A172481 -id:A172481 - cp's OEIS frontend

A059570 Number of fixed points in all 231-avoiding involutions in S_n.

1, 2, 6, 14, 34, 78, 178, 398, 882, 1934, 4210, 9102, 19570, 41870, 89202, 189326, 400498, 844686, 1776754, 3728270, 7806066, 16311182, 34020466, 70837134, 147266674, 305718158, 633805938, 1312351118, 2714180722, 5607318414, 11572550770, 23860929422

Offset: 1

Emeric Deutsch, Feb 16 2001

Number of odd parts in all compositions (ordered partitions) of n: a(3)=6 because in 3=2+1=1+2=1+1+1 we have 6 odd parts. Number of even parts in all compositions (ordered partitions) of n+1: a(3)=6 because in 4=3+1=1+3=2+2=2+1+1=1+2+1=1+1+2=1+1+1+1 we have 6 even parts.
Convolved with (1, 2, 2, 2, ...) = A001787: (1, 4, 12, 32, 80, ...). - Gary W. Adamson, May 23 2009
An elephant sequence, see A175654. For the corner squares 36 A[5] vectors, with decimal values between 15 and 480, lead to this sequence. For the central square these vectors lead to the companion sequence 4*A172481, for n>=-1. - Johannes W. Meijer, Aug 15 2010
a(n) is the total number of runs of equal parts in the compositions of n. a(5) = 34 because there are 34 runs of equal parts in the compositions of 5, with parentheses enclosing each run: (5), (4)(1), (1)(4), (3)(2), (2)(3), (3)(1,1), (1)(3)(1), (1,1)(3), (2,2)(1), (2)(1)(2), (1)(2,2), (2)(1,1,1), (1)(2)(1,1), (1,1)(2)(1), (1,1,1)(2), (1,1,1,1,1). - Gregory L. Simay, Apr 28 2017
a(n) - a(n-2) is the number of 1's in all compositions of n and more generally, the number of k's in all compositions of n+k-1. - Gregory L. Simay, May 01 2017

			a(3) = 6 because in the 231-avoiding involutions of {1,2,3}, i.e., in 123, 132, 213, 321, we have altogether 6 fixed points (3+1+1+1).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Roland Bacher, Chebyshev polynomials, quadratic surds and a variation of Pascal's triangle, arXiv:1509.09054 [math.CO], 2015.
S. Heubach and T. Mansour, Counting rises, levels and drops in compositions, arXiv:math/0310197 [math.CO], 2003.
Brian Hopkins, Andrew V. Sills, Thotsaporn "Aek" Thanatipanonda, and Hua Wang, Parts and subword patterns in compositions, Preprint 2015.
Silvana Ramaj, New Results on Cyclic Compositions and Multicompositions, Master's Thesis, Georgia Southern Univ., 2021. See p. 30.
Index entries for linear recurrences with constant coefficients, signature (3,0,-4).

Cf. A001787, A027934, A045623, A127984, A128255, A172481, A225084.

[(3*n+4)*2^n/18-2*(-1)^n/9: n in [1..40]]; // Vincenzo Librandi, May 01 2017

LinearRecurrence[{3,0,-4},{1,2,6},30] (* Harvey P. Dale, Dec 29 2013 *)
Table[(3 n + 4) 2^n/18 - 2 (-1)^n/9, {n, 30}] (* Vincenzo Librandi, May 01 2017 *)

