A033129 -id:A033129 - cp's OEIS frontend

A262849 T(n,k)=Number of (n+2)X(k+2) 0..1 arrays with each row divisible by 7 and column not divisible by 7, read as a binary number with top and left being the most significant bits.

6, 6, 13, 12, 34, 27, 318, 196, 132, 54, 900, 3181, 1336, 396, 109, 4536, 31050, 37635, 5184, 1264, 219, 34782, 352880, 771084, 420654, 31512, 3962, 438, 178926, 4679725, 17912392, 14762016, 3896365, 175820, 11886, 877, 1042284, 58693450, 481968171

Offset: 1

R. H. Hardin, Oct 03 2015

Table starts
....6......6.......12........318..........900..........4536.........34782
...13.....34......196.......3181........31050........352880.......4679725
...27....132.....1336......37635.......771084......17912392.....481968171
...54....396.....5184.....420654.....14762016.....661066920...35819485902
..109...1264....31512....3896365....290338650...26232879096.2864161217701
..219...3962...175820...39348387...5692555116.1007501698644
..438..11886...793812..417279054.108936025308
..877..35914..4140908.3999504445
.1755.108556.21744992
.3510.325668

			Some solutions for n=3 k=4
..0..1..1..1..0..0....0..0..0..1..1..1....0..0..1..1..1..0....0..0..0..0..0..0
..0..0..0..1..1..1....1..1..1..0..0..0....0..1..1..1..0..0....1..1..1..1..1..1
..0..0..1..1..1..0....0..1..0..1..0..1....1..1..0..0..0..1....1..0..0..0..1..1
..0..0..1..1..1..0....0..1..0..1..0..1....0..1..1..1..0..0....1..0..0..0..1..1
..1..1..1..1..1..1....0..1..0..1..0..1....1..1..1..0..0..0....1..0..0..0..1..1

R. H. Hardin, Table of n, a(n) for n = 1..70

Column 1 is A033129(n+2).

Empirical for column k:
k=1: a(n) = 2*a(n-1) +a(n-3) -2*a(n-4)
k=2: [order 15]
k=3: [order 43]
k=4: [order 29]
Empirical for row n:
n=1: [linear recurrence of order 16]

A083713 a(n) = (8^n - 1)*3/7.

Original entry on oeis.org

0, 3, 27, 219, 1755, 14043, 112347, 898779, 7190235, 57521883, 460175067, 3681400539, 29451204315, 235609634523, 1884877076187, 15079016609499, 120632132875995, 965057063007963, 7720456504063707, 61763652032509659

Offset: 0

Klaus Brockhaus, Jun 14 2003

Fixed points of the mapping defined by A067585. In binary these numbers show a regular pattern: 0, 11, 11011, 11011011, 11011011011, etc.
From Reinhard Zumkeller, Feb 22 2010: (Start)
a(n) = A173593(6*n-5) for n > 0:
terms of A173593 beginning and ending with digits '11' in binary representation;
for n > 0: a(n) = A033129(3*n-1); a(n) - a(n-1) = A103333(n). (End)

			From _Zerinvary Lajos_, Jan 14 2007: (Start)
Octal..........decimal:
0....................0
3....................3
33..................27
333................219
3333..............1755
33333............14043
333333..........112347
3333333.........898779
33333333.......7190235
333333333.....57521883
3333333333...460175067
etc. (End)

Index entries for linear recurrences with constant coefficients, signature (9,-8).

Cf. A067585, A023001.

(3/7)(8^Range[0,20]-1) (* or *) LinearRecurrence[{9,-8},{0,3},30] (* or *) NestList[8#+3&,0,30] (* Harvey P. Dale, Jun 06 2013 *)

a(n)=(8^n-1)*3/7 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 3*A023001(n).
Recursion: a(0) = 0, a(n+1) = (((a(n)*2)*2+1)*2+1).
a(n) = 8*a(n-1) + 3 (with a(0)=0). - Vincenzo Librandi, Aug 08 2010
a(0)=0, a(1)=3, a(n) = 9*a(n-1) - 8*a(n-2). - Harvey P. Dale, Jun 06 2013
From Stefano Spezia, Feb 23 2025: (Start)
G.f.: 3*x/((1 - x)*(1 - 8*x)).
E.g.f.: 3*exp(x)*(exp(7*x) - 1)/7. (End)

A294627 Expansion of x(1 + x)/((1-2x)*(1+x+x^2)).

