A007283 -id:A007283 - cp's OEIS frontend

A347789 a(n) is the number of times that only 2 pegs have disks on them during the optimal solution to a Towers of Hanoi problem with n disks.

0, 2, 4, 8, 12, 20, 28, 44, 60, 92, 124, 188, 252, 380, 508, 764, 1020, 1532, 2044, 3068, 4092, 6140, 8188, 12284, 16380, 24572, 32764, 49148, 65532, 98300, 131068, 196604, 262140, 393212, 524284, 786428, 1048572, 1572860, 2097148, 3145724, 4194300, 6291452

Offset: 1

John Bonomo, Sep 13 2021

Zero together with the partial sum of the even terms of A016116. - Omar E. Pol, Sep 14 2021

For n >= 2, a(n+1) is the number of n X n arrays of 0's and 1's with every 2 X 2 square having density exactly 1. - David desJardins, Oct 27 2022

J. Bonomo, Back to the Tower, The College Mathematics Journal 52(2021), 265-273.
Index entries for sequences related to Towers of Hanoi
Index entries for linear recurrences with constant coefficients, signature (1,2,-2).

Cf. A016116, A027383, A052955, A341579.

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

a:= proc(n) option remember;
      `if`(n<3, 2*n-2, 2*(a(n-2)+2))
    end:
seq(a(n), n=1..42);  # Alois P. Heinz, Sep 14 2021

LinearRecurrence[{1, 2, -2}, {0, 2, 4}, 42] (* Jean-François Alcover, May 14 2022 *)

a(n) = (3+(n % 2))*(2^(n\2)) - 4; \\ Michel Marcus, Sep 14 2021

def a(n): return (3 + n%2) * 2**(n//2) - 4
print([a(n) for n in range(1, 43)]) # Michael S. Branicky, Sep 14 2021

a(n) = (3+(n mod 2))*(2^floor(n/2)) - 4.
a(n) = 4 * A052955(n-3) for n >= 3. - Joerg Arndt, Sep 14 2021
a(n) = A027383(n) - 2. - Omar E. Pol, Sep 14 2021
a(n) = 2 * A027383(n-2) for n >= 2. - Alois P. Heinz, Sep 14 2021
From Stefano Spezia, Sep 14 2021: (Start)
G.f.: 2*x^2*(1+x)/((1-x)*(1-2*x^2)).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) for n > 3. (End)

A067771 Number of vertices in Sierpiński triangle of order n.

Original entry on oeis.org

3, 6, 15, 42, 123, 366, 1095, 3282, 9843, 29526, 88575, 265722, 797163, 2391486, 7174455, 21523362, 64570083, 193710246, 581130735, 1743392202, 5230176603, 15690529806, 47071589415, 141214768242, 423644304723, 1270932914166

Offset: 0

Martin Wessendorf (martinw(AT)mail.ahc.umn.edu), Feb 09 2002

An order 0 Sierpiński triangle is a triangle. Order n+1 is formed from three copies of order n by identifying pairs of corner vertices of each pair of triangles. - Allan Bickle, Aug 03 2024
This sequence represents another link from the product factor space Q X Q / {(1,1), (-1, -1)} to Sierpiński's triangle. The first "link" found was to sequence A048473. - Creighton Dement, Aug 05 2004
a(n) equals the number of orbits of the finite group PSU(3,3^n) on subsets of size 3 of the 3^(3n)+1 isotropic points of a unitary 3 space. - Paul M. Bradley, Jan 31 2017
For n >= 1, number of edges in a planar Apollonian graph at iteration n. - Andrew D. Walker, Jul 08 2017
Also the total domination number of the (n+3)-Dorogovtsev-Goltsev-Mendes graph, using the convention DGM(0) = P_2. - Eric W. Weisstein, Jan 14 2024

			Order 0 is a triangle, so a(0) = 3.
Order 1 has three corners (degree 2) and three other vertices, so a(1) = 6.
3 example graphs:                        o
                                        / \
                                       o---o
                                      / \ / \
                        o            o---o---o
                       / \          / \     / \
             o        o---o        o---o   o---o
            / \      / \ / \      / \ / \ / \ / \
           o---o    o---o---o    o---o---o---o---o
Graph:      S_1        S_2              S_3
Vertices:    3          6                15
Edges:       3          9                27

Peter Wessendorf and Kristina Downing, personal communication.