a(n) = (3*n+4)*2^n/18 - 2*(-1)^n/9.
G.f.: z*(1-z)/((1+z)*(1-2*z)^2).
a(n) = Sum_{j=0..n} Sum_{k=0..n} binomial(n-k, k+j)*2^k. - Paul Barry, Aug 29 2004
a(n) = Sum_{k=0..n+1} (-1)^(k+1)*binomial(n+1, k+j)*A001045(k). - Paul Barry, Jan 30 2005
Convolution of "Expansion of (1-x)/(1-x-2*x^2)" (A078008) with "Powers of 2" (A000079), treating the result as if offset=1. - Graeme McRae, Jul 12 2006
Convolution of "Difference sequence of A045623" (A045891) with "Positive integers repeated" (A008619), treating the result as if offset=1. - Graeme McRae, Jul 12 2006
a(n) = 3*a(n-1)-4*a(n-3); a(1)=1,a(2)=2,a(3)=6. - Philippe Deléham, Aug 30 2006
Equals row sums of A128255. (1, 2, 6, 14, 34, ...) - (0, 0, 1, 2, 6, 14, 34, ...) = A045623: (1, 2, 5, 12, 28, 64, ...). - Gary W. Adamson, Feb 20 2007
Equals triangle A059260 * [1, 2, 3, ...] as a vector. - Gary W. Adamson, Mar 06 2012
a(n) + a(n-1) = A001792(n-1). - Gregory L. Simay, Apr 30 2017
a(n) - a(n-2) = A045623(n-1). - Gregory L. Simay, May 01 2017
a(n) = A045623(n-1) + A045623(n-3) + A045623(n-5) + ... - Gregory L. Simay, Feb 19 2018
a(n) = A225084(2n,n). - Alois P. Heinz, Aug 30 2018

More terms from Eugene McDonnell (eemcd(AT)mac.com), Jan 13 2005

A175654 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1 - x - x^2)/(1 - 3x - x^2 + 6x^3).

Original entry on oeis.org

1, 2, 6, 14, 36, 86, 210, 500, 1194, 2822, 6660, 15638, 36642, 85604, 199626, 464630, 1079892, 2506550, 5811762, 13462484, 31159914, 72071654, 166599972, 384912086, 888906306, 2052031172, 4735527306, 10925175254, 25198866036, 58108609526, 133973643090

Offset: 0

Johannes W. Meijer, Aug 06 2010; edited Jun 21 2013

a(n) represents the number of n-move routes of a fairy chess piece starting in a given corner square (m = 1, 3, 7 or 9) on a 3 X 3 chessboard. This fairy chess piece behaves like a bishop on the eight side and corner squares but on the center square the bishop flies into a rage and turns into a raging elephant.
In chaturanga, the old Indian version of chess, one of the pieces was called gaja, elephant in Sanskrit. The Arabs called the game shatranj and the elephant became el fil in Arabic. In Spain chess became chess as we know it today but surprisingly in Spanish a bishop isn't a Christian bishop but a Moorish elephant and it still goes by its original name of el alfil.
On a 3 X 3 chessboard there are 2^9 = 512 ways for an elephant to fly into a rage on the central square (off the center the piece behaves like a normal bishop). The elephant is represented by the A[5] vector in the fifth row of the adjacency matrix A, see the Maple program and A180140. For the corner squares the 512 elephants lead to 46 different elephant sequences, see the overview of elephant sequences and the crossreferences.
The sequence above corresponds to 16 A[5] vectors with decimal values 71, 77, 101, 197, 263, 269, 293, 323, 326, 329, 332, 353, 356, 389, 449 and 452. These vectors lead for the side squares to A000079 and for the central square to A175655.

Gary Chartrand, Introductory Graph Theory, pp. 217-221, 1984.
David Hooper and Kenneth Whyld, The Oxford Companion to Chess, pp. 74, 366, 1992.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Viswanathan Anand, The Indian Defense, Time, Jun 19 2008.
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
Johannes W. Meijer, The elephant sequences.
Wikipedia, War Elephant.
Index entries for linear recurrences with constant coefficients, signature (3,1,-6).

Cf. Elephant sequences corner squares [decimal value A[5]]: A040000 [0], A000027 [16], A000045 [1], A094373 [2], A000079 [3], A083329 [42], A027934 [11], A172481 [7], A006138 [69], A000325 [26], A045623 [19], A000129 [21], A095121 [170], A074878 [43], A059570 [15], A175654 [71, this sequence], A026597 [325], A097813 [58], A057711 [27], 2*A094723 [23; n>=-1], A002605 [85], A175660 [171], A123203 [186], A066373 [59], A015518 [341], A134401 [187], A093833 [343].

Cf. A000244, A006130, A006138, A075188, A175655, A180140.