Original entry on oeis.org

0, 1, 2, 3, 7, 14, 27, 55, 110, 219, 439, 878, 1755, 3511, 7022, 14043, 28087, 56174, 112347, 224695, 449390, 898779, 1797559, 3595118, 7190235, 14380471, 28760942, 57521883, 115043767, 230087534, 460175067, 920350135, 1840700270, 3681400539, 7362801079, 14725602158, 29451204315

Offset: 0

Wojciech Florek, Feb 12 2018

A generalized tribonacci (A001590) sequence.
For n > 2, 6*a(n) is the number of quaternary sequences of length n in which all triples (q(i),q(i+1),q(i+2)) contain two (arbitrarily chosen) digits, e.g., 0 and 3.
Similarly, recurrences a(n) = a(n-1) + a(n-2) + k*a(n-3) are related to binary (k=0, the Fibonacci sequence A000045), ternary (k=1, the tribonacci sequence A001590), quinary (k=3) and so on sequences with all triples (t(i),t(i+1),t(i+2)) containing two (arbitrarily chosen) digits (usually 0 and k+1).
For n > 0, a(n) is the number of ways to tile a strip of length n with squares, dominoes, and two colors of trominoes, with the restriction that the first tile cannot be a tromino. - Greg Dresden and Bora Bursalı, Aug 31 2023
For n > 1, a(n) is the number of ways to tile a strip of length n-2 with squares, dominoes, and two colors of trominoes, where the strip begins with an upper level of two cells. For example, when n=7 we have this strip of length 5:
 _
|||_____
|||_|||. - Guanji Chen and Greg Dresden, Jun 17 2024

			For n=4 there are 6*7=42 quaternary sequences of length 4 such that each triple (i.e., exactly two of them: q1,q2,q3 and q2,q3,q4) contain both 0 and 3. They are 003x, 030x, 03y0, 0330, 330x, 303x, 30y3, 3003, 0y30, 3y03, y03x, y30x, where x=0,1,2,3 and y=1,2.

Wojciech Florek, Table of n, a(n) for n = 0..500
Wojciech Florek, A class of generalized Tribonacci sequences applied to counting problems, Appl. Math. Comput., 338 (2018), 809-821.
Index entries for linear recurrences with constant coefficients, signature (1,1,2).

Cf. A000045, A001590, A007040, A033129 (partial sums), A078043, A167373.

LinearRecurrence[{1, 1, 2}, {0, 1, 2}, 50] (* Paolo Xausa, Aug 28 2024 *)

my(x='x+O('x^99)); concat(0, Vec(x*(1+x)/(1-x-x^2-2*x^3))) \\ Altug Alkan, Mar 03 2018

a(n) = a(n-1) + a(n-2) + 2*a(n-3) for n > 2.
a(n+1)/a(n) tends to 2, the unique real root of x^3 - x^2 - x - 2 = 0.
a(n+1) = abs(A078043(n)).
7*a(n) = 3*2^n - A167373(n+1). - R. J. Mathar, Mar 24 2018
E.g.f.: exp(-x/2)*(9*exp(5*x/2) - 9*cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2))/21. - Stefano Spezia, Aug 29 2024

A078043 Expansion of (1 - x)/(1 + x - x^2 + 2*x^3).

Original entry on oeis.org

1, -2, 3, -7, 14, -27, 55, -110, 219, -439, 878, -1755, 3511, -7022, 14043, -28087, 56174, -112347, 224695, -449390, 898779, -1797559, 3595118, -7190235, 14380471, -28760942, 57521883, -115043767, 230087534, -460175067, 920350135, -1840700270, 3681400539, -7362801079

Offset: 0

N. J. A. Sloane, Nov 17 2002

abs(a(n)) = A033129(n+1) - A033129(n). - Reinhard Zumkeller, Feb 22 2010

Robert Israel, Table of n, a(n) for n = 0..3318
Index entries for linear recurrences with constant coefficients, signature (-1,1,-2).

f:= gfun:-rectoproc({a(0)=1, a(1)=-2, a(2)=3, a(n) = -a(n-1) + a(n-2) - 2*a(n-3)},a(n),remember):
map(f, [$0..40]); # Robert Israel, Mar 28 2018