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Allan Bickle, Properties of Sierpinski Triangle Graphs, Springer PROMS 448 (2021) 295-303.
Paul Bradley and Peter Rowley, Orbits on k-subsets of 2-transitive Simple Lie-type Groups, 2014.
A. Hinz, S. Klavzar, and S. Zemljic, A survey and classification of Sierpinski-type graphs, Discrete Applied Mathematics 217 3 (2017), 565-600.
András Kaszanyitzky, Triangular fractal approximating graphs and their covering paths and cycles, arXiv:1710.09475 [math.CO], 2017. See Table 2.
C. Lanius, Fractals.
Eric Weisstein's World of Mathematics, Dorogovtsev-Goltsev-Mendes Graph.
Eric Weisstein's World of Mathematics, Sierpiński Gasket Graph.
Eric Weisstein's World of Mathematics, Total Domination Number.
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A048473.
Cf. A003462, A007051, A034472, A024023.
Cf. A007283, A029858, A067771, A193277, A233774, A233775, A246959 (Sierpiński triangle graphs).

[(3/2)*(1+3^n): n in [0..30]]; // Vincenzo Librandi, Jun 20 2011

LinearRecurrence[{4, -3}, {3, 6}, 26] (* or *)
CoefficientList[Series[3 (1 - 2 x)/((1 - x) (1 - 3 x)), {x, 0, 25}], x] (* Michael De Vlieger, Feb 02 2017 *)
Table[3/2 (3^n + 1), {n, 0, 20}] (* Eric W. Weisstein, Jan 14 2024 *)
3/2 (3^Range[0, 20] + 1) (* Eric W. Weisstein, Jan 14 2024 *)

a(n) = (3/2)*(3^n + 1).
a(n) = 3 + 3^1 + 3^2 + 3^3 + 3^4 + ... + 3^n = 3 + Sum_{k=1..n} 3^n.
a(n) = 3*A007051(n).
a(0) = 3, a(n) = a(n-1) + 3^n. a(n) = (3/2)*(1+3^n). - Zak Seidov, Mar 19 2007
a(n) = 4*a(n-1) - 3*a(n-2).
G.f.: 3*(1-2*x)/((1-x)*(1-3*x)). - Colin Barker, Jan 10 2012
a(n) = A233774(2^n). - Omar E. Pol, Dec 16 2013
a(n) = 3*a(n-1) - 3. - Zak Seidov, Oct 26 2014
E.g.f.: 3*(exp(x) + exp(3*x))/2. - Stefano Spezia, Feb 09 2021
a(n) = A029858(n+1) + 3. - Allan Bickle, Aug 03 2024

More terms from Benoit Cloitre, Feb 22 2002

A209721 1/4 the number of (n+1) X 3 0..2 arrays with every 2 X 2 subblock having distinct clockwise edge differences.

Original entry on oeis.org

3, 4, 5, 7, 9, 13, 17, 25, 33, 49, 65, 97, 129, 193, 257, 385, 513, 769, 1025, 1537, 2049, 3073, 4097, 6145, 8193, 12289, 16385, 24577, 32769, 49153, 65537, 98305, 131073, 196609, 262145, 393217, 524289, 786433, 1048577, 1572865, 2097153, 3145729

Offset: 1

R. H. Hardin, Mar 12 2012

nonn

Column 2 of A209727.
From Richard Locke Peterson, Apr 26 2020: (Start)
The formula a(n) = 2*a(n-2)-1 also fits empirically. With the given initial numbers a(1)=3, a(2)=4, a(3)=5, this new formula implies the old empirical formula. (But it is not established that the old empirical formula is true, so it is not established that the new formula is true either.) Furthermore, if the initial numbers had somehow, for example, been 3,4,6 instead, the new formula no longer implies the old formula.
If the new formula actually is true, it follows that a(n) is the number of distinct integer triangles that can be formed with sides of length a(n-1) and a(n-2), since the greatest length the third side can have is a(n-1)+a(n-2)-1, and the least length would be a(n-1)-a(n-2)+1. (End)
Conjectures: a(n) = A029744(n+1)+1. Also, a(n) = positions of the zeros in A309019(n+2) - A002487(n+2). - George Beck, Mar 26 2022

			Some solutions for n=4
..2..1..2....1..2..1....0..2..1....2..0..1....1..2..0....2..1..2....0..1..0
..0..2..0....2..0..2....1..0..2....1..2..0....2..0..1....0..2..0....2..0..2
..1..0..1....0..1..0....0..2..1....2..0..1....1..2..0....1..0..1....1..2..1
..0..2..0....2..0..2....1..0..2....1..2..0....2..0..1....0..2..0....2..0..2
..1..0..1....0..1..0....0..2..1....2..0..1....1..2..0....2..1..2....1..2..1

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A029744, A209727.

