A135530 -id:A135530 - cp's OEIS frontend

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

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

A143095 (1, 2, 4, 8, ...) interleaved with (4, 8, 16, 32, ...).

Original entry on oeis.org

1, 4, 2, 8, 4, 16, 8, 32, 16, 64, 32, 128, 64, 256, 128, 512, 256, 1024, 512, 2048, 1024, 4096, 2048, 8192, 4096, 16384, 8192, 32768, 16384, 65536, 32768, 131072, 65536, 262144, 131072, 524288, 262144, 1048576, 524288, 2097152, 1048576, 4194304

Offset: 0

Gary W. Adamson & Roger L. Bagula, Jul 23 2008

Partial sums are in A079360. a(n) = A076736(n+5). - Klaus Brockhaus, Jul 27 2009

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

Cf. A048655.

seq(coeff(series((1+4*x)/(1-2*x^2), x, n+1), x, n), n = 0..45); # G. C. Greubel, Mar 13 2020

nn=30;With[{p=2^Range[0,nn]},Riffle[Take[p,nn-2],Drop[p,2]]] (* Harvey P. Dale, Oct 03 2011 *)

A143095(n):=(5-3*(-1)^n)*2^(1/4*(2*n-1+(-1)^n))/2$
makelist(A143095(n),n,0,30); /* Martin Ettl, Nov 03 2012 */

for(n=0, 41, print1((5-3*(-1)^n)*2^(1/4*(2*n-1+(-1)^n))/2, ",")) \\ Klaus Brockhaus, Jul 27 2009

[(5 -3*(-1)^n)*2^((2*n-1+(-1)^n)/4)/2 for n in (0..45)] # G. C. Greubel, Mar 13 2020

Inverse binomial transform of A048655: (1, 5, 11, 27, 65, 157, ...).
a(n) = A135530(n+1). - R. J. Mathar, Aug 02 2008
From Klaus Brockhaus, Jul 27 2009: (Start)
a(n) = (5 - 3*(-1)^n) * 2^((2*n-1+(-1)^n)/4)/2.
a(n) = 2*a(n-2) for n > 1; a(0) = 1, a(1) = 4.
G.f.: (1+4*x)/(1-2*x^2). (End)
a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011

More terms from Klaus Brockhaus, Jul 27 2009

A161941 a(n) = ((4+sqrt(2))(2+sqrt(2))^n + (4-sqrt(2))(2-sqrt(2))^n)/4.

Original entry on oeis.org

2, 5, 16, 54, 184, 628, 2144, 7320, 24992, 85328, 291328, 994656, 3395968, 11594560, 39586304, 135156096, 461451776, 1575494912, 5379076096, 18365314560, 62703106048, 214081795072, 730920968192, 2495520282624, 8520239194112

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jun 22 2009

Second binomial transform of A135530.

G. C. Greubel, Table of n, a(n) for n = 0..1000
C. Bautista-Ramos and C. Guillen-Galvan, Fibonacci numbers of generalized Zykov sums, J. Integer Seq., 15 (2012), #12.7.8.
Index entries for linear recurrences with constant coefficients, signature (4, -2).

Cf. A135530, A161944 (third binomial transform of A135530).

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((4+r)*(2+r)^n+(4-r)*(2-r)^n)/4: n in [0..24] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 01 2009

LinearRecurrence[{4,-2},{2,5},30] (* Harvey P. Dale, May 26 2012 *)

x='x+O('x^30); Vec((2-3*x)/(1-4*x+2*x^2)) \\ G. C. Greubel, Jan 27 2018

a(n) = 4*a(n-1) - 2*a(n-2) for n>1; a(0) = 2; a(1) = 5.
G.f.: (2-3*x)/(1-4*x+2*x^2).
a(n) = 2*A007070(n) - 3*A007070(n-1). - R. J. Mathar, Oct 20 2017

Edited and extended beyond a(4) by Klaus Brockhaus, Jul 01 2009

A147675 Divide by 2, multiply by 4, repeat.

Original entry on oeis.org

10, 5, 20, 10, 40, 20, 80, 40, 160, 80, 320, 160, 640, 320, 1280, 640, 2560, 1280, 5120, 2560, 10240, 5120, 20480, 10240, 40960, 20480, 81920, 40960, 163840, 81920, 327680, 163840, 655360, 327680, 1310720, 655360, 2621440, 1310720, 5242880

Offset: 1

N. J. A. Sloane, Apr 21 2009

A147675-A147678 are from a quiz that someone asked me to help them with.

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

Cf. A020714, A135530.

LinearRecurrence[{0, 2}, {10, 5}, 50] (* Paolo Xausa, Jan 30 2024 *)

From R. J. Mathar, Apr 22 2009: (Start)
a(n) = 2*a(n-2) = 5*A135530(n-1).
G.f.: 5*x*(2+x)/(1-2*x^2). (End)
From Amiram Eldar, Feb 02 2024: (Start)
Sum_{n>=1} 1/a(n) = 3/5.
Sum_{n>=1} (-1)^n/a(n) = 1/5. (End)

More terms from R. J. Mathar, Apr 22 2009

A161944 a(n) = ((4+sqrt(2))(3+sqrt(2))^n + (4-sqrt(2))(3-sqrt(2))^n)/4.

