclic equivalence. For example, we have a(4) = DPE(5) = DPE(5,1) + DPE(5,2) + DPE(5,3) + DPE(5,4) + DPE(5,5) = 0 + 2 + 2 + 1 + 1 = 6.
The 6 double-palindromes of 5 up to cyclic equivalence are 14, 23, 113, 122, 1112, 11111. They come from cyclic equivalence classes {14,41}, {23,32}, {113,311,131}, {122,212,221}, {1112,2111,1211,1121}, and {11111}. Hence a(n)=DPE(n+1) is the total number of cyclic equivalence classes of n containing at least one double-palindrome.
(End)
From Herbert Eberle, Oct 02 2015: (Start)
For n > 0, there is a red-black tree of height n with a(n-1) internal nodes and none with less.
In order a red-black tree of given height has minimal number of nodes, it has exactly 1 path with strictly alternating red and black nodes. All nodes outside this height defining path are black.
Consider:
mrbt5                R
                    / \
                   /   \
                  /     \
                 /       B
                /       / \
mrbt4          B       /   B
              / \     B   E E
             /   B   E E
mrbt3       R   E E
           / \
          /   B
mrbt2    B   E E
        / E
mrbt1  R
      E E
(Red nodes shown as R, blacks as B, externals as E.)
Red-black trees mrbt1, mrbt2, mrbt3, mrbt4, mrbt5 of respective heights h = 1, 2, 3, 4, 5; all minimal in the number of internal nodes, namely 1, 2, 4, 6, 10.
Recursion (let n = h-1): a(-1) = 0, a(n) = a(n-1) + 2^floor(n/2), n >= 0.
(End)
Also the number of strings of length n with the digits 1 and 2 with the property that the sum of the digits of all substrings of uneven length is not divisible by 3. An example with length 8 is 21221121. - Herbert Kociemba, Apr 29 2017
a(n-2) is the number of achiral n-bead necklaces or bracelets using exactly two colors.  For n=4, the four arrangements are AAAB, AABB, ABAB, and ABBB. - Robert A. Russell, Sep 26 2018
Partial sums of powers of 2 repeated 2 times, like A200672 where is 3 times. - Yuchun Ji, Nov 16 2018
Also the number of binary words of length n with cuts-resistance <= 2, where, for the operation of shortening all runs by one, cuts-resistance is the number of applications required to reach an empty word. Explicitly, these are words whose sequence of run-lengths, all of which are 1 or 2, has no odd-length run of 1's sandwiched between two 2's. - Gus Wiseman, Nov 28 2019
Also the number of up-down paths with n steps such that the height difference between the highest and lowest points is at most 2. - Jeremy Dover, Jun 17 2020
Also the number of non-singleton integer compositions of n + 2 with no odd part other than the first or last. Including singletons gives A052955. This is an unsorted (or ordered) version of A351003. The version without even (instead of odd) interior parts is A001911, complement A232580. Note that A000045(n-1) counts compositions without odd parts, with non-singleton case A077896, and A052952/A074331 count non-singleton compositions without even parts. Also the number of compositions y of n + 1 such that y_i = y_{i+1} for all even i. - Gus Wiseman, Feb 19 2022

			After 3 folds one sees 4 fold lines.
Example: a(3) = 6 because the strings 001, 010, 100, 011, 101, 110 have the property.
Binary: 1, 10, 100, 110, 1010, 1110, 10110, 11110, 101110, 111110, 1011110, 1111110, 10111110, 11111110, 101111110, 111111110, 1011111110, 1111111110, 10111111110, ... - _Jason Kimberley_, Nov 02 2011
Example: Partial sums of powers of 2 repeated 2 times:
a(3) = 1+1+2 = 4;
a(4) = 1+1+2+2 = 6;
a(5) = 1+1+2+2+4 = 10.
_Yuchun Ji_, Nov 16 2018

John P. McSorley: Counting k-compositions of n with palindromic and related structures. Preprint, 2010. [John P. McSorley, Sep 28 2010]

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
J. Jordan and R. Southwell, Further Properties of Reproducing Graphs, Applied Mathematics, Vol. 1 No. 5, 2010, pp. 344-350. - From _N. J. A. Sloane_, Feb 03 2013
Leonard F. Klosinski, Gerald L. Alexanderson and Loren C. Larson, Under misprinted head B3, Amer Math. Monthly, 104(1997) 753-754.
Laurent Noé, Spaced seed design on profile HMMs for precise HTS read-mapping efficient sliding window product on the matrix semi-group, in Rapide Bilan 2012-2013, LIFL, Université Lille 1 - INRIA Journées au vert 11 et 12 juin 2013.
Eric Weisstein's World of Mathematics, Cage Graph
Index entries for sequences obtained by enumerating foldings
Index entries for linear recurrences with constant coefficients, signature (1,2,-2).

Cf. A132666, A152201.
Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), this sequence (k=3), A062318 (k=4), A061547 (k=5), A198306 (k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7). - Jason Kimberley, Oct 30 2011
Cf. A000066 (actual order of a (3,g)-cage).
Bisections are A033484 (even) and A000918 (odd).
a(n) = A305540(n+2,2), the second column of the triangle.
Numbers whose binary expansion is a balanced word are A330029.
Binary words counted by cuts-resistance are A319421 or A329860.
Cf. A000975, A062011, A329862, A329863.
The complementary compositions are counted by A274230(n-1) + 1, with bisections A060867 (even) and A134057 (odd).
Cf. A000346, A000984, A001405, A001700, A011782 (compositions).
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent: A029744 (s(n)); A052955 (s(n)-1), A027383 (s(n)-2), A354788 (s(n)-3), A347789 (s(n)-4), A209721 (s(n)+1), A209722 (s(n)+2), A343177 (s(n)+3), A209723 (s(n)+4); A060482, A136252 (minor differences from A354788 at the start); A354785 (3*s(n)), A354789 (3*s(n)-7). The first differences of A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences: A016116, A060546, A117575, A131572, A152166, A158780, A163403, A320770. The bisections of A029744 are A000079 and A007283. - N. J. A. Sloane, Jul 14 2022

import Data.List (transpose)
a027383 n = a027383_list !! n
a027383_list = concat $ transpose [a033484_list, drop 2 a000918_list]
-- Reinhard Zumkeller, Jun 17 2015

