A123203 -id:A123203 - cp's OEIS frontend

A036563 a(n) = 2^n - 3.

-2, -1, 1, 5, 13, 29, 61, 125, 253, 509, 1021, 2045, 4093, 8189, 16381, 32765, 65533, 131069, 262141, 524285, 1048573, 2097149, 4194301, 8388605, 16777213, 33554429, 67108861, 134217725, 268435453, 536870909, 1073741821, 2147483645

Offset: 0

N. J. A. Sloane

a(n+1) is the n-th number with exactly n 1's in binary representation. - Reinhard Zumkeller, Mar 06 2003
Berstein and Onn: "For every m = 3k+1, the Graver complexity of the vertex-edge incidence matrix of the complete bipirtite graph K(3,m) satisfies g(m) >= 2^(k+2)-3." - Jonathan Vos Post, Sep 15 2007
Row sums of triangle A135857. - Gary W. Adamson, Dec 01 2007
a(n) = A164874(n-1,n-2) for n > 2. - Reinhard Zumkeller, Aug 29 2009
Starting (1, 5, 13, ...) = eigensequence of a triangle with A016777: (1, 4, 7, 10, ...) as the left border and the rest 1's. - Gary W. Adamson, Jul 24 2010
An elephant sequence, see A175655. For the central square just one A[5] vector, with decimal value 186, leads to this sequence (n >= 2). For the corner squares this vector leads to the companion sequence A123203. - Johannes W. Meijer, Aug 15 2010
First differences of A095264: A095264(n+1) - A095264(n) = a(n+2). - J. M. Bergot, May 13 2013
a(n+2) is given by the sum of n-th row of triangle of powers of 2: 1; 2 1 2; 4 2 1 2 4; 8 4 2 1 2 4 8; ... - Philippe Deléham, Feb 24 2014
Also, the decimal representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 643", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. See A283508. - Robert Price, Mar 09 2017
a(n+3) is the value of the Ackermann function A(3,n) or ack(3,n). - Olivier Gérard, May 11 2018

			a(2) = 1;
a(3) = 2 + 1 + 2 = 5;
a(4) = 4 + 2 + 1 + 2 + 4 = 13;
a(5) = 8 + 4 + 2 + 1 + 2 + 4 + 8 = 29; etc. - _Philippe Deléham_, Feb 24 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Paul Barry, Conjectures and results on some generalized Rueppel sequences, arXiv:2107.00442 [math.CO], 2021.
Yael Berstein and Shmuel Onn, The Graver complexity of integer programming, Annals of Combinatorics, Vol. 13, No. 3 (2009), pp. 289-296; arXiv preprint, arXiv:0709.1500 [math.CO], 2007.
L' Education Mathématique, Problème 8907, 49e Annee, No 14, 15 Avril 1947, p. 113
Irving Kaplansky and John Riordan, The problem of the rooks and its applications, in Combinatorics, Duke Mathematical Journal, 13.2 (1946): 259-268. [Annotated scanned copy]
Irving Kaplansky and John Riordan, The problem of the rooks and its applications, Duke Mathematical Journal 13.2 (1946): 259-268. Sequence is on page 267.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Row sums of triangular array A027960. A column of A119725.
Cf. A081118, A130459, A131112, A050414, A050415, A135857, A000079, A016777, A283508.
Cf. A074877, A304370, A304371, A331372.

List([0..40], n-> 2^n -3); # G. C. Greubel, Nov 18 2019

[2^n-3: n in [0..40]]; // Vincenzo Librandi, May 09 2011

A036563:=n->2^n-3; seq(A036563(n), n=0..40); # Wesley Ivan Hurt, Jun 26 2014

Table[2^n - 3, {n, 0, 40}] (* Wesley Ivan Hurt, Jun 26 2014 *)
LinearRecurrence[{3,-2},{-2,-1},40] (* Harvey P. Dale, Sep 26 2018 *)