CoefficientList[Series[(1-x)/(1+x-x^2+2x^3),{x,0,40}],x] (* or *) LinearRecurrence[{-1,1,-2},{1,-2,3},40] (* Harvey P. Dale, Feb 02 2012 *)

Vec((1-x)/(1+x-x^2+2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n)=1/7*(6*(-2)^n+[1,-2,-3,-1,2,3][(n%6)+1]) /* Ralf Stephan, Aug 18 2013 */

a(n)=(6*(-2)^n+3)\7 \\ Tani Akinari, Oct 05 2014

a(0)=1, a(1)=-2, a(2)=3, a(n) = -a(n-1) + a(n-2) - 2*a(n-3). - Harvey P. Dale, Feb 02 2012 [corrected by Wojciech Florek, Feb 26 2018]
a(n) = (1/21) * (-9*2^n*e^(i*n*Pi) + 9*cos((n*Pi)/3) - sqrt(3)*sin((n*Pi)/3)). - Harvey P. Dale, Feb 02 2012
a(n) = (1/7) * (6*(-2)^n + [1,-2,-3,-1,2,3](mod 6)) = A077972(n) - A077972(n-1). - Ralf Stephan, Aug 18 2013
a(n) = floor((6*(-2)^n+3)/7). - Tani Akinari, Oct 05 2014

A173593 Numbers having in binary representation exactly two ones in three consecutive digits.

Original entry on oeis.org

3, 5, 6, 11, 13, 22, 27, 45, 54, 91, 109, 182, 219, 365, 438, 731, 877, 1462, 1755, 2925, 3510, 5851, 7021, 11702, 14043, 23405, 28086, 46811, 56173, 93622, 112347, 187245, 224694, 374491, 449389, 748982, 898779, 1497965, 1797558, 2995931, 3595117

Offset: 1

Reinhard Zumkeller, Feb 22 2010

a(2*n-1) = A033129(n+1);
a(3*n-2) = A113836(n+1);
a(6*n-5) = A083713(n);
a(2*n) - a(2*n-1) = A077947(n+1);
a(2*n+1) - a(2*n) = A077947(n).

			a(10) =   91 =      1011011_2
a(11) =  109 =      1101101_2
a(12) =  182 =     10110110_2
a(13) =  219 =     11011011_2
a(14) =  365 =    101101101_2
a(15) =  438 =    110110110_2
a(16) =  731 =   1011011011_2
a(17) =  877 =   1101101101_2
a(18) = 1462 =  10110110110_2
a(19) = 1755 =  11011011011_2
a(20) = 2925 = 101101101101_2

Index entries for linear recurrences with constant coefficients, signature (0, 2, 1, 0, -2).

Cf. A007088.

Bisections A033129, A033120.

LinearRecurrence[{0, 2, 1, 0, -2}, {3, 5, 6, 11, 13}, 50] (* Jean-François Alcover, Feb 17 2018 *)

From R. J. Mathar, Feb 24 2010: (Start)
a(n) = 2*a(n-2) + a(n-3) - 2*a(n-5).
G.f.: x*(-3-5*x+2*x^3+4*x^4)/ ((1-x) * (1+x+x^2) * (2*x^2-1)). (End)

A113836 a(n) = Sum[2^(A001651(i-1)-1), {i,1,n}].

Original entry on oeis.org

1, 3, 11, 27, 91, 219, 731, 1755, 5851, 14043, 46811, 112347, 374491, 898779, 2995931, 7190235, 23967451, 57521883, 191739611, 460175067, 1533916891, 3681400539, 12271335131, 29451204315, 98170681051, 235609634523

Offset: 1

Artur Jasinski, Jan 27 2006

nonn

From Reinhard Zumkeller, Feb 22 2010: (Start)
For n>1: a(n)=A173593(3*n-5): terms of A173593 ending with digits '11' in binary representation;
for n>0: a(n)=A033129(3*n-1); a(n+1)-a(n)=ABS(A094014(n)). (End)

			a(2) = 2^(A001651(0)-1) + 2^(A001651(1)-1) = 2^0 + 2^1 = 3

Cf. A001651.

a = {}; s = 0; For[n = 1, n < 40, n++, If[Length[Intersection[{Mod[n, 3]}, {1,2}]] > 0, s = s + 2^(n - 1); AppendTo[a, s]]]; a

Empirical g.f.: x*(2*x+1) / ((x-1)*(8*x^2-1)). - Colin Barker, Sep 01 2013

Edited by Stefan Steinerberger, Jul 23 2007

A289006 Conversion to octal of the binary expansion given by the first n terms of the period-3 sequence A011655 (repeat 0, 1, 1).

