A135318 -id:A135318 - cp's OEIS frontend

A373245 Binomial transform of A135318.

1, 2, 4, 9, 22, 55, 136, 331, 798, 1919, 4620, 11143, 26906, 64987, 156944, 378939, 914822, 2208455, 5331476, 12871151, 31073778, 75019219, 181113240, 437246723, 1055606686, 2548458047, 6152518684, 14853491319, 35859501322, 86572502155, 209004522016

Offset: 0

Paul Curtz, May 29 2024

Index entries for linear recurrences with constant coefficients, signature (4,-5,2,2).

Cf. A135318.

Cf. A114203.

CoefficientList[Series[(1 - 2*x + x^2 + x^3)/((1 - 2*x - x^2)*(1 - 2*x + 2*x^2)), {x, 0, 30}], x] (* Vaclav Kotesovec, May 29 2024 *)

G.f.: (1 - 2*x + x^2 + x^3)/((1 - 2*x - x^2)*(1 - 2*x + 2*x^2)). - Vaclav Kotesovec, May 29 2024
a(n) = A114203(n+1)/2. - Hugo Pfoertner, May 29 2024
E.g.f.: exp(x)*(2*cos(x) + 4*cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x))/6. - Stefano Spezia, May 29 2024

More terms from Vaclav Kotesovec, May 29 2024

A356050 a(n) = 2*A135318(n+1) - A135318(n).

Original entry on oeis.org

1, 1, 3, 4, 5, 6, 11, 14, 21, 26, 43, 54, 85, 106, 171, 214, 341, 426, 683, 854, 1365, 1706, 2731, 3414, 5461, 6826, 10923, 13654, 21845, 27306, 43691, 54614, 87381, 109226, 174763, 218454, 349525, 436906, 699051, 873814, 1398101, 1747626, 2796203, 3495254, 5592405

Offset: 0

Paul Curtz, Aug 19 2022

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

Cf. A001045, A084214, A135318, A230096.

Cf. A094958.

LinearRecurrence[{0, 1, 0, 2}, {1, 1, 3, 4}, 50] (* Amiram Eldar, Aug 19 2022 *)

a(n) = A135318(n) + A230096(n+1).
a(n) = a(n-8) + 5*A094958(n-5).
a(2*n) = A001045(n+2).
a(2*n+1) = A084214(n+1).
From Stefano Spezia, Aug 20 2022: (Start)
O.g.f.: (1 + x + 2*x^2 + 3*x^3)/((1 + x^2)*(1 - 2*x^2)).
E.g.f.: (8*cosh(sqrt(2)*x) - 2*cos(x) + 5*sqrt(2)*sinh(sqrt(2)*x) - 4*sin(x))/6. (End)
3*a(n) = A228826(n+1) +A094958(n+3). - R. J. Mathar, Jan 25 2023

A373392 Inverse binomial transform of A135318.

Original entry on oeis.org

1, 0, 0, 1, -2, 3, -4, 7, -18, 51, -136, 339, -814, 1935, -4620, 11111, -26842, 64923, -156944, 379067, -915078, 2208711, -5331476, 12870639, -31072754, 75018195, -181113240, 437248771, -1055610782, 2548462143, -6152518684, 14853483127, -35859484938, 86572485771

Offset: 0

Paul Curtz, Jun 03 2024

Index entries for linear recurrences with constant coefficients, signature (-4,-5,-2,2).

Cf. A000129, A009545, A135318, A373358.

LinearRecurrence[{-4, -5, -2, 2}, {1, 0, 0, 1}, 35] (* Amiram Eldar, Jun 09 2024 *)

a(n) = ((-([-2,-1;-1, 0]^(n-2))[2, 1]) - 2*((I-1)^(n-4) + (-I-1)^(n-4)))/3; \\ Thomas Scheuerle, Jun 04 2024

G.f.: (1 + 4*x + 5*x^2 + 3*x^3) / ( (1 + 2*x - x^2) * (1 + 2*x + 2*x^2) ).
E.g.f.: 1/6*exp(-x)*(2*cos(-x) + 4*cosh(sqrt(2)*-x) - 3*sqrt(2)*sinh(sqrt(2)*-x)).
a(n) = -4*a(n-1) - 5*a(n-2) - 2*a(n-3) + 2*a(n-4), for n > 4.
a(n) = (-1)^(n+1)*(A000129(n-2) + 2*A009545(n-2))/3, for n > 2. - Thomas Scheuerle, Jun 04 2024
a(n) = A373358(n-3) - (-1)^n*A009545(n+2) for n > 2.