[2^Floor((n+2)/2)+2^Floor((n+1)/2)-2: n in [0..50]]; // Vincenzo Librandi, Aug 16 2011

a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=2*a[n-2]+2 od: seq(a[n], n=1..41); # Zerinvary Lajos, Mar 16 2008

a[n_?EvenQ] := 3*2^(n/2)-2; a[n_?OddQ] := 2^(2+(n-1)/2)-2; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Oct 21 2011, after Quim Castellsaguer *)
LinearRecurrence[{1, 2, -2}, {1, 2, 4}, 41] (* Robert G. Wilson v, Oct 06 2014 *)
Table[Length[Select[Tuples[{0,1},n],And[Max@@Length/@Split[#]<=2,!MatchQ[Length/@Split[#],{_,2,ins:1..,2,_}/;OddQ[Plus[ins]]]]&]],{n,0,15}] (* Gus Wiseman, Nov 28 2019 *)

a(n)=2^(n\2+1)+2^((n+1)\2)-2 \\ Charles R Greathouse IV, Oct 21 2011

def a(n): return 2**((n+2)//2) + 2**((n+1)//2) - 2
print([a(n) for n in range(43)]) # Michael S. Branicky, Feb 19 2022

a(0)=1, a(1)=2; thereafter a(n+2) = 2*a(n) + 2.
a(2n) = 3*2^n - 2 = A033484(n);
a(2n-1) = 2^(n+1) - 2 = A000918(n+1).
G.f.: (1 + x)/((1 - x)*(1 - 2*x^2)). - David Callan, Jul 22 2008
a(n) = Sum_{k=0..n} 2^min(k, n-k).
a(n) = 2^floor((n+2)/2) + 2^floor((n+1)/2) - 2. - Quim Castellsaguer (qcastell(AT)pie.xtec.es)
a(n) = 2^(n/2)*(3 + 2*sqrt(2) + (3-2*sqrt(2))*(-1)^n)/2 - 2. - Paul Barry, Apr 23 2004
a(n) = A132340(A052955(n)). - Reinhard Zumkeller, Aug 20 2007
a(n) = A052955(n+1) - 1. - Hieronymus Fischer, Sep 15 2007
a(n) = A132666(a(n+1)) - 1. - Hieronymus Fischer, Sep 15 2007
a(n) = A132666(a(n-1)+1) for n > 0. - Hieronymus Fischer, Sep 15 2007
A132666(a(n)) = a(n-1) + 1 for n > 0. - Hieronymus Fischer, Sep 15 2007
G.f.: (1 + x)/((1 - x)*(1 - 2*x^2)). - David Callan, Jul 22 2008
a(n) = 2*( (a(n-2)+1) mod (a(n-1)+1) ), n > 1. - Pierre Charland, Dec 12 2010
a(n) = A136252(n-1) + 1, for n > 0. - Jason Kimberley, Nov 01 2011
G.f.: (1+x*R(0))/(1-x), where R(k) = 1 + 2*x/( 1 - x/(x + 1/R(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 16 2013
a(n) = 2^((2*n + 3*(1-(-1)^n))/4)*3^((1+(-1)^n)/2) - 2. - Luce ETIENNE, Sep 01 2014
a(n) = a(n-1) + 2^floor((n-1)/2) for n>0, a(0)=1. - Yuchun Ji, Nov 23 2018
E.g.f.: 3*cosh(sqrt(2)*x) - 2*cosh(x) + 2*sqrt(2)*sinh(sqrt(2)*x) - 2*sinh(x). - Stefano Spezia, Apr 06 2022

More terms from Larry Reeves (larryr(AT)acm.org), Mar 24 2000

Replaced definition with a simpler one. - N. J. A. Sloane, Jul 09 2022

A029744 Numbers of the form 2^n or 3*2^n.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 768, 1024, 1536, 2048, 3072, 4096, 6144, 8192, 12288, 16384, 24576, 32768, 49152, 65536, 98304, 131072, 196608, 262144, 393216, 524288, 786432, 1048576, 1572864, 2097152, 3145728, 4194304

Offset: 1

N. J. A. Sloane

This entry is a list, and so has offset 1. WARNING: However, in this entry several comments, formulas and programs seem to refer to the original version of this sequence which had offset 0. - M. F. Hasler, Oct 06 2014
Number of necklaces with n-1 beads and two colors that are the same when turned over and hence have reflection symmetry. [edited by Herbert Kociemba, Nov 24 2016]
The subset {a(1),...,a(2k)} contains all proper divisors of 3*2^k. - Ralf Stephan, Jun 02 2003
Let k = any nonnegative integer and j = 0 or 1. Then n+1 = 2k + 3j and a(n) = 2^k*3^j. - Andras Erszegi (erszegi.andras(AT)chello.hu), Jul 30 2005
Smallest number having no fewer prime factors than any predecessor, a(0)=1; A110654(n) = A001222(a(n)); complement of A116451. - Reinhard Zumkeller, Feb 16 2006
A093873(a(n)) = 1. - Reinhard Zumkeller, Oct 13 2006
a(n) = a(n-1) + a(n-2) - gcd(a(n-1), a(n-2)), n >= 3, a(1)=2, a(2)=3. - Ctibor O. Zizka, Jun 06 2009
Where records occur in A048985: A193652(n) = A048985(a(n)) and A193652(n) < A048985(m) for m < a(n). - Reinhard Zumkeller, Aug 08 2011
A002348(a(n)) = A000079(n-3) for n > 2. - Reinhard Zumkeller, Mar 18 2012
Without initial 1, third row in array A228405. - Richard R. Forberg, Sep 06 2013
Also positions of records in A048673. A246360 gives the record values. - Antti Karttunen, Sep 23 2014
Known in numerical mathematics as "Bulirsch sequence", used in various extrapolation methods for step size control. - Peter Luschny, Oct 30 2019
For n > 1, squares of the terms can be expressed as the sum of two powers of two: 2^x + 2^y. - Karl-Heinz Hofmann, Sep 08 2022

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Spencer Daugherty, Pamela E. Harris, Ian Klein, and Matt McClinton, Metered Parking Functions, arXiv:2406.12941 [math.CO], 2024. See pp. 11, 22.
Michael De Vlieger, Thomas Scheuerle, Rémy Sigrist, N. J. A. Sloane, and Walter Trump, The Binary Two-Up Sequence, arXiv:2209.04108 [math.CO], Sep 11 2022.
David Eppstein, Making Change in 2048, arXiv:1804.07396 [cs.DM], 2018.
Guo-Niu Han, Enumeration of Standard Puzzles. [Cached copy]
John P. McSorley and Alan H. Schoen, Rhombic tilings of (n,k)-ovals, (n, k, lambda)-cyclic difference sets, and related topics, Discrete Math., 313 (2013), 129-154. - From _N. J. A. Sloane_, Nov 26 2012
Index entries for linear recurrences with constant coefficients, signature (0,2).
Index entries for sequences related to necklaces

Cf. A056493, A038754, A063759. Union of A000079 and A007283.
First differences are in A016116(n-1).
Cf. A082125, A094958, A048739, A048985, A193652, A048673, A064216, A246360, A354785.
Row sums of the triangle in sequence A119963. - John P. McSorley, Aug 31 2010
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent. There may be minor differences from (s(n)) at the start, and a shift of indices. A029744 (s(n)); A052955 (s(n)-1), A027383 (s(n)-2), A354788 (s(n)-3), A060482 (s(n)-3); A136252 (s(n)-3); A347789 (s(n)-4), A209721 (s(n)+1), A209722 (s(n)+2), A343177 (s(n)+3), A209723 (s(n)+4);   A354785 (3*s(n)), A061776 (3*s(n)-6); A354789 (3*s(n)-7). The first differences of A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences: A016116, A060546, A117575, A131572, A152166, A158780, A163403, A320770. The bisections of A029744 are A000079 and A007283. - N. J. A. Sloane, Jul 14 2022

a029744 n = a029744_list !! (n-1)
a029744_list = 1 : iterate
   (\x -> if x `mod` 3 == 0 then 4 * x `div` 3 else 3 * x `div` 2) 2
-- Reinhard Zumkeller, Mar 18 2012