Original entry on oeis.org

2, 7, 28, 119, 518, 2275, 10024, 44219, 195146, 861343, 3802036, 16782815, 74082638, 327016123, 1443518272, 6371996771, 28127352722, 124160138935, 548069364556, 2419295214791, 10679285736854, 47140647917587

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jun 22 2009

Third binomial transform of A135530.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (6,-7).

Cf. A135530, A161941 (second binomial transform of A135530).

a:=[2,7];; for n in [3..25] do a[n]:=6*a[n-1]-7*a[n-2]; od; a; # Muniru A Asiru, Apr 04 2018

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((4+r)*(3+r)^n+(4-r)*(3-r)^n)/4: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 01 2009

LinearRecurrence[{6,-7}, {2,7}, 50] (* G. C. Greubel, Apr 03 2018 *)
Table[((4+Sqrt[2])(3+Sqrt[2])^n+(4-Sqrt[2])(3-Sqrt[2])^n)/4,{n,0,30}]// Simplify (* Harvey P. Dale, Jun 03 2020 *)

x='x+O('x^30); Vec((2-5*x)/(1-6*x+7*x^2)) \\ G. C. Greubel, Apr 03 2018

a(n) = 6*a(n-1) - 7*a(n-2) for n > 1; a(0) = 2; a(1) = 7.
G.f.: (2-5*x)/(1-6*x+7*x^2).
E.g.f.: exp(3*x)*(4*cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x))/2. - G. C. Greubel, Apr 03 2018

Edited and extended beyond a(4) by Klaus Brockhaus, Jul 01 2009

A135536 a(n) = 8*a(n-2), with a(0) = 7, a(1) = 14.

Original entry on oeis.org

7, 14, 56, 112, 448, 896, 3584, 7168, 28672, 57344, 229376, 458752, 1835008, 3670016, 14680064, 29360128, 117440512, 234881024, 939524096, 1879048192, 7516192768, 15032385536, 60129542144, 120259084288, 481036337152

Offset: 0

Paul Curtz, Feb 22 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,8).

Table[(7/4)*( (2 + Sqrt[2]) + (-1)^n*(2 - Sqrt[2]) )*(Sqrt[2])^(3*n), {n,0,25}] (* or *) LinearRecurrence[{0,8},{7,14}, 25] (* G. C. Greubel, Oct 18 2016 *)

a(n)=([0,1; 8,0]^n*[7;14])[1,1] \\ Charles R Greathouse IV, Oct 18 2016

a(n) = b(3*n) + b(3*n + 1) + b(3*n + 2), where b(n) = A135530(n) [previous name].
a(n) = 7 * abs(A094014(n)).
O.g.f.: 7*(1 + 2*x)/(1 - 8*x^2). - R. J. Mathar, Feb 23 2008
From G. C. Greubel, Oct 18 2016: (Start)
a(n) = (7/4)*( (2 + sqrt(2)) + (-1)^n*(2 - sqrt(2)) )*(sqrt(2))^(3*n).
a(n) = 8*a(n-2).
E.g.f.: (7/2)*( 2*cosh(2*sqrt(2)*x) + sqrt(2)*sinh(2*sqrt(2)*x) ). (End)

More terms from R. J. Mathar, Feb 23 2008

New name from G. C. Greubel, Oct 18 2016

A161947 a(n) = ((4+sqrt(2))(5+sqrt(2))^n + (4-sqrt(2))(5-sqrt(2))^n)/4.

Original entry on oeis.org

2, 11, 64, 387, 2398, 15079, 95636, 609543, 3895802, 24938531, 159781864, 1024232427, 6567341398, 42116068159, 270111829436, 1732448726703, 11111915190002, 71272831185851, 457154262488464, 2932267507610067, 18808127038865998, 120639117713628439, 773804255242366436

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jun 22 2009

Fifth binomial transform of A135530.

Muniru A Asiru, Table of n, a(n) for n = 0..252
Index entries for linear recurrences with constant coefficients, signature (10,-23).

Cf. A135530.

a := [2, 11];; for n in [3..10^2] do a[n] := 10*a[n-1] - 23*a[n-2]; od; a; # Muniru A Asiru, Feb 02 2018

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((4+r)*(5+r)^n+(4-r)*(5-r)^n)/4: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 01 2009

seq(simplify(((4+sqrt(2))*(5+sqrt(2))^n+(4-sqrt(2))*(5-sqrt(2))^n)*1/4), n = 0 .. 20); # Emeric Deutsch, Jun 28 2009

LinearRecurrence[{10,-23}, {2,11}, 50] (* G. C. Greubel, Aug 17 2018 *)
Table[(((4+Sqrt[2])(5+Sqrt[2])^n)+((4-Sqrt[2])(5-Sqrt[2])^n))/4,{n,0,20}]//Simplify (* Harvey P. Dale, Mar 07 2020 *)

x='x+O('x^30); Vec((2-9*x)/(1-10*x+23*x^2)) \\ G. C. Greubel, Aug 17 2018

a(n) = 10*a(n-1) - 23*a(n-2) for n>1; a(0) = 2; a(1) = 11.
G.f.: (2-9*x)/(1-10*x+23*x^2).
E.g.f.: exp(5*x)*(4*cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x))/2. - Stefano Spezia, Oct 24 2023

Edited and extended beyond a(4) by Klaus Brockhaus, Jul 01 2009

Extended by Emeric Deutsch, Jun 28 2009

A137206 First differences of A074323.