A374927 a(n) = A135318(n)^2.

Original entry on oeis.org

1, 1, 1, 4, 9, 16, 25, 64, 121, 256, 441, 1024, 1849, 4096, 7225, 16384, 29241, 65536, 116281, 262144, 466489, 1048576, 1863225, 4194304, 7458361, 16777216, 29822521, 67108864, 119311929, 268435456, 477204025, 1073741824, 1908903481, 4294967296, 7635439161

Offset: 0

Paul Curtz, Jul 24 2024

A374098 terms swapped by pairs.

Index entries for linear recurrences with constant coefficients, signature (0,3,0,6,0,-8).

Cf. A000302, A112387, A135318, A139818, A374098.

a(2*n) = A139818(n+1).
a(2*n+1) = 4^n = A000302(n).
a(n) = 3*a(n-2) +6*a(n-4) -8*a(n-6).

A379530 a(n) = (A135318(3n) + A135318(3n+1) + A135318(3*n+2))/3.

Original entry on oeis.org

1, 3, 8, 23, 64, 185, 512, 1479, 4096, 11833, 32768, 94663, 262144, 757305, 2097152, 6058439, 16777216, 48467513, 134217728, 387740103, 1073741824, 3101920825, 8589934592, 24815366599, 68719476736, 198522932793, 549755813888, 1588183462343, 4398046511104, 12705467698745

Offset: 0

Paul Curtz, Dec 24 2024

Index entries for linear recurrences with constant coefficients, signature (0,7,0,8).

Cf. A001018, A013730, A015565, A135318.

LinearRecurrence[{0, 7, 0, 8}, {1, 3, 8, 23}, 30] (* Amiram Eldar, Dec 31 2024 *)

a(n) = 7*a(n-2) + 8*a(n-4) with a(0)=1, a(1)=3, a(2)=8, a(3)=23 for n >= 4.
a(2*n) = A001018(n).
a(2*n+1) = A015565(n+1) + A013730(n).

A048573 a(n) = a(n-1) + 2*a(n-2), a(0)=2, a(1)=3.

Original entry on oeis.org

2, 3, 7, 13, 27, 53, 107, 213, 427, 853, 1707, 3413, 6827, 13653, 27307, 54613, 109227, 218453, 436907, 873813, 1747627, 3495253, 6990507, 13981013, 27962027, 55924053, 111848107, 223696213, 447392427, 894784853, 1789569707, 3579139413, 7158278827, 14316557653

Offset: 0

Michael Somos, Jun 17 1999

Number of positive integers requiring exactly n signed bits in the modified non-adjacent form representation. - Ralf Stephan, Aug 02 2003
The n-th entry (n>1) of the sequence is equal to the 1,1-entry of the n-th power of the unnormalized 4 X 4 Haar matrix: [1 1 1 0 / 1 1 -1 0 / 1 1 0 1 / 1 1 0 -1]. - Simone Severini, Oct 27 2004
Pisano period lengths:  1, 1, 6, 2, 2, 6, 6, 2, 18, 2, 10, 6, 12, 6, 6, 2, 8, 18, 18, 2, ... - R. J. Mathar, Aug 10 2012
For n >= 1, a(n) is the number of ways to tile a strip of length n+2 with blue squares and blue and red dominos, with the restriction that the first two tiles must be the same color. - Guanji Chen and Greg Dresden, Jul 15 2024

			G.f. = 2 + 3*x + 7*x^2 + 13*x^3 + 27*x^4 + 53*x^5 + 107*x^6 + 213*x^7 + 427*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Wieb Bosma, Signed bits and fast exponentiation, Journal de théorie des nombres de Bordeaux, Vol. 13, No. 1 (2001), pp. 27-41.
Karl Dilcher and Larry Ericksen, Continued fractions and Stern polynomials, Ramanujan Journal 45.3 (2018): 659-681. See Table 2.
Karl Dilcher and Hayley Tomkins, Square classes and divisibility properties of Stern polynomials, Integers, Vol. 18 (2018), Article #A29.
Petro Kosobutskyy, The Collatz problem as a reverse n->0 problem on a graph tree formed from theta*2^n Jacobsthal-type numbers, arXiv:2306.14635 [math.GM], 2023.
Petro Kosobutskyy and Dariia Rebot, Collatz conjecture 3n+/-1 as a Newton binomial problem, Comp. Des. Sys. Theor. Prac., Lviv Nat'l Polytech. Univ. (Ukraine 2023) Vol. 5, No. 1, 137-145. See p. 140.
Saad Mneimneh, Simple Variations on the Tower of Hanoi to Guide the Study of Recurrences and Proofs by Induction, Department of Computer Science, Hunter College, CUNY, 2019.
Sam Northshield, Stern's diatomic sequence 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, ..., Amer. Math. Monthly, Vol. 117, No. 7 (2010), pp. 581-598.
Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A084214 (first differences).
Cf. A001045, A014551, A024493, A062510.
Cf. A007283, A007679, A078008, A081254.
Cf. A083322, A083944, A083581, A084170.
Cf. A102713, A127978, A130755, A139763.
Cf. A140360, A140966, A166249, A168642.
Cf. A280560, A290604, A340627.
Cf. A112387, A135318.