[n le 3 select Factorial(n) else 3*Self(n-1) +Self(n-2) -6*Self(n-3): n in [1..41]]; // G. C. Greubel, Dec 08 2021

nmax:=28; m:=1; A[1]:=[0,0,0,0,1,0,0,0,1]: A[2]:=[0,0,0,1,0,1,0,0,0]: A[3]:=[0,0,0,0,1,0,1,0,0]: A[4]:=[0,1,0,0,0,0,0,1,0]: A[5]:=[0,0,1,0,0,0,1,1,1]: A[6]:=[0,1,0,0,0,0,0,1,0]: A[7]:=[0,0,1,0,1,0,0,0,0]: A[8]:=[0,0,0,1,0,1,0,0,0]: A[9]:=[1,0,0,0,1,0,0,0,0]: A:=Matrix([A[1], A[2], A[3], A[4], A[5], A[6], A[7], A[8], A[9]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax);

LinearRecurrence[{3,1,-6}, {1,2,6}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2012 *)

a(n)=([0,1,0; 0,0,1; -6,1,3]^n*[1;2;6])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

[( (1-x-x^2)/((1-2*x)*(1-x-3*x^2)) ).series(x,n+1).list()[n] for n in (0..40)] # G. C. Greubel, Dec 08 2021

G.f.: (1 - x - x^2)/(1 - 3*x - x^2 + 6*x^3).
a(n) = 3*a(n-1) + a(n-2) - 6*a(n-3) with a(0)=1, a(1)=2 and a(2)=6.
a(n) = ((6+10*A)*A^(-n-1) + (6+10*B)*B^(-n-1))/13 - 2^n with A = (-1+sqrt(13))/6 and B = (-1-sqrt(13))/6.
Limit_{k->oo} a(n+k)/a(k) = (-1)^(n)*2*A000244(n)/(A075118(n) - A006130(n-1)*sqrt(13)).
a(n) = b(n) - b(n-1) - b(n-2), where b(n) = Sum_{k=1..n} Sum_{j=0..k} binomial(j,n-3*k+2*j)*(-6)^(k-j)*binomial(k,j)*3^(3*k-n-j), n>0, b(0)=1, with a(0) = b(0), a(1) = b(1) - b(0). - Vladimir Kruchinin, Aug 20 2010
a(n) = 2*A006138(n) - 2^n = 2*(A006130(n) + A006130(n-1)) - 2^n. - G. C. Greubel, Dec 08 2021
E.g.f.: 2*exp(x/2)*(13*cosh(sqrt(13)*x/2) + 3*sqrt(13)*sinh(sqrt(13)*x/2))/13 - cosh(2*x) - sinh(2*x). - Stefano Spezia, Feb 12 2023

A175655 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1+x-5x^2)/(1-3x-x^2+6*x^3).

Original entry on oeis.org

1, 4, 8, 22, 50, 124, 290, 694, 1628, 3838, 8978, 21004, 48962, 114022, 265004, 615262, 1426658, 3305212, 7650722, 17697430, 40911740, 94528318, 218312114, 503994220, 1163124866, 2683496134, 6189647948, 14273690782

Offset: 0

Johannes W. Meijer, Aug 06 2010, Aug 10 2010

a(n) represents the number of n-move routes of a fairy chess piece starting in the central square (m = 5) on a 3 X 3 chessboard. This fairy chess piece behaves like a bishop on the eight side and corner squares but on the central square the bishop turns into a raging elephant, see A175654.
For the central square the 512 elephants lead to 46 different elephant sequences, see the cross-references for examples.
The sequence above corresponds to 16 A[5] vectors with decimal values 71, 77, 101, 197, 263, 269, 293, 323, 326, 329, 332, 353, 356, 389, 449 and 452. These vectors lead for the side squares to A000079 and for the corner squares to A175654.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,1,-6).

Cf. Elephant sequences central square [decimal value A[5]]: A000007 [0], A000012 [16], A000045 [1], A011782 [2], A000079 [3], A003945 [42], A099036 [11], A175656 [7], A105476 [69], A168604 [26], A045891 [19], A078057 [21], A151821 [170], A175657 [43], 4*A172481 [15; n>=-1], A175655 [71, this sequence], 4*A026597 [325; n>=-1], A033484 [58], A087447 [27], A175658 [23], A026150 [85], A175661 [171], A036563 [186], A098156 [59], A046717 [341], 2*A001792 [187; n>=1 with a(0)=1], A175659 [343].

Cf. A000244, A006130, A075118, A175654.

