A083420 -id:A083420 - cp's OEIS frontend

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

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

A096053 a(n) = (3*9^n - 1)/2.

Original entry on oeis.org

1, 13, 121, 1093, 9841, 88573, 797161, 7174453, 64570081, 581130733, 5230176601, 47071589413, 423644304721, 3812798742493, 34315188682441, 308836698141973, 2779530283277761, 25015772549499853, 225141952945498681

Offset: 0

Benoit Cloitre, Jun 18 2004

Generalized NSW numbers. - Paul Barry, May 27 2005
Counts total area under elevated Schroeder paths of length 2n+2, where area under a horizontal step is weighted 3. Case r=4 for family (1+(r-1)x)/(1-2(1+r)x+(1-r)^2*x^2). Case r=2 gives NSW numbers A002315. Fifth binomial transform of (1+8x)/(1-16x^2), A107906. - Paul Barry, May 27 2005
Primes in this sequence include: a(2) = 13, a(4) = 1093, a(7) = 797161. Semiprimes in this sequence include: a(3) = 121 = 11^2, a(5) = 9841 = 13 * 757, a(6) = 88573 = 23 * 3851, a(9) = 64570081 = 1871 * 34511, a(10) = 581130733 = 1597 * 363889, a(12) = 47071589413 = 47 * 1001523179, a(19) = 225141952945498681 = 13097927 * 17189128703.
Sum of divisors of 9^n. - Altug Alkan, Nov 10 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Cf. A083420, A096045, A096046, A096047, A096054.

Cf. A107903, A138894 ((5*9^n-1)/4).

[(3*9^n-1)/2: n in [0..20]]; // Vincenzo Librandi, Nov 01 2011

Table[(3*9^n - 1)/2, {n, 0, 18}] (* L. Edson Jeffery, Feb 13 2015 *)

a(n)=(3*9^n-1)/2 \\ Charles R Greathouse IV, Sep 28 2015

vector(30, n, n--; sigma(9^n)) \\ Altug Alkan, Nov 10 2015

From Paul Barry, May 27 2005: (Start)
G.f.: (1+3*x)/(1-10*x+9*x^2);
a(n) = Sum_{k=0..n} binomial(2n+1, 2k)*4^k;
a(n) = ((1+sqrt(4))*(5+2*sqrt(4))^n+(1-sqrt(4))*(5-2*sqrt(4))^n)/2. (End)
a(n-1) = (-9^n/3)*B(2n,1/3)/B(2n) where B(n,x) is the n-th Bernoulli polynomial and B(k)=B(k,0) is the k-th Bernoulli number.
a(n) = 10*a(n-1) - 9*a(n-2).
a(n) = 9*a(n-1) + 4. - Vincenzo Librandi, Nov 01 2011
a(n) = A000203(A001019(n)). - Altug Alkan, Nov 10 2015
a(n) = A320030(3^n-1). - Nathan M Epstein, Jan 02 2019

Edited by N. J. A. Sloane, at the suggestion of Andrew S. Plewe, Jun 15 2007

A289481 Number A(n,k) of Dyck paths of semilength k*n and height n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 7, 1, 0, 1, 1, 31, 57, 1, 0, 1, 1, 127, 1341, 484, 1, 0, 1, 1, 511, 26609, 59917, 4199, 1, 0, 1, 1, 2047, 497845, 5828185, 2665884, 36938, 1, 0, 1, 1, 8191, 9096393, 517884748, 1244027317, 117939506, 328185, 1, 0

Offset: 0

Alois P. Heinz, Jul 06 2017

For fixed k > 1, A(n,k) ~ 2^(2*k*n + 3) * k^(2*k*n + 1/2) / ((k-1)^((k-1)*n + 1/2) * (k+1)^((k+1)*n + 7/2) * sqrt(Pi*n)). - Vaclav Kotesovec, Jul 14 2017

			Square array A(n,k) begins:
  1, 1,    1,       1,          1,            1, ...
  0, 1,    1,       1,          1,            1, ...
  0, 1,    7,      31,        127,          511, ...
  0, 1,   57,    1341,      26609,       497845, ...
  0, 1,  484,   59917,    5828185,    517884748, ...
  0, 1, 4199, 2665884, 1244027317, 517500496981, ...