[(5*2^n+(-1)^n)/3: n in [0..35]]; // Vincenzo Librandi, Jul 05 2011

LinearRecurrence[{1,2},{2,3},40] (* Harvey P. Dale, Dec 11 2017 *)

{a(n) = if( n<0, 0, (5*2^n + (-1)^n) / 3)};

{a(n) = if (n<0 ,0, if( n<2, n+2, a(n-1) + 2*a(n-2)))};

[(5*2^n+(-1)^n)/3 for n in range(35)] # G. C. Greubel, Apr 10 2019

G.f.: (2 + x) / (1 - x - 2*x^2).
a(n) = (5*2^n + (-1)^n) / 3.
a(n) = 2^(n+1) - A001045(n).
a(n) = A084170(n)+1 = abs(A083581(n)-3) = A081254(n+1) - A081254(n) = A084214(n+2)/2.
a(n) = 2*A001045(n+1) + A001045(n) (note that 2 is the limit of A001045(n+1)/A001045(n)). - Paul Barry, Sep 14 2009
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-3, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=-charpoly(A,-1). - Milan Janjic, Jan 27 2010
Equivalently, with different offset, a(n) = b(n+1) with b(0)=1 and b(n) = Sum_{i=0..n-1} (-1)^i (1 + (-1)^i b(i)). - Olivier Gérard, Jul 30 2012
a(n) = A000975(n-2)*10 + 5 + 2*(-1)^(n-2), a(0)=2, a(1)=3. - Yuchun Ji, Mar 18 2019
a(n+1) = Sum_{i=0..n} a(i) + 1 + (1-(-1)^n)/2, a(0)=2. - Yuchun Ji, Apr 10 2019
a(n) = 2^n + J(n+1) = J(n+2) + J(n+1) - J(n), where J is A001045. - Yuchun Ji, Apr 10 2019
a(n) = A001045(n+2) + A078008(n) = A062510(n+1) - A078008(n+1) = (A001045(n+2) + A062510(n+1))/2 = A014551(n) + 2*A001045(n). - Paul Curtz, Jul 14 2021
From Thomas Scheuerle, Jul 14 2021: (Start)
a(n) = A083322(n) + A024493(n).
a(n) = A127978(n) - A102713(n).
a(n) = A130755(n) - A166249(n).
a(n) = A007679(n) + A139763(n).
a(n) = A168642(n) XOR A007283(n).
a(n) = A290604(n) + A083944(n). (End)
From Paul Curtz, Jul 21 2021: (Start)
a(n) = 5*A001045(n) - A280560(n+1) = abs(A140360(n+1)) - A280560(n+1).
a(n) = 2^n + A001045(n+1) = A001045(n+3) - A000079(n).
a(n) = A001045(n+4) - A340627(n). (End)
a(n) = A001045(n+5) - A005010(n).
a(n+1) + a(n) = a(n+2) - a(n) = 5*2^n. - Michael Somos, Feb 22 2023
a(n) = A135318(2*n) + A135318(2*n+1) = A112387(2*n) + A112387(2*n+1). - Paul Curtz, Jun 26 2024
E.g.f.: (cosh(x) + 5*cosh(2*x) - sinh(x) + 5*sinh(2*x))/3. - Stefano Spezia, May 18 2025

Formula of Milan Janjic moved here from wrong sequence by Paul D. Hanna, May 29 2010

A112387 a(n) = 2^(n/2) if n is even and a(n-1) - a(n-2) if n is odd, a(1) = 1.