I:=[1, 4, 8]; [n le 3 select I[n] else 3*Self(n-1)+Self(n-2)-6*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jul 21 2013

with(LinearAlgebra): nmax:=27; m:=5; A[5]:= [0,0,1,0,0,0,1,1,1]: A:=Matrix([[0,0,0,0,1,0,0,0,1], [0,0,0,1,0,1,0,0,0], [0,0,0,0,1,0,1,0,0], [0,1,0,0,0,0,0,1,0], A[5], [0,1,0,0,0,0,0,1,0], [0,0,1,0,1,0,0,0,0], [0,0,0,1,0,1,0,0,0], [1,0,0,0,1,0,0,0,0]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax);

CoefficientList[Series[(1 + x - 5 x^2) / (1 - 3 x - x^2 + 6 x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 21 2013 *)
LinearRecurrence[{3,1,-6},{1,4,8},40] (* Harvey P. Dale, Dec 25 2024 *)

a(n)=([0,1,0; 0,0,1; -6,1,3]^n*[1;4;8])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

G.f.: (1+x-5*x^2)/(1-3*x-x^2+6*x^3).
a(n) = 3*a(n-1) + a(n-2) - 6*a(n-3) with a(0)=1, a(1)=4 and a(2)=8.
a(n) = ((10+8*A)*A^(-n-1) + (10+8*B)*B^(-n-1))/13 - 2^n with A = (-1+sqrt(13))/6 and B = (-1-sqrt(13))/6.
Limit_{k->oo} a(n+k)/a(k) = (-1)^(n)*2*A000244(n)/(A075118(n)-A006130(n-1)*sqrt(13)).
E.g.f.: 2*exp(x/2)*(13*cosh(sqrt(13)*x/2) + 5*sqrt(13)*sinh(sqrt(13)*x/2))/13 - cosh(2*x) - sinh(2*x). - Stefano Spezia, Jan 31 2023

A127984 a(n) = (n/3 + 7/9)*2^(n - 1) + (-1)^n/9.

Original entry on oeis.org

1, 3, 7, 17, 39, 89, 199, 441, 967, 2105, 4551, 9785, 20935, 44601, 94663, 200249, 422343, 888377, 1864135, 3903033, 8155591, 17010233, 35418567, 73633337, 152859079, 316902969, 656175559, 1357090361, 2803659207, 5786275385, 11930464711, 24576757305

Offset: 1

Artur Jasinski, Feb 09 2007

a(n) is the number of runs of strictly increasing parts in all compositions of n. a(3) = 7: (1)(1)(1), (12), (2)(1), (3). - Alois P. Heinz, Apr 30 2017
From Hugo Pfoertner, Feb 19 2020: (Start)
a(n)/2^(n-2) apparently is the expected number of flips of a fair coin to completion of a game where the player advances by 1 for heads and by 2 for tails, starting at position 0 and repeating to flip until the target n+1 is exactly reached. If the position n (1 below the target) is reached, the player stays at this position and continues to flip the coin and count the flips until he can advance by 1.
The expected number of flips for targets 1, 2, 3,... , found by inversion of the corresponding Markov matrices, is 2, 2, 3, 7/2, 17/4, 39/8, 89/16, 199/32, 441/64, ...
Target 1 needs an expected number of 2 flips and would require a(0) = 1/2.
  n=1, target n+1 = 2: 1 / 2^(1-2) = 2;
  n=2, target n+1 = 3: 3 / 2^(2-2) = 3;
  n=3, target n+1 = 4: 7 / 2^(3-2) = 7/2.
(End)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
W. Bosma, Signed bits and fast exponentiation, J. Th. des Nombres de Bordeaux Vol.13, Fasc. 1, 2001.
Aruna Gabhe, Problem 11623, Am. Math. Monthly 119 (2012) 161.
Index entries for linear recurrences with constant coefficients, signature (3,0,-4).

Cf. A059570, A073371, A127976, A127978, A127979, A127980, A127981, A127982, A127983, A073371, A000337, A172481.