a(n)= 2^n-3 \\ Charles R Greathouse IV, Dec 22 2011

def A036563(n): return (1<Chai Wah Wu, Sep 27 2024

[gaussian_binomial(n,1,2)-2 for n in range(0,40)] # Zerinvary Lajos, May 31 2009

a(n) = 2*a(n-1) + 3.
The sequence 1, 5, 13, ... has a(n) = 4*2^n-3. These are the partial sums of A151821. - Paul Barry, Aug 25 2003
a(n) = A118654(n-3, 6), for n > 2. - N. J. A. Sloane, Sep 29 2006
Row sums of triangle A130459 starting (1, 5, 13, 29, 61, ...). - Gary W. Adamson, May 26 2007
Row sums of triangle A131112. - Gary W. Adamson, Jun 15 2007
Binomial transform of [1, 4, 4, 4, ...] = (1, 5, 13, 29, 61, ...). - Gary W. Adamson, Sep 20 2007
a(n) = 2*StirlingS2(n,2) - 1, for n > 0. - Ross La Haye, Jul 05 2008
a(n) = A000079(n) - 3. - Omar E. Pol, Dec 21 2008
From Mohammad K. Azarian, Jan 14 2009: (Start)
G.f.: 1/(1-2*x) - 3/(1-x).
E.g.f.: exp(2*x) - 3*exp(x). (End)
For n >= 3, a(n) = 2<+>n, where operation <+> is defined in A206853. - Vladimir Shevelev, Feb 17 2012
a(n) = 3*a(n-1) - 2*a(n-2) for n > 1, a(0)=-2, a(1)=-1. - Philippe Deléham, Dec 23 2013
Sum_{n>=1} 1/a(n) = A331372. - Amiram Eldar, Nov 18 2020

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

A131061 Triangle read by rows: T(n,k) = 4*binomial(n,k) - 3 for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 9, 9, 1, 1, 13, 21, 13, 1, 1, 17, 37, 37, 17, 1, 1, 21, 57, 77, 57, 21, 1, 1, 25, 81, 137, 137, 81, 25, 1, 1, 29, 109, 221, 277, 221, 109, 29, 1, 1, 33, 141, 333, 501, 501, 333, 141, 33, 1, 1, 37, 177, 477, 837, 1005, 837, 477, 177, 37, 1

Offset: 0

Gary W. Adamson, Jun 13 2007

Row sums = A131062: (1, 2, 7, 20, 49, 110, 235, ...); the binomial transform of (1, 1, 4, 4, 4, ...).

Triangle equals 4*A007318 - 3*A000012 as infinite lower triangular matrices. - Emeric Deutsch, Jun 21 2007

			First few rows of the triangle are
  1;
  1,  1;
  1,  5,  1;
  1,  9,  9,  1;
  1, 13, 21, 13,  1;
  1, 17, 37, 37, 17,  1;
  1, 21, 57, 77, 57, 21, 1;
  ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A109128, A123203, A131060, A131063, A131064, A131065, A131066, A131067, A131068.

[4*Binomial(n, k) -3: k in [0..n], n in [0..10]]; // G. C. Greubel, Mar 12 2020

T := proc (n, k) if k <= n then 4*binomial(n, k)-3 else 0 end if end proc; for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form - Emeric Deutsch, Jun 21 2007

Table[4*Binomial[n, k] -3, {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 12 2020 *)

[[4*binomial(n, k) -3 for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 12 2020

G.f.:(1 - z - t*z + 4*t*z^2)/((1-z)*(1-t*z)*(1-z-t*z)). - Emeric Deutsch, Jun 21 2007

More terms from Emeric Deutsch, Jun 21 2007

A131065 Triangle read by rows: T(n,k) = 6*binomial(n,k) - 5 for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 13, 13, 1, 1, 19, 31, 19, 1, 1, 25, 55, 55, 25, 1, 1, 31, 85, 115, 85, 31, 1, 1, 37, 121, 205, 205, 121, 37, 1, 1, 43, 163, 331, 415, 331, 163, 43, 1, 1, 49, 211, 499, 751, 751, 499, 211, 49, 1, 1, 55, 265, 715, 1255, 1507, 1255, 715, 265, 55, 1

Offset: 0

Gary W. Adamson, Jun 13 2007

Row sums = A131066.
The matrix inverse starts:
       1;
      -1,       1;
       6,      -7,      1;
     -66,      78,    -13,      1;
    1086,   -1284,    216,    -19,   1;
  -23826,   28170,  -4740,    420, -25,   1;
  653406, -772536, 129990, -11520, 690, -31, 1; - R. J. Mathar, Mar 12 2013

			First few rows of the triangle are:
  1;
  1,  1;
  1,  7,  1;
  1, 13, 13,  1;
  1, 19, 31, 19,  1;
  1, 25, 55, 55, 25, 1;
...

Indranil Ghosh, Rows 0..120 of triangle, flattened

Cf. A109128, A123203, A131060, A131061, A131063, A131064, A131066, A131067, A131068.