1,seq(op([2^i,3*2^(i-1)]),i=1..100); # Robert Israel, Sep 23 2014

CoefficientList[Series[(-x^2 - 2*x - 1)/(2*x^2 - 1), {x, 0, 200}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)
Function[w, DeleteCases[Union@ Flatten@ w, k_ /; k > Max@ First@ w]]@ TensorProduct[{1, 3}, 2^Range[0, 22]] (* Michael De Vlieger, Nov 24 2016 *)
LinearRecurrence[{0,2},{1,2,3},50] (* Harvey P. Dale, Jul 04 2017 *)

PARI
```
a(n)=if(n%2,3/2,2)<<((n-1)\2)\1
```

def A029744(n):
    if n == 1: return 1
    elif n % 2 == 0: return 2**(n//2)
    else: return 3 * 2**((n-3)//2) # Karl-Heinz Hofmann, Sep 08 2022

(define (A029744 n) (cond ((<= n 1) n) ((even? n) (expt 2 (/ n 2))) (else (* 3 (expt 2 (/ (- n 3) 2)))))) ;; Antti Karttunen, Sep 23 2014

a(n) = 2*A000029(n) - A000031(n).
For n > 2, a(n) = 2*a(n - 2); for n > 3, a(n) = a(n - 1)*a(n - 2)/a(n - 3). G.f.: (1 + x)^2/(1 - 2*x^2). - Henry Bottomley, Jul 15 2001, corrected May 04 2007
a(0)=1, a(1)=1 and a(n) = a(n-2) * ( floor(a(n-1)/a(n-2)) + 1 ). - Benoit Cloitre, Aug 13 2002
(3/4 + sqrt(1/2))*sqrt(2)^n + (3/4 - sqrt(1/2))*(-sqrt(2))^n. a(0)=1, a(2n) = a(n-1)*a(n), a(2n+1) = a(n) + 2^floor((n-1)/2). - Ralf Stephan, Apr 16 2003 [Seems to refer to the original version with offset=0. - M. F. Hasler, Oct 06 2014]
Binomial transform is A048739. - Paul Barry, Apr 23 2004
E.g.f.: (cosh(x/sqrt(2)) + sqrt(2)sinh(x/sqrt(2)))^2.
a(1) = 1; a(n+1) = a(n) + A000010(a(n)). - Stefan Steinerberger, Dec 20 2007
u(2)=1, v(2)=1, u(n)=2*v(n-1), v(n)=u(n-1), a(n)=u(n)+v(n). - Jaume Oliver Lafont, May 21 2008
For n => 3, a(n) = sqrt(2*a(n-1)^2 + (-2)^(n-3)). - Richard R. Forberg, Aug 20 2013
a(n) = A064216(A246360(n)). - Antti Karttunen, Sep 23 2014
a(n) = sqrt((17 - (-1)^n)*2^(n-4)) for n >= 2. - Anton Zakharov, Jul 24 2016
Sum_{n>=1} 1/a(n) = 8/3. - Amiram Eldar, Nov 12 2020
a(n) = 2^(n/2) if n is even. a(n) = 3 * 2^((n-3)/2) if n is odd and for n>1. - Karl-Heinz Hofmann, Sep 08 2022

Corrected and extended by Joe Keane (jgk(AT)jgk.org), Feb 20 2000

A225546 Tek's flip: Write n as the product of distinct factors of the form prime(i)^(2^(j-1)) with i and j integers, and replace each such factor with prime(j)^(2^(i-1)).

Original entry on oeis.org

1, 2, 4, 3, 16, 8, 256, 6, 9, 32, 65536, 12, 4294967296, 512, 64, 5, 18446744073709551616, 18, 340282366920938463463374607431768211456, 48, 1024, 131072, 115792089237316195423570985008687907853269984665640564039457584007913129639936, 24, 81, 8589934592, 36, 768

Offset: 1

Paul Tek, May 10 2013

This is a multiplicative self-inverse permutation of the integers.
A225547 gives the fixed points.
From Antti Karttunen and Peter Munn, Feb 02 2020: (Start)
This sequence operates on the Fermi-Dirac factors of a number. As arranged in array form, in A329050, this sequence reflects these factors about the main diagonal of the array, substituting A329050[j,i] for A329050[i,j], and this results in many relationships including significant homomorphisms.
This sequence provides a relationship between the operations of squaring and prime shift (A003961) because each successive column of the A329050 array is the square of the previous column, and each successive row is the prime shift of the previous row.
A329050 gives examples of how significant sets of numbers can be formed by choosing their factors in relation to rows and/or columns. This sequence therefore maps equivalent derived sets by exchanging rows and columns. Thus odd numbers are exchanged for squares, squarefree numbers for powers of 2 etc.
Alternative construction: For n > 1, form a vector v of length A299090(n), where each element v[i] for i=1..A299090(n) is a product of those distinct prime factors p(i) of n whose exponent e(i) has the bit (i-1) "on", or 1 (as an empty product) if no such exponents are present. a(n) is then Product_{i=1..A299090(n)} A000040(i)^A048675(v[i]). Note that because each element of vector v is squarefree, it means that each exponent A048675(v[i]) present in the product is a "submask" (not all necessarily proper) of the binary string A087207(n).
This permutation effects the following mappings:
A000035(a(n)) = A010052(n), A010052(a(n)) = A000035(n). [Odd numbers <-> Squares]
A008966(a(n)) = A209229(n), A209229(a(n)) = A008966(n). [Squarefree numbers <-> Powers of 2]
(End)
From Antti Karttunen, Jul 08 2020: (Start)
Moreover, we see also that this sequence maps between A016825 (Numbers of the form 4k+2) and A001105 (2*squares) as well as between A008586 (Multiples of 4) and A028983 (Numbers with even sum of the divisors).
(End)

			  7744  = prime(1)^2^(2-1)*prime(1)^2^(3-1)*prime(5)^2^(2-1).
a(7744) = prime(2)^2^(1-1)*prime(3)^2^(1-1)*prime(2)^2^(5-1) = 645700815.

