A029744 -id:A029744 - cp's OEIS frontend

A060546 a(n) = 2^ceiling(n/2).

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

Offset: 0

André Barbé (Andre.Barbe(AT)esat.kuleuven.ac.be), Apr 03 2001

a(n) is also the number of median-reflective (palindrome) symmetric patterns in a top-down equilateral triangular arrangement of closely packed black and white cells satisfying the local matching rule of Pascal's triangle modulo 2, where n is the number of cells in each edge of the arrangement. The matching rule is such that any elementary top-down triangle of three neighboring cells in the arrangement contains either one or three white cells.
The number of possibilities for an n-game (sub)set of tennis with neither player gaining a 2-game advantage. (Motivated by the marathon Isner-Mahut match at Wimbledon, 2010.) - Barry Cipra, Jun 28 2010
Number of achiral rows of n colors using up to two colors. For a(3)=4, the rows are AAA, ABA, BAB, and BBB. - Robert A. Russell, Nov 07 2018
Also the number of walks of length n on the graph x--y--z starting at y. - Sean A. Irvine, May 30 2025

Harry J. Smith, Table of n, a(n) for n = 0..500
A. Barbé, Symmetric patterns in the cellular automaton that generates Pascal's triangle modulo 2, Discr. Appl. Math. 105(2000), 1-38.
Index entries for sequences related to cellular automata
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (0,2).

Column k=2 of A321391.
Cf. A016116, A008619.
Cf. A000079 (oriented), A005418(n+1) (unoriented), A122746(n-2) (chiral).
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^Ceiling(n/2): n in [0..50]]; // G. C. Greubel, Nov 07 2018

for n from 0 to 100 do printf(`%d,`,2^ceil(n/2)) od:

2^Ceiling[Range[0,50]/2] (* or *) Riffle[2^Range[0, 25], 2^Range[25]] (* Harvey P. Dale, Mar 05 2013 *)
LinearRecurrence[{0, 2}, {1, 2}, 40] (* Robert A. Russell, Nov 07 2018 *)

a(n) = { 2^ceil(n/2) } \\ Harry J. Smith, Jul 06 2009

a(n) = 2^ceiling(n/2).
a(n) = A016116(n+1) for n >= 1.
a(n) = 2^A008619(n-1) for n >= 1.
G.f.: (1 + 2*x) / (1 - 2*x^2). - Ralf Stephan, Jul 15 2013
E.g.f.: cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x). - Stefano Spezia, Feb 02 2023

More terms from James Sellers, Apr 04 2001
a(0)=1 prepended by Robert A. Russell, Nov 07 2018
Edited by N. J. A. Sloane, Nov 10 2018

A163403 a(n) = 2*a(n-2) for n > 2; a(1) = 1, a(2) = 2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Jul 26 2009

a(n+1) is the number of palindromic words of length n using a two-letter alphabet. - Michael Somos, Mar 20 2011

			x + 2*x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 8*x^6 + 8*x^7 + 16*x^8 + 16*x^9 + 32*x^10 + ...

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

Equals A016116 without initial 1. Unsigned version of A152166.
Partial sums are in A136252.
Binomial transform is A078057, second binomial transform is A007070, third binomial transform is A102285, fourth binomial transform is A163350, fifth binomial transform is A163346.
Cf. A000079 (powers of 2), A009116, A009545, A051032.
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

[ n le 2 select n else 2*Self(n-2): n in [1..43] ];

LinearRecurrence[{0, 2}, {1, 2}, 50] (* Paolo Xausa, Feb 02 2024 *)

{a(n) = if( n<1, 0, 2^(n\2))} /* Michael Somos, Mar 20 2011 */

def A163403():
    x, y = 1, 1
    while True:
        yield x
        x, y = x + y, x - y
a = A163403(); [next(a) for i in range(40)]  # Peter Luschny, Jul 11 2013

a(n) = 2^((1/4)*(2*n - 1 + (-1)^n)).
G.f.: x*(1 + 2*x)/(1 - 2*x^2).
a(n) = A051032(n) - 1.
G.f.: x / (1 - 2*x / (1 + x / (1 + x))) = x * (1 + 2*x / (1 - x / (1 - x / (1 + 2*x)))). - Michael Somos, Jan 03 2013
From R. J. Mathar, Aug 06 2009: (Start)
a(n) = A131572(n).
a(n) = A060546(n-1), n > 1. (End)
a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011
a(n) = |A009116(n-1)| + |A009545(n-1)|. - Bruno Berselli, May 30 2011
E.g.f.: cosh(sqrt(2)*x) + sinh(sqrt(2)*x)/sqrt(2) - 1. - Stefano Spezia, Feb 05 2023

A060482 New record highs reached in A060030.