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

Empirical: a(n) = a(n-1) +2*a(n-2) -2*a(n-3).

Empirical g.f.: x*(3+x-5*x^2)/((1-x)*(1-2*x^2)). [Colin Barker, Mar 23 2012]

A209722 1/4 the number of (n+1) X 4 0..2 arrays with every 2 X 2 subblock having distinct clockwise edge differences.

Original entry on oeis.org

4, 5, 6, 8, 10, 14, 18, 26, 34, 50, 66, 98, 130, 194, 258, 386, 514, 770, 1026, 1538, 2050, 3074, 4098, 6146, 8194, 12290, 16386, 24578, 32770, 49154, 65538, 98306, 131074, 196610, 262146, 393218, 524290, 786434, 1048578, 1572866, 2097154, 3145730

Offset: 1

R. H. Hardin, Mar 12 2012

nonn

Column 3 of A209727.

			Some solutions for n=4:
..2..1..2..1....2..1..2..1....1..2..1..2....1..0..2..0....2..1..2..1
..0..2..0..2....0..2..0..2....2..0..2..0....0..2..1..2....0..2..0..2
..2..1..2..1....1..0..1..0....0..1..0..1....1..0..2..0....1..0..1..0
..0..2..0..2....0..2..0..2....2..0..2..0....0..2..1..2....0..2..0..2
..2..1..2..1....2..1..2..1....0..1..0..1....1..0..2..0....1..0..1..0

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A209727.

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

Empirical: a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3).
Conjectures from Colin Barker, Jul 12 2018: (Start)
G.f.: x*(4 + x - 7*x^2) / ((1 - x)*(1 - 2*x^2)).
a(n) = 3*2^(n/2 - 1) + 2 for n even.
a(n) = 2^((n + 1)/2) + 2 for n odd.
(End)

A209723 1/4 the number of (n+1) X 5 0..2 arrays with every 2 X 2 subblock having distinct clockwise edge differences.

Original entry on oeis.org

6, 7, 8, 10, 12, 16, 20, 28, 36, 52, 68, 100, 132, 196, 260, 388, 516, 772, 1028, 1540, 2052, 3076, 4100, 6148, 8196, 12292, 16388, 24580, 32772, 49156, 65540, 98308, 131076, 196612, 262148, 393220, 524292, 786436, 1048580, 1572868, 2097156

Offset: 1

R. H. Hardin, Mar 12 2012

nonn

Column 4 of A209727.

			Some solutions for n=4:
..2..1..2..0..2....0..2..0..1..0....0..1..0..1..0....0..1..0..1..0
..0..2..0..1..0....2..1..2..0..2....2..0..2..0..2....2..0..2..0..2
..2..1..2..0..2....0..2..0..1..0....0..1..0..1..0....0..1..0..1..0
..0..2..0..1..0....2..1..2..0..2....2..0..2..0..2....2..0..2..0..2
..2..1..2..0..2....0..2..0..1..0....1..2..1..2..1....0..1..0..1..0

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A209727.

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

Empirical: a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3).
Conjectures from Colin Barker, Jul 12 2018: (Start)
G.f.: x*(6 + x - 11*x^2) / ((1 - x)*(1 - 2*x^2)).
a(n) = 3*2^(n/2 - 1) + 4 for n even.
a(n) = 2^((n + 1)/2) + 4 for n odd.
(End)

A320770 a(n) = (-1)^floor(n/4) * 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

Offset: 0

Michael Somos, Oct 20 2018

			G.f. = 1 + x + 2*x^2 + 2*x^3 - 4*x^4 - 4*x^5 - 8*x^6 - 8*x^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,-4).

Cf. A016116.

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

[(-1)^Floor(n/4)* 2^Floor(n/2): n in [0..50]]; // G. C. Greubel, Oct 27 2018

a[ n_] := (-1)^Quotient[n, 4] * 2^Quotient[n, 2];

{a(n) = (-1)^floor(n/4) * 2^floor(n/2)};

def A320770(n): return -(1<<(n>>1)) if n&4 else 1<<(n>>1) # Chai Wah Wu, Jan 18 2023

G.f.: (1 + x) * (1 + 2*x^2) / (1 + 4*x^4).
G.f.: A(x) = 1/(1 - x/(1 - x/(1 + 2*x/(1 - 4*x/(1 + 3*x/(1 + 5*x/(3 - 2*x))))))).
a(n) = (-1)^floor(n/2) * 2 * a(n-2) = -4 * a(n-4) for all n in Z.
a(n) = c(n) * (-2)^n * a(-n) for all n in Z where c(4*k+2) = -1 else 1.
a(n) = a(n+1) = (1+I)^n * (-I)^(n/2) * (-1)^floor(n/4) if n = 2*k.
a(n) =  (-1)^floor(n/4) * A016116(n).
E.g.f.: cosh(x)*(cos(x) + sin(x)) + sin(x)*sinh(x). - Stefano Spezia, Feb 04 2023

A343177 a(0)=4; if n > 0 is even then a(n) = 2^(n/2+1)+3, otherwise a(n) = 3*(2^((n-1)/2)+1).