Cf. A225547 (fixed points) and the subsequences listed there.
Transposes A329050, A329332.
An automorphism of positive integers under the binary operations A059895, A059896, A059897, A306697, A329329.
An automorphism of A059897 subgroups: A000379, A003159, A016754, A122132.
Permutes lists where membership is determined by number of Fermi-Dirac factors: A000028, A050376, A176525, A268388.
Also permutes: A036554, A059404, A066427, A066428, A072774, A331593.
Sequences f that satisfy f(a(n)) = f(n): A048675, A064179, A064547, A097248, A302777, A331592.
Pairs of sequences (f,g) that satisfy a(f(n)) = g(a(n)): (A000265,A008833), (A000290,A003961), (A005843,A334747), (A006519,A007913), (A008586,A334748).
Pairs of sequences (f,g) that satisfy a(f(n)) = g(n), possibly with offset change: (A000040,A001146), (A000079,A019565).
Pairs of sequences (f,g) that satisfy f(a(n)) = g(n), possibly with offset change: (A000035, A010052), (A008966, A209229), (A007814, A248663), (A061395, A299090), (A087207, A267116), (A225569, A227291).
Cf. A331287 [= gcd(a(n),n)].
Cf. A331288 [= min(a(n),n)], see also A331301.
Cf. A331309 [= A000005(a(n)), number of divisors].
Cf. A331590 [= a(a(n)*a(n))].
Cf. A331591 [= A001221(a(n)), number of distinct prime factors], see also A331593.
Cf. A331740 [= A001222(a(n)), number of prime factors with multiplicity].
Cf. A331733 [= A000203(a(n)), sum of divisors].
Cf. A331734 [= A033879(a(n)), deficiency].
Cf. A331735 [= A009194(a(n))].
Cf. A331736 [= A000265(a(n)) = a(A008833(n)), largest odd divisor].
Cf. A335914 [= A038040(a(n))].
A self-inverse isomorphism between pairs of A059897 subgroups: (A000079,A005117), (A000244,A062503), (A000290\{0},A005408), (A000302,A056911), (A000351,A113849 U {1}), (A000400,A062838), (A001651,A252895), (A003586,A046100), (A007310,A000583), (A011557,A113850 U {1}), (A028982,A042968), (A053165,A065331), (A262675,A268390).
A bijection between pairs of sets: (A001248,A011764), (A007283,A133466), (A016825, A001105), (A008586, A028983).
Cf. also A336321, A336322 (compositions with another involution, A122111).

Array[If[# == 1, 1, Times @@ Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]] &, 28] (* Michael De Vlieger, Jan 21 2020 *)

A019565(n) = factorback(vecextract(primes(logint(n+!n, 2)+1), n));
a(n) = {my(f=factor(n)); for (i=1, #f~, my(p=f[i,1]); f[i,1] = A019565(f[i,2]); f[i,2] = 2^(primepi(p)-1);); factorback(f);} \\ Michel Marcus, Nov 29 2019

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A225546(n) = if(1==n,1,my(f=factor(n),u=#binary(vecmax(f[, 2])),prods=vector(u,x,1),m=1,e); for(i=1,u,for(k=1,#f~, if(bitand(f[k,2],m),prods[i] *= f[k,1])); m<<=1); prod(i=1,u,prime(i)^A048675(prods[i]))); \\ Antti Karttunen, Feb 02 2020

from math import prod
from sympy import prime, primepi, factorint
def A225546(n): return prod(prod(prime(i) for i, v in enumerate(bin(e)[:1:-1],1) if v == '1')**(1<Chai Wah Wu, Mar 17 2023

Multiplicative, with a(prime(i)^j) = A019565(j)^A000079(i-1).
a(prime(i)) = 2^(2^(i-1)).
From Antti Karttunen and Peter Munn, Feb 06 2020: (Start)
a(A329050(n,k)) = A329050(k,n).
a(A329332(n,k)) = A329332(k,n).
Equivalently, a(A019565(n)^k) = A019565(k)^n. If n = 1, this gives a(2^k) = A019565(k).
a(A059897(n,k)) = A059897(a(n), a(k)).
The previous formula implies a(n*k) = a(n) * a(k) if A059895(n,k) = 1.
a(A000040(n)) = A001146(n-1); a(A001146(n)) = A000040(n+1).
a(A000290(a(n))) = A003961(n); a(A003961(a(n))) = A000290(n) = n^2.
a(A000265(a(n))) = A008833(n); a(A008833(a(n))) = A000265(n).
a(A006519(a(n))) = A007913(n); a(A007913(a(n))) = A006519(n).
A007814(a(n)) = A248663(n); A248663(a(n)) = A007814(n).
A048675(a(n)) = A048675(n) and A048675(a(2^k * n)) = A048675(2^k * a(n)) = k + A048675(a(n)).
(End)
From Antti Karttunen and Peter Munn, Jul 08 2020: (Start)
For all n >= 1, a(2n) = A334747(a(n)).
In particular, for n = A003159(m), m >= 1, a(2n) = 2*a(n). [Note that A003159 includes all odd numbers]
(End)

Name edited by Peter Munn, Feb 14 2020

"Tek's flip" prepended to the name by Antti Karttunen, Jul 08 2020

A243071 Permutation of nonnegative integers: a(1) = 0, a(2) = 1, a(2n) = 2a(n), a(2n+1) = 1 + 2a(A064989(2n+1)).

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 15, 4, 5, 14, 31, 12, 63, 30, 13, 8, 127, 10, 255, 28, 29, 62, 511, 24, 11, 126, 9, 60, 1023, 26, 2047, 16, 61, 254, 27, 20, 4095, 510, 125, 56, 8191, 58, 16383, 124, 25, 1022, 32767, 48, 23, 22, 253, 252, 65535, 18, 59, 120, 509, 2046, 131071

Offset: 1

Antti Karttunen, Jun 20 2014

nonn

Note the indexing: the domain starts from 1, while the range includes also zero.

See also the comments at A163511, which is the inverse permutation to this one.

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A163511.
Cf. A000040, A000225, A007814, A054429, A064989, A064216, A122111, A209229, A245611 (= (a(2n-1)-1)/2, for n > 1), A245612, A292383, A292385, A297171 (Möbius transform).
Cf. A007283 (known positions where a(n)=n), A364256 [= gcd(n,a(n))], A364288 [= n-a(n)], A364289 [where a(n)>=n], A364290 [where a(n)A364291 [where a(n)<=n], A364497 [where n|a(n)].
Cf. A156552 (variant with inverted binary code), A253566, A332215, A332811, A334859 (other variants).

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A243071(n) = if(n<=2, n-1, if(!(n%2), 2*A243071(n/2), 1+(2*A243071(A064989(n))))); \\ Antti Karttunen, Jul 18 2020

A243071(n) = if(n<=2, n-1, my(f=factor(n), p, p2=1, res=0); for(i=1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p*p2*(2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); ((3<<#binary(res\2))-res-1)); \\ (Combining programs given in A156552 and A054429) - Antti Karttunen, Jul 28 2023

from functools import reduce
from sympy import factorint, prevprime
from operator import mul
def a064989(n):
    f = factorint(n)
    return 1 if n==1 else reduce(mul, (1 if i==2 else prevprime(i)**f[i] for i in f))
def a(n): return n - 1 if n<3 else 2*a(n//2) if n%2==0 else 1 + 2*a(a064989(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 15 2017

;; With memoizing definec-macro from Antti Karttunen's IntSeq-library.
(definec (A243071 n) (cond ((<= n 2) (- n 1)) ((even? n) (* 2 (A243071 (/ n 2)))) (else (+ 1 (* 2 (A243071 (A064989 n)))))))

a(1) = 0, a(2) = 1, a(2n) = 2*a(n), a(2n+1) = 1 + 2*a(A064989(2n+1)).
For n >= 1, a(A000040(n)) = A000225(n).
For n >= 1, a(2n+1) = 1 + 2*a(A064216(n+1)).
From Antti Karttunen, Jul 18 2020: (Start)
a(n) = A245611(A048673(n)).
a(n) = A253566(A122111(n)).
a(n) = A334859(A225546(n)).
For n >= 2, a(n) = A054429(A156552(n)).
a(n) = A292383(n) + A292385(n) = A292383(n) OR A292385(n).
For n > 1, A007814(a(n)) = A007814(n) - A209229(n). [This map  preserves the 2-adic valuation of n, except when n is a power of two, in which cases it is decremented by one.]
(End)

A033484 a(n) = 3*2^n - 2.