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 8, 5, 16, 11, 32, 21, 64, 43, 128, 85, 256, 171, 512, 341, 1024, 683, 2048, 1365, 4096, 2731, 8192, 5461, 16384, 10923, 32768, 21845, 65536, 43691, 131072, 87381, 262144, 174763, 524288, 349525, 1048576, 699051, 2097152, 1398101, 4194304

Offset: 0

Edwin F. Sampang, Dec 05 2005

This sequence originated from the Fibonacci sequence, but instead of adding the last two terms, you get the average. Example, if you have the initial condition a(1)=x and a(2)=y, a(3)=(x+y)/2, a(4)=(x+3y)/4, a(5)=(3x+5y)/8, a(6)=(5x+11y)/16 and so on and so forth. I used the coefficients of x and y as well as the denominator.

As n approaches infinity a(n)/a(n+1) oscillates between the values 3/2 and 1/3.

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

Cf. A000079, A001045, A048573, A135318.

a:= proc(n) option remember;
      `if`(n::even, 2^(n/2), a(n-1)-a(n-2))
    end: a(1):=1:
seq(a(n), n=0..50);  # Alois P. Heinz, Sep 27 2023

a[1] = 1; a[2] = 2; a[n_] := a[n] = If[ EvenQ[n], 2^(n/2), a[n - 1] - a[n - 2]]; Array[a, 43] (* Robert G. Wilson v, Dec 05 2005 *)
nxt[{n_,a_,b_}]:={n+1,b,If[OddQ[n],2^((n+1)/2),b-a]}; NestList[nxt,{2,1,2},50][[All,2]] (* Harvey P. Dale, Jul 08 2019 *)

a(n) = 2^(n/2) if n is even, a(n) = a(n-1) - a(n-2) if n is odd, and a(1) = 1.
a(2n) = A000079(n), a(2n-1) = A001045(n).
G.f.: (1+x+x^2)/((1+x^2)*(1-2*x^2)). - Joerg Arndt, Apr 25 2021
a(n) = A135318(n + (-1)^n). - Paul Curtz, Sep 27 2023
E.g.f.: (3*cosh(sqrt(2)*x) + sin(x) + sqrt(2)*sinh(sqrt(2)*x))/3. - Stefano Spezia, Jun 30 2024
a(2*n) + a(2*n+1) = A048573(n); a(2*n+1) + a(2*n+2) = A001045(n+3). - Paul Curtz, Jan 03 2025

Edited and extended by Robert G. Wilson v, Dec 05 2005

a(0)=1 prepended by Alois P. Heinz, Sep 27 2023

A133684 a(2n) = A001045(n); a(1)=1; a(2n+1) = 2*A001045(n-1) for n >= 1.

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 3, 2, 5, 6, 11, 10, 21, 22, 43, 42, 85, 86, 171, 170, 341, 342, 683, 682, 1365, 1366, 2731, 2730, 5461, 5462, 10923, 10922, 21845, 21846, 43691, 43690, 87381, 87382, 174763, 174762, 349525, 349526, 699051, 699050, 1398101, 1398102, 2796203, 2796202

Offset: 0

Paul Curtz, Jan 04 2008

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

Cf. A001045, A016116, A133730 (first differences).

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; 2,0,1,0]^n*[0;1;1;0])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

a(4*n) + a(4*n+1) = a(4*n+2).
a(n) + a(n+2) = 2^floor(n/2) = A016116(n).
O.g.f.: x - x^2*(1 + 2*x^3)/((2*x^2-1)*(x^2+1)) . - R. J. Mathar, Feb 23 2008
a(n+1) = 2*A135318(n) - A135318(n+1). - Paul Curtz, May 27 2024

Terms a(24) and beyond from Andrew Howroyd, Feb 02 2020

A366143 a(n) = a(n-2) + 2*a(n-4) - a(n-10), with a[0..9] = [1, 1, 1, 1, 1, 2, 3, 5, 6, 9].

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 5, 6, 9, 11, 18, 22, 35, 43, 69, 84, 134, 164, 263, 321, 513, 627, 1004, 1226, 1961, 2396, 3835, 4684, 7494, 9155, 14651, 17896, 28635, 34980, 55976, 68376, 109411, 133652, 213869, 261249, 418040, 510657, 817143, 998175, 1597247, 1951113

Offset: 0

Greg Dresden and Ziyi Xie, Sep 30 2023

a(n) is the number of ways to tile a zig-zag strip of n cells using squares (of 1 cell) and strips (of 3 cells). Here is the zig-zag strip corresponding to n=11, with 11 cells:
       _     _
   _|   |_|   |_
  |   |_|   |_|   |_
  |_|   |_|   |_|   |
  |   |_|   |_|   |_|
  |_|   |_|   |_|,
and here is the strip of 3 cells (which can be reflected)
           _
       _|   |
   _|    _|
  |    _|
  |_|
As an example, here is one of the a(11) = 18 ways to tile the zig-zag strip of 11 cells:
       _     _
   _|   |_|   |_
  |   |_|   |_    |_
  |_|    _|   |_    |
  |    _|   |_|   |_|
  |_|   |_|   |_|

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

Cf. A135318.