Original entry on oeis.org

0, 2, -1, 4, -2, 8, -4, 16, -8, 32, -16, 64, -32, 128, -64, 256, -128, 512, -256, 1024, -512, 2048, -1024, 4096, -2048, 8192, -4096, 16384, -8192, 32768, -16384, 65536, -32768, 131072, -65536, 262144, -131072, 524288, -262144, 1048576, -524288

Offset: 0

Paul Curtz, Mar 05 2008

sign

Matthew House, Table of n, a(n) for n = 0..6605
Index entries for linear recurrences with constant coefficients, signature (0,2).

Cf. A000079, A131577, A135530.

From R. J. Mathar, Apr 22 2009: (Start)
G.f.: x*(2-x)/(1-2*x^2).
a(n) = (-1)^(n+1)*A135530(n-1). (End)
a(n) = 2*a(n-2). - Matthew House, Jan 15 2017

More terms from R. J. Mathar, Apr 22 2009

A140407 A000225 interleaved with A000051.

Original entry on oeis.org

1, 2, 3, 3, 7, 5, 15, 9, 31, 17, 63, 33, 127, 65, 255, 129, 511, 257, 1023, 513, 2047, 1025, 4095, 2049, 8191, 4097, 16383, 8193, 32767, 16385, 65535, 32769, 131071, 65537, 262143, 131073, 524287, 262145, 1048575, 524289, 2097151, 1048577, 4194303

Offset: 0

Paul Curtz, Jun 16 2008

nonn

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

Cf. A000051, A000225, A000079, A135530.

LinearRecurrence[{-1,2,2},{1,2,3},50] (* Harvey P. Dale, Apr 03 2013 *)

def A140407(n): return 2 if n == 1 else (1<<(n>>1))|1 if n&1 else -1^(-2<<(n>>1)) # Chai Wah Wu, Dec 21 2022

a(2n) = A000225(n+1) = A135530(2n) - 1. a(2n+1) = A000051(n) = 1 + A135530(2n+1).
a(n) = -a(n-1) + 2*a(n-2) + 2*a(n-3). a(2n) + a(2n+1) = 3*A000079(n).
O.g.f.: (1 + 3x + 3x^2)/((1+x)*(1-2x^2)). - R. J. Mathar, Jul 08 2008

Edited and extended by R. J. Mathar, Jul 08 2008

A162356 a(n) = 8a(n-1)-14a(n-2) for n>1; a(0) = 2; a(1) = 9.

Original entry on oeis.org

2, 9, 44, 226, 1192, 6372, 34288, 185096, 1000736, 5414544, 29306048, 158644768, 858873472, 4649961024, 25175459584, 136304222336, 737977344512, 3995559643392, 21632794323968, 117124519584256, 634137036138496

Offset: 0

Klaus Brockhaus, Jul 01 2009

Fourth binomial transform of A135530.

Index entries for linear recurrences with constant coefficients, signature (8,-14).

Cf. A135530, A161944 (third binomial transform of A135530), A161947 (fifth binomial transform of A135530).

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((4+r)^(n+1)+(4-r)^(n+1))/4: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ];

G.f.: (2-7*x)/(1-8*x+14*x^2).

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

cp's OEIS Frontend

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

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A143095 (1, 2, 4, 8, ...) interleaved with (4, 8, 16, 32, ...).

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A161941 a(n) = ((4+sqrt(2))*(2+sqrt(2))^n + (4-sqrt(2))*(2-sqrt(2))^n)/4.

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A147675 Divide by 2, multiply by 4, repeat.

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A161944 a(n) = ((4+sqrt(2))*(3+sqrt(2))^n + (4-sqrt(2))*(3-sqrt(2))^n)/4.

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A135536 a(n) = 8*a(n-2), with a(0) = 7, a(1) = 14.

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A161947 a(n) = ((4+sqrt(2))*(5+sqrt(2))^n + (4-sqrt(2))*(5-sqrt(2))^n)/4.

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

A161941 a(n) = ((4+sqrt(2))(2+sqrt(2))^n + (4-sqrt(2))(2-sqrt(2))^n)/4.

A161944 a(n) = ((4+sqrt(2))(3+sqrt(2))^n + (4-sqrt(2))(3-sqrt(2))^n)/4.

A161947 a(n) = ((4+sqrt(2))(5+sqrt(2))^n + (4-sqrt(2))(5-sqrt(2))^n)/4.

A162356 a(n) = 8a(n-1)-14a(n-2) for n>1; a(0) = 2; a(1) = 9.