[(n/3+7/9)*2^(n-1)+(-1)^n/9: n in [1..35]]; // Vincenzo Librandi, Jun 15 2017

A127984:=n->(n/3 + 7/9)*2^(n - 1) + (-1)^n/9; seq(A127984(n), n=1..50); # Wesley Ivan Hurt, Mar 14 2014

Table[(n/3 + 7/9)2^(n - 1) + (-1)^n/9, {n, 50}] (* Artur Jasinski *)
CoefficientList[Series[(1 - 2 x^2) / ((-1 + 2 x)^2 (1 + x)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 15 2017 *)

a(n) = (n/3 + 7/9)*2^(n - 1) + (-1)^n/9.
From R. J. Mathar, Apr 04 2008: (Start)
O.g.f.: -x*(-1+2x^2)/((-1+2x)^2*(1+x)).
a(n) = 3*a(n-1) - 4*a(n-3). (End)
a(n) + a(n+1) = A087447(n+1). - R. J. Mathar, Feb 21 2009
A172481(n) = a(n) + 2^(n-1). Application: Problem 11623, AMM 119 (2012) 161. - Stephen J. Herschkorn, Feb 11 2012
From Wolfdieter Lang, Jun 14 2017: (Start)
a(n) = f(n+1)*2^(n-1), where f(n) is a rational Fibonacci type sequence based on fuse(a,b) = (a+b+1)/2 with f(0) = 0, f(1) = 1 and f(n) = fuse(f(n-1),f(n-2)), for n >= 2. For fuse(a,b) see the Jeff Erickson link under A188545. Proof: f(n) = (3*n+4 - (-1)^n/2^(n-2))/9, n >= 0, by induction.
a(n) = a(n-1) + a(n-2) + 2^(n-2), n >= 1, with inputs a(-1) = 0, a(0) = 1/2.
(End)
E.g.f.: (2*exp(-x) + exp(2*x)*(7 + 6*x) - 9)/18. - Stefano Spezia, Feb 19 2020

A175656 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1-3x^2)/(1-3x+4*x^3).

Original entry on oeis.org

1, 3, 6, 14, 30, 66, 142, 306, 654, 1394, 2958, 6258, 13198, 27762, 58254, 121970, 254862, 531570, 1106830, 2301042, 4776846, 9903218, 20505486, 42409074, 87614350, 180821106, 372827022, 768023666, 1580786574, 3251051634

Offset: 0

Johannes W. Meijer, Aug 06 2010

The a(n) represent the number of n-move routes of a fairy chess piece starting in the central square (m = 5) on a 3 X 3 chessboard. This fairy chess piece behaves like a bishop on the eight side and corner squares but on the central square the bishop turns into a raging elephant, see A175654.

The sequence above corresponds to 24 A[5] vectors with decimal values 7, 13, 37, 67, 70, 73, 76, 97, 100, 133, 193, 196, 259, 262, 265, 268, 289, 292, 322, 328, 352, 385, 388 and 448. These vectors lead for the side squares to A000079 and for the corner squares to A172481.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,-4).

Cf. A175655 (central square).

[((3*n+22)*2^n-4*(-1)^n)/18: n in [0..40]]; // Vincenzo Librandi, Aug 04 2011

with(LinearAlgebra): nmax:=29; m:=5; A[5]:= [0,0,0,0,0,0,1,1,1]: A:=Matrix([[0,0,0,0,1,0,0,0,1], [0,0,0,1,0,1,0,0,0], [0,0,0,0,1,0,1,0,0], [0,1,0,0,0,0,0,1,0], A[5], [0,1,0,0,0,0,0,1,0], [0,0,1,0,1,0,0,0,0], [0,0,0,1,0,1,0,0,0], [1,0,0,0,1,0,0,0,0]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax);

CoefficientList[Series[(1 - 3 x^2)/(1 - 3 x + 4 x^3), {x, 0, 29}], x] (* Michael De Vlieger, Nov 02 2018 *)
LinearRecurrence[{3,0,-4},{1,3,6},30] (* Harvey P. Dale, Aug 12 2020 *)

vector(40, n, n--; ((3*n+22)*2^n - 4*(-1)^n)/18) \\ G. C. Greubel, Nov 03 2018

G.f.: (1-3*x^2)/(1 - 3*x + 4*x^3).
a(n) = 3*a(n-1) - 4*a(n-3) with a(0)=1, a(1)=3 and a(2)=6.
a(n) = ((3*n+22)*2^n - 4*(-1)^n)/18.

A126184 Number of hex trees with n edges and having no nonroot nodes of outdegree 2.