Original entry on oeis.org

1, 4, 10, 22, 46, 94, 190, 382, 766, 1534, 3070, 6142, 12286, 24574, 49150, 98302, 196606, 393214, 786430, 1572862, 3145726, 6291454, 12582910, 25165822, 50331646, 100663294, 201326590, 402653182, 805306366, 1610612734, 3221225470

Offset: 0

N. J. A. Sloane

Number of nodes in rooted tree of height n in which every node (including the root) has valency 3.
Pascal diamond numbers: reflect Pascal's n-th triangle vertically and sum all elements. E.g., a(3)=1+(1+1)+(1+2+1)+(1+1)+1. - Paul Barry, Jun 23 2003
Number of 2 X n binary matrices avoiding simultaneously the right-angled numbered polyomino patterns (ranpp) (00;1), (10;0) and (11;0). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1 < i2 and j1 < j2 and these elements are in the same relative order as those in the triple (x,y,z). - Sergey Kitaev, Nov 11 2004
Binomial and inverse binomial transform are in A001047 (shifted) and A122553. - R. J. Mathar, Sep 02 2008
a(n) = (Sum_{k=0..n-1} a(n)) + (2*n + 1); e.g., a(3) = 22 = (1 + 4 + 10) + 7. - Gary W. Adamson, Jan 21 2009
Let P(A) be the power set of an n-element set A and R be a relation on P(A) such that for all x, y of P(A), xRy if either 0) x is a proper subset of y or y is a proper subset of x and x and y are disjoint, or 1) x equals y. Then a(n) = |R|. - Ross La Haye, Mar 19 2009
Equals the Jacobsthal sequence A001045 convolved with (1, 3, 4, 4, 4, 4, 4, ...). - Gary W. Adamson, May 24 2009
Equals the eigensequence of a triangle with the odd integers as the left border and the rest 1's. - Gary W. Adamson, Jul 24 2010
An elephant sequence, see A175655. For the central square four A[5] vectors, with decimal values 58, 154, 178 and 184, lead to this sequence. For the corner squares these vectors lead to the companion sequence A097813. - Johannes W. Meijer, Aug 15 2010
a(n+2) is the integer with bit string "10" * "1"^n * "10".
a(n) = A027383(2n). - Jason Kimberley, Nov 03 2011
a(n) = A153893(n)-1 = A083416(2n+1). - Philippe Deléham, Apr 14 2013
a(n) = A082560(n+1,A000079(n)) = A232642(n+1,A128588(n+1)). - Reinhard Zumkeller, May 14 2015
a(n) is the sum of the entries in the n-th and (n+1)-st rows of Pascal's triangle minus 2. - Stuart E Anderson, Aug 27 2017
Also the number of independent vertex sets and vertex covers in the complete tripartite graph K_{n,n,n}. - Eric W. Weisstein, Sep 21 2017
Apparently, a(n) is the least k such that the binary expansion of A000045(k) ends with exactly n+1 ones. - Rémy Sigrist, Sep 25 2021
a(n) is the number of root ancestral configurations for a pair consisting of a matching gene tree and species tree with the modified lodgepole shape and n+1 cherry nodes. - Noah A Rosenberg, Jan 16 2025

			Binary: 1, 100, 1010, 10110, 101110, 1011110, 10111110, 101111110, 1011111110, 10111111110, 101111111110, 1011111111110, 10111111111110,
G.f. = 1 + 4*x + 10*x^2 + 22*x^3 + 46*x^4 + 94*x^5 + 190*x^6 + 382*x^7 + ...

J. Riordan, Series-parallel realization of the sum modulo 2 of n switching variables, in Claude Elwood Shannon: Collected Papers, edited by N. J. A. Sloane and A. D. Wyner, IEEE Press, NY, 1993, pp. 877-878.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, The Triple Riordan Group, arXiv:2412.05461 [math.CO], 2024. See pp. 3, 10.
Dennis E. Davenport, Shakuan K. Frankson, Louis W. Shapiro, and Leon C. Woodson, An Invitation to the Riordan Group, Enum. Comb. Appl. (2024) Vol. 4, No. 3, Art. #S2S1. See p. 22.
Erik D. Demaine et al., Picture-Hanging Puzzles, arXiv:1203.3602 [cs.DS], 2012, 2014. See p. 8, actually length(Sn) is 2^n+2^(n-1)-2, that is, a(n-1).
Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Egor Lappo and Noah A. Rosenberg, A lattice structure for ancestral configurations arising from the relationship between gene trees and species trees, Adv. Appl. Math. 343 (2024), 65-81.
Eric Weisstein's World of Mathematics, Complete Tripartite Graph
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, Vertex Cover
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A000045, A007283, A036563, A131110, A051597, A132776, A001045.
Cf. A000918.
Cf. A112468, A112739.
Cf. A082560, A000079, A232642, A128588.

List([0..35], n-> 3*2^n -2); # G. C. Greubel, Nov 18 2019

a033484 = (subtract 2) . (* 3) . (2 ^)
a033484_list = iterate ((subtract 2) . (* 2) . (+ 2)) 1
-- Reinhard Zumkeller, Apr 23 2013

[3*2^n-2: n in [1..36]]; // Vincenzo Librandi, Nov 22 2010

with(combinat):a:=n->stirling2(n,2)+stirling2(n+1,2): seq(a(n), n=1..35); # Zerinvary Lajos, Oct 07 2007
a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=(a[n-1]+1)*2 od: seq(a[n], n=1..35); # Zerinvary Lajos, Feb 22 2008

Table[3 2^n - 2, {n, 0, 35}] (* Vladimir Joseph Stephan Orlovsky, Dec 16 2008 *)
(* Start from Eric W. Weisstein, Sep 21 2017 *)
3*2^Range[0, 35] - 2
LinearRecurrence[{3, -2}, {1, 4}, 36]
CoefficientList[Series[(1+x)/(1-3x+2x^2), {x, 0, 35}], x] (* End *)

a(n) = 3<Charles R Greathouse IV, Nov 02 2011

[3*2^n -2 for n in (0..35)] # G. C. Greubel, Nov 18 2019

G.f.: (1+x)/(1-3*x+2*x^2).
a(n) = 2*(a(n-1) + 1) for n>0, with a(0)=1.
a(n) = A007283(n) - 2.
G.f. is equivalent to (1-2*x-3*x^2)/((1-x)*(1-2*x)*(1-3*x)). - Paul Barry, Apr 28 2004
From Reinhard Zumkeller, Oct 09 2004: (Start)
A099257(a(n)) = A099258(a(n)) = a(n).
a(n) = 2*A055010(n) = (A068156(n) - 1)/2. (End)
Row sums of triangle A130452. - Gary W. Adamson, May 26 2007
Row sums of triangle A131110. - Gary W. Adamson, Jun 15 2007
Binomial transform of (1, 3, 3, 3, ...). - Gary W. Adamson, Oct 17 2007
Row sums of triangle A051597 (a triangle generated from Pascal's rule given right and left borders = 1, 2, 3, ...). - Gary W. Adamson, Nov 04 2007
Equals A132776 * [1/1, 1/2, 1/3, ...]. - Gary W. Adamson, Nov 16 2007
a(n) = Sum_{k=0..n} A112468(n,k)*3^k. - Philippe Deléham, Feb 23 2014
a(n) = -(2^n) * A036563(1-n) for all n in Z. - Michael Somos, Jul 04 2017
E.g.f.: 3*exp(2*x) - 2*exp(x). - G. C. Greubel, Nov 18 2019

A024495 a(n) = C(n,2) + C(n,5) + ... + C(n, 3*floor(n/3)+2).