Original entry on oeis.org

4, 6, 7, 9, 11, 15, 19, 27, 35, 51, 67, 99, 131, 195, 259, 387, 515, 771, 1027, 1539, 2051, 3075, 4099, 6147, 8195, 12291, 16387, 24579, 32771, 49155, 65539, 98307, 131075, 196611, 262147, 393219, 524291, 786435, 1048579, 1572867, 2097155, 3145731, 4194307, 6291459

Offset: 0

N. J. A. Sloane, Apr 26 2021

Number of edges along the boundary of the graph G(n) described in A342759.

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2).

Cf. A342759.

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

f:=n->if n = 0 then 4 elif (n mod 2) = 0 then 2^(n/2+1)+3 else 3*(2^((n-1)/2)+1); fi;
[seq(f(n),n=0..40)];

LinearRecurrence[{1, 2, -2}, {4, 6, 7, 9}, 50] (* or *)
A343177[n_] := Which[n == 0, 4, OddQ[n], 3*(2^((n-1)/2)+1), True, 2^(n/2+1)+3];
Array[A343177, 50, 0] (* Paolo Xausa, Feb 02 2024 *)

G.f.: (4 + 2*x - 7*x^2 - 2*x^3)/((1 - x)*(1 - 2*x^2)). - Stefano Spezia, Feb 04 2023

E.g.f.: 3*cosh(x) + 2*cosh(sqrt(2)*x) + 3*sinh(x) + 3*sinh(sqrt(2)*x)/sqrt(2) - 1. - Stefano Spezia, Jul 25 2024

A354785 Numbers of the form 32^k or 92^k.

Original entry on oeis.org

3, 6, 9, 12, 18, 24, 36, 48, 72, 96, 144, 192, 288, 384, 576, 768, 1152, 1536, 2304, 3072, 4608, 6144, 9216, 12288, 18432, 24576, 36864, 49152, 73728, 98304, 147456, 196608, 294912, 393216, 589824, 786432, 1179648, 1572864, 2359296, 3145728, 4718592, 6291456, 9437184, 12582912, 18874368, 25165824, 37748736, 50331648

Offset: 1

N. J. A. Sloane, Jul 12 2022

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

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.

seq[max_] := Union[Table[3*2^n, {n, 0, Floor[Log2[max/3]]}], Table[9*2^n, {n, 0, Floor[Log2[max/9]]}]]; seq[10^8] (* Amiram Eldar, Jan 16 2024 *)

Sum_{n>=1} 1/a(n) = 8/9. - Amiram Eldar, Jan 16 2024

G.f.: (3*x^2+6*x+3)/(1-2*x^2). - Georg Fischer, Apr 10 2025

A173787 Triangle read by rows: T(n,k) = 2^n - 2^k, 0 <= k <= n.

Original entry on oeis.org

0, 1, 0, 3, 2, 0, 7, 6, 4, 0, 15, 14, 12, 8, 0, 31, 30, 28, 24, 16, 0, 63, 62, 60, 56, 48, 32, 0, 127, 126, 124, 120, 112, 96, 64, 0, 255, 254, 252, 248, 240, 224, 192, 128, 0, 511, 510, 508, 504, 496, 480, 448, 384, 256, 0, 1023, 1022, 1020, 1016, 1008, 992, 960, 896, 768, 512, 0

Offset: 0

Reinhard Zumkeller, Feb 28 2010

			Triangle begins as:
   0;
   1,  0;
   3,  2,  0;
   7,  6,  4,  0;
  15, 14, 12,  8,  0;
  31, 30, 28, 24, 16, 0;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

[2^n -2^k: k in [0..n], n in [0..15]]; // G. C. Greubel, Jul 13 2021

Table[2^n -2^k, {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 13 2021 *)

flatten([[2^n -2^k for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jul 13 2021