Alois P. Heinz, Antidiagonals n = 0..80, flattened

Columns k=0..10 give: A000007, A000012, A268316, A289473, A289474, A289475, A289476, A289477, A289478, A289479, A289480.
Rows n=0-2 give: A000012, A057427, A083420(k+1).
Main diagonal gives A289482.
Cf. A080936.

b:= proc(x, y, k) option remember;
      `if`(x=0, 1, `if`(y>0, b(x-1, y-1, k), 0)+
      `if`(y <  min(x-1, k), b(x-1, y+1, k), 0))
    end:
A:= (n, k)-> `if`(n=0, 1, b(2*n*k, 0, n)-b(2*n*k, 0, n-1)):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[x_, y_, k_]:=b[x, y, k]=If[x==0, 1, If[y>0, b[x - 1, y - 1, k], 0] + If[yIndranil Ghosh, Jul 07 2017, after Maple code *)

A147590 Numbers whose binary representation is the concatenation of 2n-1 digits 1 and n-1 digits 0.

Original entry on oeis.org

1, 14, 124, 1016, 8176, 65504, 524224, 4194176, 33554176, 268434944, 2147482624, 17179867136, 137438949376, 1099511619584, 8796093005824, 70368744144896, 562949953355776, 4503599627239424, 36028797018701824, 288230376151187456

Offset: 1

Omar E. Pol, Nov 08 2008

a(n) is the number whose binary representation is A147589(n).

			     1_10 is 1_2;
    14_10 is 1110_2;
   124_10 is 1111100_2;
  1016_10 is 1111111000_2.

Nathaniel Johnston, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (10,-16).

Cf. A020540, A138118, A147537, A147589.

List([1..25], n-> 2^(n-2)*(4^n-2)); # G. C. Greubel, Jul 27 2019

[8^n/4-2^(n-1): n in [1..25]]; // Vincenzo Librandi, Jul 27 2019

seq(8^n/4-2^(n-1),n=1..25); # Nathaniel Johnston, Apr 30 2011

LinearRecurrence[{10,-16},{1,14},30] (* Harvey P. Dale, Oct 10 2014 *)
Table[8^n / 4 - 2^(n - 1), {n, 25}] (* Vincenzo Librandi, Jul 27 2019 *)

vector(25, n, 2^(n-2)*(4^n-2)) \\ G. C. Greubel, Jul 27 2019

[2^(n-2)*(4^n-2) for n in (1..25)] # G. C. Greubel, Jul 27 2019

a(n) = A147537(n)/2.
From R. J. Mathar, Jul 13 2009: (Start)
a(n) = 8^n/4 - 2^(n-1) = A083332(2n-2).
a(n) = 10*a(n-1) - 16*a(n-2).
G.f.: x*(1+4*x)/((1-2*x)*(1-8*x)). (End)
From César Aguilera, Jul 26 2019: (Start)
Lim_{n->infinity} a(n)/a(n-1) = 8;
a(n)/a(n-1) = 8 + 6/A083420(n). (End)
E.g.f.: (1/4)*(exp(2*x)*(-2 + exp(6*x)) + 1). - Stefano Spezia, Aug 05 2019
a(n) = A020540(n - 1)/4. - Jon Maiga, Aug 05 2019

More terms from R. J. Mathar, Jul 13 2009

Typo in a(12) corrected by Omar E. Pol, Jul 20 2009

A164908 a(n) = (3*4^n - 0^n)/2.

Original entry on oeis.org

1, 6, 24, 96, 384, 1536, 6144, 24576, 98304, 393216, 1572864, 6291456, 25165824, 100663296, 402653184, 1610612736, 6442450944, 25769803776, 103079215104, 412316860416, 1649267441664, 6597069766656, 26388279066624, 105553116266496, 422212465065984, 1688849860263936

Offset: 0

Klaus Brockhaus, Aug 31 2009

Binomial transform of A164907. Inverse binomial transform of A057651.
Partial sums are in A083420.
Decimal representations of the n-th iterations of elementary cellular automata rules 14, 46, 142 and 174 generate this sequence (see A266298 and A266299). - Karl V. Keller, Jr., Aug 31 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.
N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2.
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Index entries for sequences related to cellular automata.
Index entries for linear recurrences with constant coefficients, signature (4).

Equals 1 followed by A002023 (6*4^n). Essentially the same as A084509.

Cf. A164907, A057651, A083420 (2*4^n-1), A247640, A266298, A266299.

Magma
```
[ (3*4^n-0^n)/2: n in [0..22] ];
```

a[n_]:=(MatrixPower[{{2,2},{2,2}},n].{{2},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
Join[{1},(3*4^Range[25])/2] (* or *) Join[{1},NestList[4#&,6,25]] (* Harvey P. Dale, Feb 14 2012 *)

a(n)=3*4^n\2 \\ Charles R Greathouse IV, Oct 12 2015

print([int(6*4**(n-1)) for n in range(50)]) # Karl V. Keller, Jr., Aug 30 2021

a(n) = 4*a(n-1) for n > 1; a(0) = 1, a(1) = 6.
G.f.: (1+2*x)/(1-4*x).
a(n) = floor(6*4^(n-1)). - Karl V. Keller, Jr., Aug 30 2021
E.g.f.: (3*exp(4*x) - 1)/2. - Elmo R. Oliveira, Mar 31 2025

A101622 A Horadam-Jacobsthal sequence.