Original entry on oeis.org

0, 0, 1, 3, 6, 11, 21, 42, 85, 171, 342, 683, 1365, 2730, 5461, 10923, 21846, 43691, 87381, 174762, 349525, 699051, 1398102, 2796203, 5592405, 11184810, 22369621, 44739243, 89478486, 178956971, 357913941, 715827882, 1431655765, 2863311531, 5726623062

Offset: 0

Clark Kimberling

Trisections give A082365, A132804, A132805. - Paul Curtz, Nov 18 2007
If the offset is changed to 1, this is the maximal number of closed regions bounded by straight lines after n straight line cuts in a plane: a(n) = a(n-1) + n - 3, a(1)=0; a(2)=0; a(3)=1; and so on. - Srikanth K S, Jan 23 2008
M^n * [1,0,0] = [A024493(n), a(n), A024494(n)]; where M = a 3x3 matrix [1,1,0; 0,1,1; 1,0,1]. Sum of terms = 2^n. Example: M^5 * [1,0,0] = [11, 11, 10], sum = 2^5 = 32. - Gary W. Adamson, Mar 13 2009
For n>=1, a(n-1) is the number of generalized compositions of n when there are i^2/2 - 3*i/2 + 1 different types of i, (i=1,2,...). - Milan Janjic, Sep 24 2010
Let M be any endomorphism on any vector space, such that M^3 = 1 (identity). Then (1+M)^n = A024493(n) + A024494(n)*M + a(n)*M^2. - Stanislav Sykora, Jun 10 2012
{A024493, A131708, A024495} is the difference analog of the hyperbolic functions {h_1(x), h_2(x), h_3(x)} of order 3. For the definitions of {h_i(x)} and the difference analog {H_i(n)} see [Erdelyi] and the Shevelev link respectively. - Vladimir Shevelev, Aug 01 2017
This is the p-INVERT of (1,1,1,1,1,...) for p(S) = 1 - S^3; see A291000.  - Clark Kimberling, Aug 24 2017

A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, 2nd. ed., Problem 38, p. 70.

Seiichi Manyama, Table of n, a(n) for n = 0..3000
Paul Barry, A note on Krawtchouk Polynomials and Riordan Arrays, JIS 11 (2008) 08.2.2, example 9.
Antoine-Augustin Cournot, Solution d'un problème d'analyse combinatoire, Bulletin des Sciences Mathématiques, Physiques et Chimiques, item 34, volume 11, 1829, pages 93-97. Also at Google Books. Page 97 case p=3 formula y^(2) = a(n).
Christian Ramus, Solution générale d'un problème d'analyse combinatoire, Journal für die Reine und Angewandte Mathematik (Crelle's journal), volume 11, 1834, pages 353-355. Page 353 case p=3 formula y^(2) = a(n).
Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.
Eric Weisstein's World of Mathematics, Plane division by lines
Index entries for linear recurrences with constant coefficients, signature (3,-3,2).

Cf. A007283, A010892, A011655, A024493, A024494, A024495, A057079, A082365.
Cf. A083322, A139469, A111927, A113405, A131577, A131708, A132804, A132805.
Cf. A175805, A291000.
Sequences of the form 1/((1-x)^m - x^m): A000079 (m=1,2), this sequence (m=3), A000749 (m=4), A049016 (m=5), A192080 (m=6), A049017 (m=7), A290995 (m=8), A306939 (m=9).

R:=PowerSeriesRing(Integers(), 30); [0,0] cat Coefficients(R!( x^2/((1-x)^3-x^3) )); // G. C. Greubel, Apr 11 2023

a:= proc(n) option remember; `if`(n=0, 0, 2*a(n-1)+
      [-1, 0, 1, 1, 0, -1, -1][1+(n mod 6)])
    end:
seq(a(n), n=0..33); # Paul Weisenhorn, May 17 2020

LinearRecurrence[{3,-3,2},{0,0,1},40] (* Harvey P. Dale, Sep 20 2016 *)

a(n) = sum(k=0,n\3,binomial(n,3*k+2)) /* Michael Somos, Feb 14 2006 */

a(n)=if(n<0, 0, ([1,0,1;1,1,0;0,1,1]^n)[3,1]) /* Michael Somos, Feb 14 2006 */

def A024495(n): return (2^n - chebyshev_U(n, 1/2) - chebyshev_U(n-1, 1/2))/3
[A024495(n) for n in range(41)] # G. C. Greubel, Apr 11 2023

a(n) = ( 2^n + 2*cos((n-4)*Pi/3) )/3 = (2^n - A057079(n))/3.
a(n) = 2*a(n-1) + A010892(n-2) = a(n-1) + A024494(n-1). With initial zero, binomial transform of A011655 which is effectively A010892 unsigned. - Henry Bottomley, Jun 04 2001
a(2) = 1, a(3) = 3, a(n+2) = a(n+1) - a(n) + 2^n. - Benoit Cloitre, Sep 04 2002
a(n) = Sum_{k=0..n} 2^k*2*sin(Pi*(n-k)/3 + Pi/3)/sqrt(3) (offset 0). - Paul Barry, May 18 2004
G.f.: x^2/((1-x)^3 - x^3) =  x^2 / ( (1-2*x)*(1-x+x^2) ).
a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3). - Paul Curtz, Nov 18 2007
a(n) + A024493(n-1) = A131577(n). - Paul Curtz, Jan 24 2008
From Paul Curtz, May 29 2011: (Start)
a(n) + a(n+3) = 3*2^n = A007283(n).
a(n+6) - a(n) = 21*2^n = A175805(n).
a(n) + a(n+9) = 171*2^n.
a(n+12) - a(n) = 1365*2^n. (End)
a(n) = A113405(n) + A113405(n+1). - Paul Curtz, Jun 05 2011
Start with x(0)=1, y(0)=0, z(0)=0 and set x(n+1) = x(n) + z(n), y(n+1) = y(n) + x(n), z(n+1) = z(n) + y(n). Then a(n) = z(n). - Stanislav Sykora, Jun 10 2012
G.f.: -x^2/( x^3 - 1 + 3*x/Q(0) ) where Q(k) = 1 + k*(x+1) + 3*x - x*(k+1)*(k+4)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Mar 15 2013
a(n) = 1/18*(-4*(-1)^floor((n - 1)/3) - 6*(-1)^floor(n/3) - 3*(-1)^floor((n + 1)/3) + (-1)^(1 + floor((n + 2)/3)) + 3*2^(n + 1)). - John M. Campbell, Dec 23 2016
a(n) = (1/63)*(-40 + 21*2^n - 42*floor(n/6) + 32*floor((n+3)/6) + 16*floor((n+ 4)/6) - 24*floor((n+5)/6) - 22*floor((n+7)/6) + 21*floor((n+8)/6) + 10*floor((n+9)/6) + 5*floor((n+10)/6) + 3*floor((n+11)/6) + floor((n+ 13)/6)). - John M. Campbell, Dec 24 2016
a(n+m) = a(n)*A024493(m) + A131708(n)*A131708(m) + A024493(n)*a(m). - Vladimir Shevelev, Aug 01 2017
From Kevin Ryde, Sep 24 2020: (Start)
a(n) = (1/3)*2^n - (1/3)*cos((1/3)*Pi*n) - (1/sqrt(3))*sin((1/3)*Pi*n). [Cournot]
a(n) + A111927(n) + A131708(n) = 2^n - 1. [Cournot, page 96 last formula, but misprint should be 2^x - 1 rather than 2^p - 1] (End)
E.g.f.: (exp(2*x) - exp(x/2)*(cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)))/3. - Stefano Spezia, Feb 06 2025