Original entry on oeis.org

1, 2, 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

Offset: 1

Henry Bottomley, Mar 19 2001

Harry J. Smith, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2).

Cf. A060013, A060030.

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

LinearRecurrence[{1,2,-2},{1,2,3,5,9},50] (* Harvey P. Dale, Sep 11 2016 *)

{ for (n=1, 1000, if (n%2==0, m=n/2; a=2^(m + 1) - 3, m=(n - 1)/2; a=3*2^m - 3); if (n<3, a=n); write("b060482.txt", n, " ", a); ) } \\ Harry J. Smith, Jul 05 2009

a(n) = a(n-1) + 2^((n-1)/2) = 2*a(n-2) + 3 = a(n-1) + A016116(n-1) = A027383(n-1) - 1 = 2*A027383(n-3) + 1 = 4*A052955(n-4) + 1. a(2n) = 2^(n+1) - 3; a(2n+1) = 3*2^n - 3.
From Colin Barker, Jan 12 2013: (Start)
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) for n > 5.
G.f.: x*(2*x^4-x^2+x+1) / ((x-1)*(2*x^2-1)). (End)
E.g.f.: 1 + x + x^2/2 - 3*cosh(x) + 2*cosh(sqrt(2)*x) - 3*sinh(x) + 3*sinh(sqrt(2)*x)/sqrt(2). - Stefano Spezia, Jul 25 2024

A136252 a(n) = a(n-1) + 2a(n-2) - 2a(n-3).

Original entry on oeis.org

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

Offset: 0

Paul Curtz, Mar 17 2008

For n >= 2, number of n X n arrays with values that are squares of integers, having all 2 X 2 subblocks summing to 4. - R. H. Hardin, Apr 03 2009

Number of moves required in 4-peg Tower of Hanoi solution using a (suboptimal) recursive algorithm: Move (n-2) disks, move bottom 2 disks, move (n-2) disks. Cf. A007664. - Toby Gottfried, Nov 29 2010

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

Same recurrence as in A135530.
Partial sums of A163403.
A060482 without the term 2.
Cf. A007664 (Optimal 4-peg Tower of Hanoi).
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) options operator,arrow: 2^((1/2)*n-1)*(4+4*(-1)^n+3*sqrt(2)*(1-(-1)^n))-3 end proc: seq(a(n),n=0..40); # Emeric Deutsch, Mar 31 2008

LinearRecurrence[{1, 2, -2}, {1, 3, 5}, 100] (* G. C. Greubel, Feb 18 2017 *)

x='x+O('x^50); Vec((1+2*x)/((1-x)*(1-2*x^2))) \\ G. C. Greubel, Feb 18 2017

a(n) = 2^((1/2)*n-1)*(4 + 4(-1)^n + 3*sqrt(2)*(1-(-1)^n)) - 3. - Emeric Deutsch, Mar 31 2008
G.f.: (1+2*x)/((1-x)*(1-2*x^2)). - Jaume Oliver Lafont, Aug 30 2009
a(n) = 2*a(n-2) + 3; first differences are powers of 2, occurring in pairs. - Toby Gottfried, Nov 29 2010
a(n) = A027383(n+1) - 1. - Jason Kimberley, Nov 01 2011
a(2n+1) = (a(2n) + a(2n+2))/2. - Richard R. Forberg, Nov 30 2013
E.g.f.: 4*cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x) - 3*cosh(x) - 3*sinh(x). - Stefano Spezia, May 13 2023

Edited by N. J. A. Sloane, Apr 18 2008

More terms from Emeric Deutsch, Mar 31 2008

A152166 a(2n) = 2^n; a(2n+1) = -(2^(n+1)).

Original entry on oeis.org

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

Philippe Deléham, Nov 27 2008

Ratios of successive terms are -2,-1,-2,-1,-2,-1,-2,-1,... - Philippe Deléham, Dec 12 2008

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

Cf. A000079, A016116, A147703.