Original entry on oeis.org

0, 1, 6, 13, 30, 61, 126, 253, 510, 1021, 2046, 4093, 8190, 16381, 32766, 65533, 131070, 262141, 524286, 1048573, 2097150, 4194301, 8388606, 16777213, 33554430, 67108861, 134217726, 268435453, 536870910, 1073741821, 2147483646, 4294967293, 8589934590

Offset: 0

Paul Barry, Dec 10 2004

Companion sequence to A084639.
This is the sequence A(0,1;1,2;5) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - Wolfdieter Lang, Oct 18 2010
Except for the initial three terms, the decimal representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 961", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, Mar 27 2017
Named after the Australian mathematician Alwyn Francis Horadam (1923-2016) and the German mathematician Ernst Jacobsthal (1882-1965). - Amiram Eldar, Jun 10 2021

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
A. F. Horadam, Jacobsthal Representation Numbers, Fib Quart., Vol. 34, No. 1 (1996), pp. 40-54.
Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences. [_Wolfdieter Lang_, Oct 18 2010]
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.
Stephen 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 (2,1,-2).

Cf. A131953.

[(2^(n+2)+(-1)^n-5)/2: n in [0..35]]; // Vincenzo Librandi, Aug 12 2011

LinearRecurrence[{2,1,-2},{0,1,6},40] (* Harvey P. Dale, Jul 08 2014 *)

concat(0, Vec(x*(1+4*x)/((1-x)*(1+x)*(1-2*x)) + O(x^30))) \\ Colin Barker, Mar 28 2017

a(n) = (2^(n+2) + (-1)^n - 5)/2.
G.f.: x*(1+4*x)/((1-x)*(1+x)*(1-2*x)).
a(n) = (A014551(n+2)-5)/2.
(1, 6, 13, 30, 61, ...) are the row sums of A131953. - Gary W. Adamson, Jul 31 2007
From Paul Curtz, Jan 01 2009: (Start)
a(n) = a(n-1) + 2*a(n-2) + 5.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3).
a(n) = A000079(n+1) - A010693(n).
a(n+1) = A141722(n) + 5 = A141722(n) + A010716(n).
a(2n+1) - a(2n) = 1, 7, 31, ... = A083420.
a(2n+1) - 2*a(2n) = 1.
a(2n) = A002446 = 6*A002450, a(2n+1) = A141725. (End)
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>2. - Colin Barker, Mar 28 2017
a(n) = (1/2) * Sum_{k=1..n} binomial(n+1,k) * (2+(-1)^k). - Wesley Ivan Hurt, Sep 23 2017

A331891 Negabinary palindromes: nonnegative numbers whose negabinary expansion (A039724) is palindromic.

Original entry on oeis.org

0, 1, 3, 5, 7, 11, 17, 21, 23, 31, 43, 51, 57, 65, 77, 85, 87, 103, 127, 143, 155, 171, 195, 211, 217, 233, 257, 273, 285, 301, 325, 341, 343, 375, 423, 455, 479, 511, 559, 591, 603, 635, 683, 715, 739, 771, 819, 851, 857, 889, 937, 969, 993, 1025, 1073, 1105, 1117

Offset: 1

Amiram Eldar, Jan 30 2020

Numbers of the form 2^(2*m-1) - 1 (A083420) and 2^(2*m) + 1 (A052539) are terms.

			5 is a term since its negabinary representation is 101 which is palindromic.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002113, A006995, A014190, A039724, A094202, A331191.

negabin[n_] := negabin[n] = If[n==0, 0, negabin[Quotient[n-1, -2]]*10 + Mod[n, 2]]; Select[Range[0, 1200], PalindromeQ @ negabin[#] &]

A140529 a(n) = 6*4^n - 1.

Original entry on oeis.org

5, 23, 95, 383, 1535, 6143, 24575, 98303, 393215, 1572863, 6291455, 25165823, 100663295, 402653183, 1610612735, 6442450943, 25769803775, 103079215103, 412316860415, 1649267441663, 6597069766655, 26388279066623, 105553116266495

Offset: 0

Paul Curtz, Jul 03 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A028894.

[6*4^n-1: n in [0..30]]; // Vincenzo Librandi, Aug 08 2011

NestList[4#+3&,5,30] (* Harvey P. Dale, Jan 14 2022 *)

a(n) = 4*a(n-1) + 3, a(0)=5.
a(n) = A002023(n) - 1 = A000302(n+1) + A083420(n).
G.f.: ( 5-2*x ) / ( (4*x-1)*(x-1) ). - R. J. Mathar, Jul 08 2022

A099856 Expansion of (1+3x)/(1-3x).