A052955 a(2n) = 22^n - 1, a(2n+1) = 32^n - 1.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 15, 23, 31, 47, 63, 95, 127, 191, 255, 383, 511, 767, 1023, 1535, 2047, 3071, 4095, 6143, 8191, 12287, 16383, 24575, 32767, 49151, 65535, 98303, 131071, 196607, 262143, 393215, 524287, 786431, 1048575, 1572863, 2097151, 3145727

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

a(n) is the least k such that A056792(k) = n.
One quarter of the number of positive integer (n+2) X (n+2) arrays with every 2 X 2 subblock summing to 1. - R. H. Hardin, Sep 29 2008
Number of length n+1 left factors of Dyck paths having no DUU's (here U=(1,1) and D=(1,-1)). Example: a(4)=7 because we have UDUDU, UUDDU, UUDUD, UUUDD, UUUDU, UUUUD, and UUUUU (the paths UDUUD, UDUUU, and UUDUU do not qualify).
Number of binary palindromes < 2^n (see A006995). - Hieronymus Fischer, Feb 03 2012
Partial sums of A016116 (omitting the initial term). - Hieronymus Fischer, Feb 18 2012
a(n - 1), n > 1, is the number of maximal subsemigroups of the monoid of order-preserving or -reversing partial injective mappings on a set with n elements. - Wilf A. Wilson, Jul 21 2017
Number of monomials of the algebraic normal form of the Boolean function representing the n-th bit of the product 3x in terms of the bits of x. - Sebastiano Vigna, Oct 04 2020

			G.f. = 1 + 2*x + 3*x^2 + 5*x^3 + 7*x^4 + 11*x^5 + 15*x^6 + 23*x^7 + ... - _Michael Somos_, Jun 24 2018

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Andrei Asinowski, Cyril Banderier, and Benjamin Hackl, On extremal cases of pop-stack sorting, Permutation Patterns (Zürich, Switzerland, 2019).
Andrei Asinowski, Cyril Banderier, and Benjamin Hackl, Flip-sort and combinatorial aspects of pop-stack sorting, arXiv:2003.04912 [math.CO], 2020.
J.-L. Baril, T. Mansour, and A. Petrossian, Equivalence classes of permutations modulo excedances, preprint, 2014.
J.-L. Baril, T. Mansour, and A. Petrossian, Equivalence classes of permutations modulo excedances, Journal of Combinatorics 5 (2014), 453-469.
David Blackman and Sebastiano Vigna, Scrambled Linear Pseudorandom Number Generators, ACM Transactions on Mathematical Software, Vol. 47, No. 4, p. 1-32, 2021; arXiv preprint, arXiv:1805.01407 [cs.DS], 2018.
James East, Jitender Kumar, James D. Mitchell, and Wilf A. Wilson, Maximal subsemigroups of finite transformation and partition monoids, arXiv:1706.04967 [math.GR], 2017. [_Wilf A. Wilson_, Jul 21 2017]
Brian Hopkins and Aram Tangboonduangjit, Water Cells in Compositions of 1s and 2s, arXiv:2412.11528 [math.CO], 2024. See p. 9.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1026
Mohammed A. Raouf, Fazirulhisyam Hashim, Jiun Terng Liew, and Kamal Ali Alezabi, Pseudorandom sequence contention algorithm for IEEE 802.11ah based internet of things network, PLoS ONE (2020) Vol. 15, No. 8, e0237386.
Mark Shattuck, Further Results for the Capacity Statistic Distribution on Compositions of 1's and 2's, arXiv:2501.09931 [math.CO], 2025. See p. 3.
Index to sequences related to the complexity of n
Index entries for linear recurrences with constant coefficients, signature (1,2,-2).

Cf. A000225 for even terms, A055010 for odd terms. See also A056792.
Essentially 1 more than A027383, 2 more than A060482. [Comment corrected by Klaus Brockhaus, Aug 09 2009]
Union of A000225 & A055010.
For partial sums see A027383.
See A016116 for the first differences.
Cf. A083329, A107659, A132666.
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent: A029744 (s(n)); A052955 (s(n)-1), A027383 (s(n)-2), A354788 (s(n)-3), A347789 (s(n)-4), A209721 (s(n)+1), A209722 (s(n)+2), A343177 (s(n)+3), A209723 (s(n)+4); A060482, A136252 (minor differences from A354788 at the start); A354785 (3*s(n)), A354789 (3*s(n)-7). The first differences of A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences: A016116, A060546, A117575, A131572, A152166, A158780, A163403, A320770. The bisections of A029744 are A000079 and A007283. - N. J. A. Sloane, Jul 14 2022

List([0..45], n-> ((5-(-1)^n)/2)*2^((2*n-1+(-1)^n)/4)-1); # G. C. Greubel, Oct 22 2019

a052955 n = a052955_list !! n
a052955_list = 1 : 2 : map ((+ 1) . (* 2)) a052955_list
-- Reinhard Zumkeller, Feb 22 2012