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

LinearRecurrence[{0, 2}, {1, -2}, 50] (* Paolo Xausa, Jul 19 2024 *)

G.f.: (1 - 2*x)/(1 - 2*x^2).
a(n) = 2*a(n-2); a(0)=1, a(1)=-2.
a(n) = Sum_{k=0..n} A147703(n,k)*(-3)^k.
E.g.f.: cosh(sqrt(2)*x) - sqrt(2)*sinh(sqrt(2)*x). - Stefano Spezia, Feb 05 2023

A117575 Expansion of (1-x^3)/((1-x)(1+2x^2)).

Original entry on oeis.org

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

Offset: 0

Paul Barry, Mar 29 2006

Row sums of A116949.
From Paul Curtz, Oct 24 2012: (Start)
b(n) = abs(a(n)) = A158780(n+1) = 1,1,1,2,2,4,4,8,8,8,... .
Consider the autosequence (that is a sequence whose inverse binomial transform is equal to the signed sequence) of the first kind of the example. Its numerator is A046978(n), its denominator is b(n). The numerator of the first column is A075553(n).
The denominator corresponding to the 0's is a choice.
The classical denominator is 1,1,1,2,1,4,4,8,1,16,16,32,1,... . (End)

			   0/1,  1/1    1/1,   1/2,   0/2,  -1/4,  -1/4,  -1/8, ...
   1/1,  0/1,  -1/2,  -1/2,  -1/4,   0/4,   1/8,   1/8, ...
  -1/1, -1/2,   0/2,   1/4,   1/4,   1/8,   0/8, -1/16, ...
   1/2,  1/2,   1/4,   0/4   -1/8,  -1/8, -1/16,  0/16, ...
   0/2, -1/4,  -1/4,  -1/8,   0/8,  1/16,  1/16,  1/32, ...
  -1/4,  0/4,   1/8,   1/8,  1/16,  0/16, -1/32, -1/32, ...
   1/4,  1/8,   0/8, -1/16, -1/16, -1/32,  0/32,  1/64, ...
  -1/8, -1/8, -1/16,  0/16,  1/32,  1/32,  1/64,  0/64. - _Paul Curtz_, Oct 24 2012

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,-2).

Cf. A016116, A046978, A075553, A116949, A122016, A191754/A192366.

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

CoefficientList[Series[(1-x^3)/((1-x)(1+2x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{0,-2},{1,1,-1},45] (* Harvey P. Dale, Apr 09 2018 *)

a(n)=if(n,(-1)^(n\2)<<((n-1)\2),1) \\ Charles R Greathouse IV, Jan 31 2012

def A117575(n): return 1 if (n==0) else (-1)^(n//2)*2^((n-1)//2)
[A117575(n) for n in range(51)] # G. C. Greubel, Apr 19 2023

a(n) = a(n-1) - 2*a(n-2) + 2*a(n-3) for n >= 3.
a(n) = (cos(Pi*n/2) + sin(Pi*n/2)) * (2^((n-1)/2)*(1-(-1)^n)/2 + 2^((n-2)/2)*(1+(-1)^n)/2 + 0^n/2).
a(n+1) = Sum_{k=0..n} A122016(n,k)*(-1)^k. - Philippe Deléham, Jan 31 2012
E.g.f.: (1 + cos(sqrt(2)*x) + sqrt(2)*sin(sqrt(2)*x))/2. - Stefano Spezia, Feb 05 2023
a(n) = (-1)^floor(n/2)*2^floor((n-1)/2), with a(0) = 1. - G. C. Greubel, Apr 19 2023

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

Original entry on oeis.org

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

cp's OEIS Frontend

A060546 a(n) = 2^ceiling(n/2).

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A163403 a(n) = 2*a(n-2) for n > 2; a(1) = 1, a(2) = 2.

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A060482 New record highs reached in A060030.

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A136252 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3).

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A152166 a(2*n) = 2^n; a(2*n+1) = -(2^(n+1)).

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A117575 Expansion of (1-x^3)/((1-x)*(1+2*x^2)).

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

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A136252 a(n) = a(n-1) + 2a(n-2) - 2a(n-3).

A152166 a(2n) = 2^n; a(2n+1) = -(2^(n+1)).

A117575 Expansion of (1-x^3)/((1-x)(1+2x^2)).

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.