Original entry on oeis.org

1, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366, 118098, 354294, 1062882, 3188646, 9565938, 28697814, 86093442, 258280326, 774840978, 2324522934, 6973568802, 20920706406, 62762119218, 188286357654, 564859072962, 1694577218886, 5083731656658, 15251194969974, 45753584909922

Offset: 0

Paul Barry, Oct 28 2004

A099858 gives a Chebyshev transform. Binomial transform is A083420.

Hankel transform is 1, -18, 0, 0, 0, 0, 0, 0, 0, ... - Philippe Deléham, Dec 13 2011

Index entries for linear recurrences with constant coefficients, signature (3).

Cf. A025192, A083420, A093561, A099858.

CoefficientList[Series[(1+3x)/(1-3x),{x,0,30}],x] (* or *) Join[{1}, NestList[3#&,6,30]] (* Harvey P. Dale, Nov 08 2011 *)

Vec((1+3*x)/(1-3*x) + O(x^40)) \\ Michel Marcus, Dec 11 2015

a(n) = 2*3^n - 0^n.
a(n) = A025192(n+1), n > 0. - R. J. Mathar, Sep 02 2008
a(n) = Sum_{k=0..n} A093561(n,k)*2^k. - Philippe Deléham, Dec 13 2011
From Elmo R. Oliveira, Aug 23 2024: (Start)
E.g.f.: 2*exp(3*x) - 1.
a(n) = 3*a(n-1) for n > 1. (End)

a(26)-a(28) from Elmo R. Oliveira, Aug 23 2024

A140253 a(2n) = 2(24^(n-1)-1) and a(2n-1) = 2*4^(n-1)-1.

Original entry on oeis.org

-1, 1, 2, 7, 14, 31, 62, 127, 254, 511, 1022, 2047, 4094, 8191, 16382, 32767, 65534, 131071, 262142, 524287, 1048574, 2097151, 4194302, 8388607, 16777214, 33554431, 67108862, 134217727, 268435454, 536870911

Offset: 0

Paul Curtz, Jun 23 2008

The inverse binomial transform is 1, 1, 4, -2, 10, -14, 34, -62 which leads to (-1)^(n+1)*A135440(n).
For n > 0: A266161(a(n)) = n and A266161(m) < n for m < a(n). - Reinhard Zumkeller, Dec 22 2015
Also, the decimal representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 673", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, Jul 23 2017

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

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
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, 1, -2).

Cf. A000034, A000069, A000079, A002450, A010674, A014551, A083420, A266161.

import Data.List (transpose)
a140253 n = a140253_list !! n
a140253_list = -1 : concat
                    (transpose [a083420_list, map (* 2) a083420_list])
-- Reinhard Zumkeller, Dec 22 2015

A140253:=proc(n): if type(n, odd) then 2*4^(((n+1)/2)-1)-1 else 2*(2*4^((n/2)-1)-1) fi: end: seq(A140253(n),n=0..29); # Johannes W. Meijer, Jun 24 2011

Table[(2^(n+1) - 3 + (-1)^(n+1))/2, {n, 0, 30}] (* Jean-François Alcover, Jun 05 2017 *)

a(2*n) = 2*A083420(n-1) and a(2*n+1) = A083420(n)
a(n+1) - a(n) = A014551(n); Jacobsthal-Lucas numbers.
a(2*n) + a(2*n+1) = 9*A002450(n)
a(n+1) - 2*a(n) = A010674(n+1); repeat 3, 0.
a(n) + A000034(n+1) = A000079(n); powers of 2.
a(n)= a(n-1) + 2*a(n-2) + 3. - Gary Detlefs, Jun 22 2010
a(n+1) = A000069(2^n); odious numbers. - Johannes W. Meijer, Jun 24 2011
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>2, a(0) = -1, a(1) = 1, a(2) = 2. - Philippe Deléham, Feb 25 2012
G.f.: (x^2+3*x-1)/((1-2*x)*(1-x)*(1+x)). - Philippe Deléham, Feb 25 2012

Edited, corrected and information added by Johannes W. Meijer, Jun 24 2011

cp's OEIS Frontend

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

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A096053 a(n) = (3*9^n - 1)/2.

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A289481 Number A(n,k) of Dyck paths of semilength k*n and height n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A147590 Numbers whose binary representation is the concatenation of 2n-1 digits 1 and n-1 digits 0.

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A164908 a(n) = (3*4^n - 0^n)/2.

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A101622 A Horadam-Jacobsthal sequence.

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A099856 Expansion of (1+3x)/(1-3x).

A140253 a(2n) = 2(24^(n-1)-1) and a(2n-1) = 2*4^(n-1)-1.