[((5-(-1)^n)/2)*2^((2*n-1+(-1)^n)/4)-1: n in [0..45]]; // G. C. Greubel, Oct 22 2019

spec := [S,{S=Prod(Sequence(Prod(Union(Z,Z),Z)),Union(Sequence(Z),Z))}, unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=2*a[n-2]+2 od: seq(a[n]/2, n=2..43); # Zerinvary Lajos, Mar 16 2008

a[n_]:= If[EvenQ[n], 2^(n/2+1) -1, 3*2^((n-1)/2) -1]; Table[a[n], {n, 0, 41}] (* Robert G. Wilson v, Jun 05 2004 *)
a[0]=1; a[1]=2; a[n_]:= a[n]= 2 a[n-2] +1; Array[a, 42, 0]
a[n_]:= (2 + Mod[n, 2]) 2^Quotient[n, 2] - 1; (* Michael Somos, Jun 24 2018 *)

a(n)=(2+n%2)<<(n\2)-1 \\ Charles R Greathouse IV, Jun 19 2011

{a(n) = (n%2 + 2) * 2^(n\2) - 1}; /* Michael Somos, Jun 24 2018 */

# command line argument tells how high to take n
# Beyond a(38) = 786431 you may need a special code to handle large integers
  $lim = shift;
  sub show{};
$n = $incr = $P = 1;
show($n, $incr, $P);
$incr = 1;
for $n (2..$lim) {
    $P += $incr;
    show($n, $P, $incr, $P);
    $incr *=2 if ($n % 2); # double the increment after an odd n
}
sub show {
    my($n, $P) = @_;
    printf("%4d\t%16g\n", $n, $P);
}
# Mark A. Mandel (thnidu aT  g ma(il) doT c0m), Dec 29 2010

def A052955(n): return ((2|n&1)<<(n>>1))-1 # Chai Wah Wu, Jul 13 2023

[((5-(-1)^n)/2)*2^((2*n-1+(-1)^n)/4)-1 for n in (0..45)] # G. C. Greubel, Oct 22 2019

a(0)=1, a(1)=2; thereafter a(n) = 2*a(n-2) + 1, n >= 2.
G.f.: (1 + x - x^2)/((1 - x)*(1 - 2*x^2)).
a(n) = -1 + Sum_{alpha = RootOf(-1 + 2*Z^2)} (1/4) * (3 + 4*alpha) * alpha^(-1-n). (That is, the sum is indexed by the roots of the polynomial -1 + 2*Z^2.)
a(n) = 2^(n/2) * (3*sqrt(2)/4 + 1 - (3*sqrt(2)/4 - 1) * (-1)^n) - 1. - Paul Barry, May 23 2004
a(n) = 1 + Sum_{k=0..n-1} A016116(k). - Robert G. Wilson v, Jun 05 2004
A132340(a(n)) = A027383(n). - Reinhard Zumkeller, Aug 20 2007
From Hieronymus Fischer, Sep 15 2007: (Start)
a(n) = A027383(n-1) + 1 for n>0.
a(n) = A132666(a(n+1)-1).
a(n) = A132666(a(n-1)) + 1 for n>0.
A132666(a(n)) = a(n+1) - 1. (End)
a(n) = A027383(n+1)/2. - Zerinvary Lajos, Mar 16 2008
a(n) = (5 - (-1)^n)/2*2^floor(n/2) - 1. - Hieronymus Fischer, Feb 03 2012
a(2n+1) = (a(2*n) + a(2*n+2))/2. Combined with a(n) = 2*a(n-2) + 1, n >= 2 and a(0) = 1, this specifies the sequence. - Richard R. Forberg, Nov 30 2013
a(n) = ((5 - (-1)^n)/2)*2^((2*n - 1 + (-1)^n)/4) - 1. - Luce ETIENNE, Sep 20 2014
a(n) = -(2^(n+1)) * A107659(-3-n) for all n in Z. - Michael Somos, Jun 24 2018
E.g.f.: (1/4)*exp(-sqrt(2)*x)*(4 - 3*sqrt(2) + (4 + 3*sqrt(2))*exp(2*sqrt(2)*x) - 4*exp(x + sqrt(2)*x)). - Stefano Spezia, Oct 22 2019
A term k appears in this sequence <=> 4 does not divide binomial(k, j) for any j in 0..k. - Peter Luschny, Jun 28 2025

Formula and more terms from Henry Bottomley, May 03 2000
Additional comments from Robert G. Wilson v, Jan 29 2001
Minor edits from N. J. A. Sloane, Jul 09 2022

A003953 Expansion of g.f.: (1+x)/(1-10*x).

Original entry on oeis.org

1, 11, 110, 1100, 11000, 110000, 1100000, 11000000, 110000000, 1100000000, 11000000000, 110000000000, 1100000000000, 11000000000000, 110000000000000, 1100000000000000, 11000000000000000

Offset: 0

N. J. A. Sloane

Coordination sequence for infinite tree with valency 11.
a(n) is sequence A003945(n-1) written in base 2: a(0)=1, a(n) for n >= 1: 2 times 1, (n-1) times 0. a(n) is also A007283(n-1) and A042950(n) for n >= 1 written in base 2. a(n) is also A098011(n+3) and A101229(n+10) for n >= 1 written in base 2. a(n) is also abs(A110164(n+1)) for n >= 1 written in base 2. - Jaroslav Krizek, Aug 17 2009
a(n) equals the numbers of words of length n on alphabet {0,1,...,10} with no two adjacent letters identical. - Milan Janjic, Jan 31 2015 [Corrected by David Nacin, Jun 02 2017]

Vincenzo Librandi, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 312
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (10).
Index entries for sequences related to trees

Cf. A003945, A003952.

k:=11;; Concatenation([1], List([1..25], n-> k*(k-1)^(n-1) )); # G. C. Greubel, Sep 24 2019

k:=11; [1] cat [k*(k-1)^(n-1): n in [1..25]]; // G. C. Greubel, Sep 24 2019

k:=11; seq(`if`(n=0, 1, k*(k-1)^(n-1)), n = 0..25); # modified by G. C. Greubel, Sep 24 2019

Join[{1}, 11*10^Range[0, 25]] (* Vladimir Joseph Stephan Orlovsky, Jul 11 2011 *)
Join[{1},NestList[10#&,11,20]] (* Harvey P. Dale, Jul 04 2025 *)

a(n)=11*10^n\10 \\ Charles R Greathouse IV, Aug 14 2015

k=11; [1]+[k*(k-1)^(n-1) for n in (1..25)] # G. C. Greubel, Sep 24 2019

a(n) = Sum_{k=0..n} A029653(n, k)*x^k for x = 9. - Philippe Deléham, Jul 10 2005
G.f.: (1+x)/(1-10*x). - Paul Barry, Mar 22 2006
a(0) = 1, a(n) = 10^n + 10^(n-1) = 11*10^(n-1) for n >= 1. - Jaroslav Krizek, Aug 17 2009
E.g.f.: (11*exp(10*x) - 1)/10. - G. C. Greubel, Sep 24 2019

Edited by N. J. A. Sloane, Dec 04 2009

cp's OEIS Frontend

A215366 Triangle T(n,k) read by rows in which n-th row lists in increasing order all partitions lambda of n encoded as Product_{i in lambda} prime(i); n>=0, 1<=k<=A000041(n).

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A016116 a(n) = 2^floor(n/2).

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A027383 a(2*n) = 3*2^n - 2; a(2*n+1) = 2^(n+2) - 2.

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A029744 Numbers of the form 2^n or 3*2^n.

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A225546 Tek's flip: Write n as the product of distinct factors of the form prime(i)^(2^(j-1)) with i and j integers, and replace each such factor with prime(j)^(2^(i-1)).

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A027383 a(2n) = 32^n - 2; a(2*n+1) = 2^(n+2) - 2.

A243071 Permutation of nonnegative integers: a(1) = 0, a(2) = 1, a(2n) = 2a(n), a(2n+1) = 1 + 2a(A064989(2n+1)).

A052955 a(2n) = 22^n - 1, a(2n+1) = 32^n - 1.