Original entry on oeis.org

1, 3, 10, 33, 108, 351, 1134, 3645, 11664, 37179, 118098, 373977, 1180980, 3720087, 11691702, 36669429, 114791256, 358722675, 1119214746, 3486784401, 10847773692, 33705582543, 104603532030, 324270949293, 1004193907488

Offset: 0

Emeric Deutsch, Dec 19 2006

nonn

A hex tree is a rooted tree where each vertex has 0, 1, or 2 children and, when only one child is present, it is either a left child, or a middle child, or a right child (name due to an obvious bijection with certain tree-like polyhexes; see the Harary-Read reference).

F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinburgh Math. Soc. (2) 17 (1970), 1-13.
Index entries for linear recurrences with constant coefficients, signature (6, -9).

Cf. A126183.

Cf. A045891, A140429, A172481.

Maple
```
1,seq(3^(n-2)*(n+8),n=1..28);
```

a(n) = A126183(n,0).
a(n) = (n+8)*3^(n-2) for n >= 1; a(0)=1.
G.f.: (1-3z+z^2)/(1-3z)^2.
From Paul Curtz, Mar 27 2022: (Start)
a(n+1) = 3*a(n) + A140429(n), for n >= 0; a(0)=1.
Binomial transform of A172481(n) for n >= 0.
Also, with a different offset, the binomial transform of A045891(n+2) for n >= 0. (End)

A348405 a(0) = 1, a(n) + a(n+1) = round(2^n/9), n >= 0.

Original entry on oeis.org

1, -1, 1, -1, 2, 0, 4, 3, 11, 17, 40, 74, 154, 301, 609, 1211, 2430, 4852, 9712, 19415, 38839, 77669, 155348, 310686, 621382, 1242753, 2485517, 4971023, 9942058, 19884104, 39768220, 79536427, 159072867, 318145721, 636291456, 1272582898, 2545165810

Offset: 0

Paul Curtz, Oct 17 2021

sign

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

Cf. A139797 (a(n) + a(n+1) = round(2^n/9) too, but a(0) = 0).
Cf. A001045, A104581, A113405, A131666.
Cf. A136298, A172481, A175656, A348407.

CoefficientList[ Series[(x^4-x^3+2x-1)/((2*x^3-3*x^2+3*x-1)*(x+1)^2), {x, 0, 40}], x] (* Thomas Scheuerle, Oct 17 2021 *)
nxt[{n_,a_}]:={n+1,Round[(2^n)/9]-a}; NestList[nxt,{0,1},40][[All,2]] (* or *) LinearRecurrence[{1,2,-1,1,2},{1,-1,1,-1,2},40] (* Harvey P. Dale, Apr 28 2022 *)

a(n+1) = 2*a(n) - A104581(n+6).
a(n) + a(n+1) = A113405(n).
a(n) + a(n+3) = A001045(n).
a(n+2) = a(n) + A131666(n).
From Thomas Scheuerle, Oct 18 2021: (Start)
G.f.: (x^4-x^3+2x-1)/((2*x^3-3*x^2+3*x-1)*(x+1)^2).
A172481(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*a(2*n-k). With negative sign for ...*a(1+2*n-k) and ...*a(3+2*n-k) too.
A175656(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*a(2+2*n-k).
A136298(n+1) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*a(4+2*n-k).
A348407(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*(a(2+2*n-k) - 2*a(1+2*n-k) - a(2*n-k)).
(End)

a(22)-a(36) from Thomas Scheuerle, Oct 17 2021

cp's OEIS Frontend

A059570 Number of fixed points in all 231-avoiding involutions in S_n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A175654 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1 - x - x^2)/(1 - 3*x - x^2 + 6*x^3).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A175655 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1+x-5*x^2)/(1-3*x-x^2+6*x^3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A127984 a(n) = (n/3 + 7/9)*2^(n - 1) + (-1)^n/9.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

A175656 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1-3*x^2)/(1-3*x+4*x^3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A126184 Number of hex trees with n edges and having no nonroot nodes of outdegree 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Formula

A348405 a(0) = 1, a(n) + a(n+1) = round(2^n/9), n >= 0.

Views

Author

A175654 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1 - x - x^2)/(1 - 3x - x^2 + 6x^3).

A175655 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1+x-5x^2)/(1-3x-x^2+6*x^3).

A175656 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1-3x^2)/(1-3x+4*x^3).