[6*Binomial(n,k) -5: k in [0..n], n in [0..10]]; // G. C. Greubel, Mar 12 2020

T := proc (n, k) if k <= n then 6*binomial(n, k)-5 else 0 end if end proc: for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # Emeric Deutsch, Jun 20 2007

Table[6*Binomial[n,k]-5,{n,0,15},{k,0,n}]//Flatten (* Harvey P. Dale, May 15 2016 *)

[[6*binomial(n,k) -5 for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 12 2020

G.f.: (1-z-t*z+6*t*z^2)/((1-z)*(1-t*z)*(1-z-t*z)). - Emeric Deutsch, Jun 20 2007

More terms from Emeric Deutsch, Jun 20 2007

A131060 3A007318 - 2A000012 as infinite lower triangular matrices.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 7, 7, 1, 1, 10, 16, 10, 1, 1, 13, 28, 28, 13, 1, 1, 16, 43, 58, 43, 16, 1, 1, 19, 61, 103, 103, 61, 19, 1, 1, 22, 82, 166, 208, 166, 82, 22, 1, 1, 25, 106, 250, 376, 376, 250, 106, 25, 1, 1, 28, 133, 358, 628, 754, 628, 358, 133, 28, 1

Offset: 0

Gary W. Adamson, Jun 13 2007

Row sums = A097813: (1, 2, 6, 16, 38, 84, 178, ...).

			First few rows of the triangle:
  1;
  1,  1;
  1,  4,  1;
  1,  7,  7,  1;
  1, 10, 16, 10,  1;
  1, 13, 28, 28, 13,  1;
  1, 16, 43, 58, 43, 16,  1;
  ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A109128, A123203, A131061, A131063, A131064, A131065, A131066, A131067, A131068.

[3*Binomial(n,k) -2: k in [0..n], n in [0..10]]; // G. C. Greubel, Mar 12 2020

A131060:= (n,k) -> 3*binomial(n, k)-2; seq(seq(A131060(n, k), k = 0..n), n = 0.. 10); # G. C. Greubel, Mar 12 2020

T[n_, k_] = 3*Binomial[n, k] -2; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* Roger L. Bagula, Aug 20 2008 *)

[[3*binomial(n,k) -2 for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 12 2020

T(n,k) = 3*binomial(n,k) - 2. - Roger L. Bagula, Aug 20 2008

More terms from Roger L. Bagula, Aug 20 2008

A131063 Triangle read by rows: T(n,k) = 5*binomial(n,k) - 4 for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 11, 11, 1, 1, 16, 26, 16, 1, 1, 21, 46, 46, 21, 1, 1, 26, 71, 96, 71, 26, 1, 1, 31, 101, 171, 171, 101, 31, 1, 1, 36, 136, 276, 346, 276, 136, 36, 1, 1, 41, 176, 416, 626, 626, 416, 176, 41, 1, 1, 46, 221, 596, 1046, 1256, 1046, 596, 221, 46, 1

Offset: 0

Gary W. Adamson, Jun 13 2007

Row sums = A131064: (1, 2, 8, 24, 60, 136, 292, ...), the binomial transform of (1, 1, 5, 5, 5, ...).

			First few rows of the triangle:
  1;
  1,  1;
  1,  6,  1;
  1, 11, 11,  1;
  1, 16, 26, 16,  1;
  1, 21, 46, 46, 21,  1;
  1, 26, 71, 96, 71, 26,  1;
  ...

Muniru A Asiru, Rows n=0..100 of triangle, flattened

Cf. A109128, A123203, A131060, A131061, A131064, A131065, A131066, A131067, A131068.

Print(Flat(List([0..10],n->List([0..n],k->5*Binomial(n,k)-4)))); # Muniru A Asiru, Feb 21 2019

[5*Binomial(n, k) -4: k in [0..n], n in [0..10]]; // G. C. Greubel, Mar 12 2020

T := proc (n, k) if k <= n then 5*binomial(n, k)-4 else 0 end if end proc: for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # Emeric Deutsch, Jun 20 2007

Table[5*Binomial[n,k] -4, {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 12 2020 *)

[[5*binomial(n, k) -4 for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 12 2020

G.f.: (1-z-t*z+5*t*z^2)/((1-z)*(1-t*z)*(1-z-t*z)). - Emeric Deutsch, Jun 20 2007

More terms from Emeric Deutsch, Jun 20 2007

A131064 Binomial transform of [1, 1, 5, 5, 5, ...].

