A209723 -id:A209723 - cp's OEIS frontend

A158780 a(2n) = A131577(n), a(2n+1) = A011782(n).

0, 1, 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, 2097152, 4194304

Offset: 0

Paul Curtz, Mar 26 2009

This construction combines the 2 basic sequences which equal their first differences in the same way as A138635 does for sequences which equal their 3rd differences and A137171 does for sequences which equal their fourth differences.
Essentially the same as A016116, A060546, and A131572. - R. J. Mathar, Apr 08 2009
Dropping a(0), this is the inverse binomial transform of A024537. - R. J. Mathar, Apr 08 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
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. - N. J. A. Sloane, Jul 14 2022

[0,1] cat [2^Floor((n-2)/2): n in [2..50]]; // G. C. Greubel, Apr 19 2023

Table[(2^Floor[n/2] +Boole[n==1] -Boole[n==0])/2, {n,0,50}] (* or *) LinearRecurrence[{0,2}, {0,1,1,1}, 51] (* G. C. Greubel, Apr 19 2023 *)

a(n)=if(n>3,([0,1; 2,0]^n*[1;1])[1,1]/2,n>0) \\ Charles R Greathouse IV, Oct 18 2022

def A158780(n): return (2^(n//2) + int(n==1) - int(n==0))/2
[A158780(n) for n in range(51)] # G. C. Greubel, Apr 19 2023

a(2n) + a(2n+1) = A000079(n).
G.f.: x*(1+x-x^2)/(1-2*x^2). - R. J. Mathar, Apr 08 2009
a(n) = (1/2)*(2^floor(n/2) + [n=1] - [n=0]). - G. C. Greubel, Apr 19 2023
E.g.f.: (2*cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x) + 2*x - 2)/4. - Stefano Spezia, May 13 2023

Edited by R. J. Mathar, Apr 08 2009

A354789 a(2n) = 92^n - 7, a(2n+1) = 32^(n+2) - 7.

Original entry on oeis.org

2, 5, 11, 17, 29, 41, 65, 89, 137, 185, 281, 377, 569, 761, 1145, 1529, 2297, 3065, 4601, 6137, 9209, 12281, 18425, 24569, 36857, 49145, 73721, 98297, 147449, 196601, 294905, 393209, 589817, 786425, 1179641, 1572857, 2359289, 3145721, 4718585, 6291449, 9437177, 12582905, 18874361, 25165817, 37748729, 50331641, 75497465

Offset: 0

N. J. A. Sloane, Jul 14 2022

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

LinearRecurrence[{1,2,-2},{2,5,11},100] (* Paolo Xausa, Oct 17 2023 *)
CoefficientList[Series[(2+3x+2x^2)/((1-x)(1-2x^2)),{x,0,50}],x] (* Harvey P. Dale, Jun 07 2024 *)

G.f.: (2 + 3*x + 2*x^2)/((1 - x)*(1 - 2*x^2)). - Stefano Spezia, Feb 05 2023

E.g.f.: - 7*cosh(x) + 9*cosh(sqrt(2)*x) - 7*sinh(x) + 6*sqrt(2)*sinh(sqrt(2)*x). - Stefano Spezia, Jul 25 2024

A131572 a(0) = 0 and a(1) = 1, continued such that absolute values of 2nd differences equal the original sequence.

Original entry on oeis.org

0, 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

Paul Curtz, Aug 28 2007

This is the main sequence of a family of sequences starting at a(0) = A and a(1) = B, continuing a(3, ...) = 2B, 2B, 4B, 4B, 8B, 8B, 16B, 16B, 32B, 32B, ... such that the absolute values of the 2nd differences, abs(a(n+2) - 2*a(n+1) + a(n)), equal the original sequence. Alternatively starting at a(0) = a(1) = 1 gives A016116.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (0,2).

Cf. A000007, A131575.

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

[2^Floor(n/2)-0^n: n in [0..50]]; // Vincenzo Librandi, Aug 18 2011

LinearRecurrence[{0,2},{0,1,2},50] (* Harvey P. Dale, Jul 10 2018 *)

[0]+[2^(n//2) for n in range(1,51)] # G. C. Greubel, Apr 22 2023

a(n) = 2*a(n-2), n>2.
O.g.f.: x*(1+2*x)/(1-2*x^2). - R. J. Mathar, Jul 16 2008
a(n) = A016116(n) - A000007(n), that is, a(0)=0, a(n) = A016116(n) for n>=1. - Bruno Berselli, Apr 13 2011
First differences: a(n+1) - a(n) = A131575(n).
Second differences: A131575(n+1) - A131575(n) = (-1)^n*a(n).
E.g.f.: -1 + cosh(sqrt(2)*x) + (1/sqrt(2))*sinh(sqrt(2)*x). - G. C. Greubel, Apr 22 2023

Edited by R. J. Mathar, Jul 16 2008

More terms from Vincenzo Librandi, Aug 18 2011

A354788 a(2k) = 32^k - 3, a(2*k+1) = 2^(k+2) - 3.

Original entry on oeis.org

0, 1, 3, 5, 9, 13, 21, 29, 45, 61, 93, 125, 189, 253, 381, 509, 765, 1021, 1533, 2045, 3069, 4093, 6141, 8189, 12285, 16381, 24573, 32765, 49149, 65533, 98301, 131069, 196605, 262141, 393213, 524285, 786429, 1048573, 1572861, 2097149, 3145725, 4194301, 6291453, 8388605, 12582909, 16777213, 25165821, 33554429, 50331645

Offset: 0

N. J. A. Sloane, Jul 13 2022

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-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. - N. J. A. Sloane, Jul 14 2022

f1:=proc(n) if (n mod 2) = 1 then 2^((n+3)/2)-3 else 3*2^(n/2)-3; fi; end;
[seq(f1(n),n=0..45)];

LinearRecurrence[{1,2,-2},{0,1,3},100] (* Paolo Xausa, Oct 17 2023 *)

a(n) = A136252(n-1). - R. J. Mathar, Jul 14 2022
G.f.: x*(1 + 2*x)/((x - 1)*(2*x^2 - 1)). - R. J. Mathar, Jul 14 2022
E.g.f.: 3*(cosh(sqrt(2)*x) - cosh(x)) - 3*sinh(x) + 2*sqrt(2)*sinh(sqrt(2)*x). - Stefano Spezia, Feb 04 2023

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.

Original entry on oeis.org

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)

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)

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

cp's OEIS Frontend

A158780 a(2n) = A131577(n), a(2n+1) = A011782(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

Extensions

A354789 a(2*n) = 9*2^n - 7, a(2*n+1) = 3*2^(n+2) - 7.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A131572 a(0) = 0 and a(1) = 1, continued such that absolute values of 2nd differences equal the original sequence.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

Extensions

A354788 a(2*k) = 3*2^k - 3, a(2*k+1) = 2^(k+2) - 3.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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

Views

A354789 a(2n) = 92^n - 7, a(2n+1) = 32^(n+2) - 7.

A354788 a(2k) = 32^k - 3, a(2*k+1) = 2^(k+2) - 3.

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