LinearRecurrence[{0, 1, 0, 2, 0, 0, 0, 0, 0, -1}, {1, 1, 1, 1, 1, 2,
  3, 5, 6, 9}, 40]

a(n) = a(n-2) + 2*a(n-4) - a(n-10).
a(2*n) = a(2*n-1) + a(2*n-4) - a(2*n-5) + a(2*n-6).
a(2*n+1) = a(2*n) + 2*a(2*n-3) - a(2*n-4) + a(2*n-6) - a(2*n-7).
G.f.: (x^8+x^7-x^5-2*x^4+x+1)/(x^10-2*x^4-x^2+1).

A365274 a(n) = a(n-2) + 4a(n-4) - 2a(n-8) - a(n-10), with a[0..9] = [1, 1, 1, 2, 3, 5, 7, 13, 18, 31].

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 7, 13, 18, 31, 43, 78, 108, 190, 263, 471, 652, 1156, 1600, 2853, 3949, 7019, 9715, 17299, 23944, 42592, 58952, 104926, 145230, 258403, 357659, 636490, 880976, 1567619, 2169764, 3861135, 5344256, 9509879, 13162764, 23423036, 32420177

Offset: 0

Greg Dresden and Ziyi Xie, Oct 01 2023

a(n) is the number of ways to tile a zig-zag strip of n cells using squares (of 1 cell) and strips and triangles (of 3 cells). Here is the zig-zag strip corresponding to n=11, with 11 cells:
       _     _
   _|   |_|   |_
  |   |_|   |_|   |_
  |_|   |_|   |_|   |
  |   |_|   |_|   |_|
  |_|   |_|   |_|,
and here are the strip and triangle of 3 cells (which can be reflected):
          _     _
      _|   |   |   |_
  _|    _|   |       |
 |    _|       |    _|
 |_|           |_|.
As an example, here is one of the a(11) = 78 ways to tile the zig-zag strip of 11 cells:
      _     _
  _|   |_|   |_
 |   |_|   |       |_
 |_|    _|    _|   |
 |    _|   |_|   |_|
 |_|   |_|   |_|.

Index entries for linear recurrences with constant coefficients, signature (0,1,0,4,0,0,0,-2,0,-1).

Cf. A135318, A366143.

LinearRecurrence[{0, 1, 0, 4, 0, 0, 0, -2, 0, -1}, {1, 1, 1, 2, 3, 5, 7, 13, 18, 31}, 40]

a(n) = a(n-2) + 4*a(n-4) - 2*a(n-8) - a(n-10).
a(2*n) = a(2*n-1) + a(2*n-2) - a(2*n-3) + a(2*n-4).
a(2*n+1) = a(2*n) + a(2*n-2) +2*a(2*n-3) - a(2*n-4) - a(2*n-7).
G.f.: (x^8-x^5-2*x^4+x^3+x+1)/(x^10+2*x^8-4*x^4-x^2+1).

cp's OEIS Frontend

A373245 Binomial transform of A135318.

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A356050 a(n) = 2*A135318(n+1) - A135318(n).

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A373392 Inverse binomial transform of A135318.

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A374927 a(n) = A135318(n)^2.

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A379530 a(n) = (A135318(3*n) + A135318(3*n+1) + A135318(3*n+2))/3.

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A048573 a(n) = a(n-1) + 2*a(n-2), a(0)=2, a(1)=3.

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A112387 a(n) = 2^(n/2) if n is even and a(n-1) - a(n-2) if n is odd, a(1) = 1.

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Maple

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A133684 a(2n) = A001045(n); a(1)=1; a(2n+1) = 2*A001045(n-1) for n >= 1.

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A379530 a(n) = (A135318(3n) + A135318(3n+1) + A135318(3*n+2))/3.

A365274 a(n) = a(n-2) + 4a(n-4) - 2a(n-8) - a(n-10), with a[0..9] = [1, 1, 1, 2, 3, 5, 7, 13, 18, 31].