Original entry on oeis.org

0, 1, 3, 6, 15, 33, 66, 155, 333, 666, 1555, 3333, 6666, 15555, 33333, 66666, 155555, 333333, 666666, 1555555, 3333333, 6666666, 15555555, 33333333, 66666666, 155555555, 333333333, 666666666, 1555555555, 3333333333, 6666666666, 15555555555, 33333333333, 66666666666, 155555555555, 333333333333, 666666666666

Offset: 1

Peter Schonefeld, Jun 21 2017

The length of the n-th term is floor((n+1)/3) digits, for all n>1. [Corrected by M. F. Hasler, Jun 23 2017]

Index entries for linear recurrences with constant coefficients, signature (0, 0, 11, 0, 0, -10).

A033129(n-1) written in base 8.

Cf. A011655. Trisections: A099915, A002277, A002280.

{ my(x='x+O('x^33)); concat([0],Vec( x*(1+x)*(1+2*x+4*x^2)/((1-x)*(1+x+x^2)*(1-10*x^3)) )) } \\ Joerg Arndt, Jun 21 2017

A289006(n)=if(n%3==2,10^(n\3+1)\6-10^(n\3)\9,10^(n\3)\3<<(n%3)) \\ M. F. Hasler, Jun 23 2017

a(3n) = floor(10^n/3) (= n times the digit '3'), a(3n+1) = floor(10^n/3)*2 (= n times the digit '6'), a(3n+2) = floor(10^(n+1)/6) - floor(10^n/9) (= digit '1' followed by n digits '5'). - M. F. Hasler, Jun 23 2017

G.f.: x^2*(1+x)*(4*x^2+2*x+1) / ( (x-1)*(1+x+x^2)*(10*x^3-1) ). - R. J. Mathar, Jun 29 2017

A057744 Expansion of (1-2x^3)/(1-2x-x^3+2*x^4).

Original entry on oeis.org

1, 2, 4, 7, 14, 28, 55, 110, 220, 439, 878, 1756, 3511, 7022, 14044, 28087, 56174, 112348, 224695, 449390, 898780, 1797559, 3595118, 7190236, 14380471, 28760942, 57521884, 115043767, 230087534, 460175068, 920350135, 1840700270, 3681400540

Offset: 0

N. J. A. Sloane, Oct 30 2000

Kevin Ryde, Iterations of the Dragon Curve, see index "JN".
Index entries for linear recurrences with constant coefficients, signature (2,0,1,-2).

Cf. A033129, A294627 (first differences).

LinearRecurrence[{2, 0, 1, -2}, {1, 2, 4, 7}, 30] (* Jinyuan Wang, Apr 07 2020 *)

Vec((1-2*x^3)/(1-2*x-x^3+2*x^4) + O(x^50)) \\ Michel Marcus, Dec 09 2014

G.f.: (1-2*x^3)/(1-2*x-x^3+2*x^4).
a(0) = 1, a(1) = 2, a(2) = 4, a(3) = 7, a(n) = 2*a(n-1) + a(n-3) - 2*a(n-4) for n > 3. - Jinyuan Wang, Apr 08 2020
a(n) = ceiling((6/7)*2^n) = (6*2^n + 2^(n mod 3))/7. - Kevin Ryde, Aug 25 2021

cp's OEIS Frontend

A262849 T(n,k)=Number of (n+2)X(k+2) 0..1 arrays with each row divisible by 7 and column not divisible by 7, read as a binary number with top and left being the most significant bits.

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A083713 a(n) = (8^n - 1)*3/7.

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A294627 Expansion of x*(1 + x)/((1-2*x)*(1+x+x^2)).

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A078043 Expansion of (1 - x)/(1 + x - x^2 + 2*x^3).

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A173593 Numbers having in binary representation exactly two ones in three consecutive digits.

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A113836 a(n) = Sum[2^(A001651(i-1)-1), {i,1,n}].

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A289006 Conversion to octal of the binary expansion given by the first n terms of the period-3 sequence A011655 (repeat 0, 1, 1).

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A294627 Expansion of x(1 + x)/((1-2x)*(1+x+x^2)).

A057744 Expansion of (1-2x^3)/(1-2x-x^3+2*x^4).