A000120(T(n,k)) = A025581(n,k).
Row sums give A000337.
Central terms give A020522.
T(2*n+1, n) = A006516(n+1).
T(2*n+3, n+2) = A059153(n).
T(n, k) = A140513(n,k) - A173786(n,k), 0 <= k <= n.
T(n, k) = A173786(n,k) - A059268(n+1,k+1), 0 < k <= n.
T(2*n, 2*k) = T(n,k) * A173786(n,k), 0 <= k <= n.
T(n, 0) = A000225(n).
T(n, 1) = A000918(n) for n>0.
T(n, 2) = A028399(n) for n>1.
T(n, 3) = A159741(n-3) for n>3.
T(n, 4) = A175164(n-4) for n>4.
T(n, 5) = A175165(n-5) for n>5.
T(n, 6) = A175166(n-6) for n>6.
T(n, n-4) = A110286(n-4) for n>3.
T(n, n-3) = A005009(n-3) for n>2.
T(n, n-2) = A007283(n-2) for n>1.
T(n, n-1) = A000079(n-1) for n>0.
T(n, n) = A000004(n).

A036284 Periodic vertical binary vectors of Fibonacci numbers.

Original entry on oeis.org

6, 24, 1440, 5728448, 92568198012160, 26494530374406845814111659520, 2095920895719545919920115988669687683503034097906010941440, 13128614603426246034591796912897206548807135027496968025827278400248602613784037111736380004928525614173642247188480

Offset: 0

Antti Karttunen, Nov 01 1998

The sequence can be also computed with a recurrence that does not explicitly refer to Fibonacci numbers. See the given Maple and C programs.

Conjecture: For n>=1, each term a(n), when considered as a GF(2)[X]-polynomial, is divisible by GF(2)[X] -polynomial (x^3 + 1) ^ A000225(n-1). If this holds, then for n>=1, a(n) = A048720bi(A136380(n),A048723bi(9,A000225(n-1))). Conjecture 2: there is also one extra (x^1 + 1) factor present, see A136384.

			When Fibonacci numbers are written in binary (see A004685), under each other as:
0000000 (0)
0000001 (1)
0000001 (1)
0000010 (2)
0000011 (3)
0000101 (5)
0001000 (8)
0001101 (13)
0010101 (21)
0100010 (34)
0110111 (55)
1011001 (89)
it can be seen that the bits in the n-th column from right repeat after a period of A007283(n): 3, 6, 12, 24, ... (See also A001175). This sequence is formed from those bits: 011, reversed is 110, is binary for 6, thus a(0) = 6. 000110, reversed is 11000, is binary for 24, thus a(1) = 24, 000001011010, reversed is 10110100000, is binary for 1440, thus a(2) = 1440.

A. Karttunen, Table of n, a(n) for n = 0..10
A. Karttunen, C program for computing this sequence

Same sequence in octal base: A036285. Bits reversed: A036286. See also A136378, A136379, A136380, A136382, A136384, A037096, A037093, A000045.

A036284:=proc(n) option remember; local a, b, c, i, j, k, l, s, x, y, z; if (0 = n) then (6) else a := 0; b := 0; s := 0; x := 0; y := 0; k := 3*(2^(n-1)); l := 3*(2^n); j := 0; for i from 0 to l do z := bit_i(A036284(n-1),(j)); c := (a + b + (`if`((x = y),x,(z+1))) mod 2); if(c <> 0) then s := s + (2^i); fi; a := b; b := c; x := y; y := z; j := j + 1; if(j = k) then j := 0; fi; od; RETURN(s); fi; end:
bit_i := (x,i) -> `mod`(floor(x/(2^i)),2);

a[n_] := Sum[Mod[Fibonacci[k]/2^n // Floor, 2]* 2^k, {k, 0, 3*2^n - 1}]; Table[a[n], {n, 0, 7}] (* Jean-François Alcover, Mar 04 2016 *)

a(n) = Sum_{k=0..A007283(n)-1} ([A000045(k)/(2^n)] mod 2) * 2^k, where [] stands for floor function, i.e. Sum (bit n of Fibonacci(k))*(2^k), k = 0 ... (3*(2^n))-1.

Entry revised Dec 29 2007

cp's OEIS Frontend

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A067771 Number of vertices in Sierpiński triangle of order n.

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A209722 1/4 the number of (n+1) X 4 0..2 arrays with every 2 X 2 subblock having distinct clockwise edge differences.

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A209723 1/4 the number of (n+1) X 5 0..2 arrays with every 2 X 2 subblock having distinct clockwise edge differences.

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A320770 a(n) = (-1)^floor(n/4) * 2^floor(n/2).

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A343177 a(0)=4; if n > 0 is even then a(n) = 2^(n/2+1)+3, otherwise a(n) = 3*(2^((n-1)/2)+1).

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A354785 Numbers of the form 3*2^k or 9*2^k.

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