Original entry on oeis.org

1, 2, 8, 24, 60, 136, 292, 608, 1244, 2520, 5076, 10192, 20428, 40904, 81860, 163776, 327612, 655288, 1310644, 2621360, 5242796, 10485672, 20971428, 41942944, 83885980, 167772056, 335544212, 671088528, 1342177164, 2684354440

Offset: 0

Gary W. Adamson, Jun 13 2007

Row sums of triangle A131063. - Emeric Deutsch, Jun 20 2007

			a(3) = 24 = sum of row 4 terms of A131063: (1 + 11 + 11 + 1).
a(3) = 24 = (1, 3, 3, 1) dot (1, 1, 5, 5).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A109128, A123203, A131060, A131061, A131063, A131065, A131066, A131067, A131068.

Print(List([0..30],n->5*2^n-4*n-4)); # Muniru A Asiru, Feb 21 2019

I:=[1, 2, 8]; [n le 3 select I[n] else 4*Self(n-1)-5*Self(n-2) + 2*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jul 05 2012

a := proc (n) options operator, arrow; 5*2^n-4*n-4 end proc: seq(a(n), n = 0 .. 30); # Emeric Deutsch, Jun 20 2007

CoefficientList[Series[(1-2x+5x^2)/((1-2x)(1-x)^2),{x,0,40}],x] (* Vincenzo Librandi, Jul 05 2012 *)
LinearRecurrence[{4,-5,2},{1,2,8},30] (* Harvey P. Dale, Dec 29 2014 *)

[5*2^n -4*(n+1) for n in (0..30)] # G. C. Greubel, Mar 12 2020

From Emeric Deutsch, Jun 20 2007: (Start)
a(n) = 5*2^n - 4*(n + 1).
G.f.: (1-2*x+5*x^2)/((1-2*x)*(1-x)^2). (End)
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3). - Vincenzo Librandi, Jul 05 2012
E.g.f.: 5*exp(2*x) - 4*(1+x)*exp(x). - G. C. Greubel, Mar 12 2020

Corrected and extended by Emeric Deutsch, Jun 20 2007

A131066 Binomial transform of [1, 1, 6, 6, 6, ...].

Original entry on oeis.org

1, 2, 9, 28, 71, 162, 349, 728, 1491, 3022, 6089, 12228, 24511, 49082, 98229, 196528, 393131, 786342, 1572769, 3145628, 6291351, 12582802, 25165709, 50331528, 100663171, 201326462, 402653049, 805306228, 1610612591, 3221225322

Offset: 0

Gary W. Adamson, Jun 13 2007

nonn

Row sums of triangle A131065. - Emeric Deutsch, Jun 20 2007

			a(3) = 28 = sum of row 4 of triangle A131065: (1 + 13 + 13 + 1).
a(3) = 28 = (1, 3, 3, 1) dot (1, 1, 6, 6) = (1 + 3 + 18 + 6).

Muniru A Asiru, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A109128, A123203, A131060, A131061, A131063, A131064, A131065, A131067, A131068.

Print(List([0..30],n->6*2^n-5*n-5)); # Muniru A Asiru, Feb 21 2019

[6*2^n -5*(n+1): n in [0..30]]; // G. C. Greubel, Mar 12 2020

a := proc (n) options operator, arrow; 6*2^n-5*n-5 end proc: seq(a(n), n = 0 .. 30); # Emeric Deutsch, Jun 20 2007

Table[6*2^n -5*(n+1), {n,0,30}] (* G. C. Greubel, Mar 12 2020 *)

[6*2^n -5*(n+1) for n in (0..30)] # G. C. Greubel, Mar 12 2020

From Emeric Deutsch, Jun 20 2007: (Start)
a(n) = 6*2^n - 5*(n + 1).
G.f.: (1 - 2*x + 6*x^2)/((1-2*x)*(1-x)^2). (End)
E.g.f.: 6*exp(2*x) - 5*(1 + x)*exp(x). - G. C. Greubel, Mar 12 2020
a(n) = 2*a(n - 1) + 5*n - 5. - Kritsada Moomuang, Jul 03 2020
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3). - Wesley Ivan Hurt, Jul 10 2020

Corrected and extended by Emeric Deutsch, Jun 20 2007

cp's OEIS Frontend

A036563 a(n) = 2^n - 3.

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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).

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A131061 Triangle read by rows: T(n,k) = 4*binomial(n,k) - 3 for 0 <= k <= n.

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A131065 Triangle read by rows: T(n,k) = 6*binomial(n,k) - 5 for 0 <= k <= n.

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A131060 3*A007318 - 2*A000012 as infinite lower triangular matrices.

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A131063 Triangle read by rows: T(n,k) = 5*binomial(n,k) - 4 for 0 <= k <= n.

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

A131060 3A007318 - 2A000012 as infinite lower triangular matrices.