A007482 -id:A007482 - cp's OEIS frontend

A180142 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + x - x^2)/(1 - 3x - 3x^2).

1, 4, 14, 54, 204, 774, 2934, 11124, 42174, 159894, 606204, 2298294, 8713494, 33035364, 125246574, 474845814, 1800277164, 6825368934, 25876938294, 98106921684, 371951579934, 1410175504854, 5346381254364, 20269670277654, 76848154596054, 291353474621124

Offset: 0

Johannes W. Meijer, Aug 13 2010

The a(n) represent the number of n-move routes of a fairy chess piece starting in a given side square (m = 2, 4, 6 or 8) on a 3 X 3 chessboard. This fairy chess piece behaves like a rook on the eight side and corner squares but on the central square the rook goes berserk and turns into a berserker, see A180140.
The sequence above corresponds to 16 A[5] vectors with decimal values between 3 and 384. These vectors lead for the corner squares to A123620 and for the central square to A155116.
This sequence appears among the members of a family of sequences with g.f. (1 + x - k*x^2)/(1 - 3*x + (k-4)*x^2). Berserker sequences that are members of this family are 4*A007482 (k=2; with leading 1 added), A180142 (k=1; this sequence), A000302 (k=0), A180140 (k=-1) and 4*A154964 (k=-2; n>=1 and a(0)=1). Some other members of this family are 2*A180148 (k=3; with leading 1 added), 4*A025192 (k=4; with leading 1 added), 2*A005248 (k=5; with leading 1 added) and A123932 (k=6).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,3).

Cf. A180141 (corner squares), A180140 (side squares), A180147 (central square).

with(LinearAlgebra): nmax:=23; m:=2; A[5]:=[0,0,0,0,0,0,0,1,1]: A:= Matrix([[0,1,1,1,0,0,1,0,0], [1,0,1,0,1,0,0,1,0], [1,1,0,0,0,1,0,0,1], [1,0,0,0,1,1,1,0,0], A[5], [0,0,1,1,1,0,0,0,1], [1,0,0,1,0,0,0,1,1], [0,1,0,0,1,0,1,0,1], [0,0,1,0,0,1,1,1,0]]): 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);
# second Maple program:
a:= n-> ceil((<<0|1>, <3|3>>^n. <<2/3, 4>>)[1,1]):
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 14 2021

LinearRecurrence[{3, 3}, {1, 4, 14}, 26] (* Jean-François Alcover, Jan 18 2025 *)

G.f.: (1 + x - x^2)/(1 - 3*x - 3*x^2).
a(n) = 3*a(n-1) + 3*a(n-2) for n >= 2 with a(0)=1, a(1)=4 and a(2)=14.
a(n) = (6-2*A)*A^(-n-1)/21 + (6-2*B)*B^(-n-1)/21 with A=(-3+sqrt(21))/6 and B=(-3-sqrt(21))/6.
Lim_{k->infinity} a(2*n+k)/a(k) = 2*A000244(n)/(A003501(n) - A004254(n)*sqrt(21)) for n >= 1.
Lim_{k->infinity} a(2*n-1+k)/a(k) = 2*A000244(n)/(A004253(n)*sqrt(21) - 3*A030221(n-1)) for n >= 1.

A007455 Number of subsequences of [ 1,...,n ] in which each odd number has an even neighbor.

Original entry on oeis.org

1, 1, 3, 5, 11, 17, 39, 61, 139, 217, 495, 773, 1763, 2753, 6279, 9805, 22363, 34921, 79647, 124373, 283667, 442961, 1010295, 1577629, 3598219, 5618809, 12815247, 20011685, 45642179, 71272673, 162557031, 253841389, 578955451, 904069513

Offset: 0

N. J. A. Sloane, Mira Bernstein

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..400
R. K. Guy, William O. J. Moser, Numbers of subsequences without isolated odd members, Fibonacci Quarterly, 34, No. 2, 152-155 (1996).
Index entries for linear recurrences with constant coefficients, signature (0,3,0,2).

Cf. A007481, A007482, A007484.

a007455_list = 1 : 1 : 3 : 5 : zipWith (+)
   (map (* 2) a007455_list) (map (* 3) $ drop 2 a007455_list)
a007455 n = a007455_list !! n
-- Reinhard Zumkeller, Jul 16 2012

CoefficientList[Series[(-1-x-2 x^3)/(-1+3 x^2+2 x^4),{x,0,40}],x]  (* Harvey P. Dale, Feb 18 2011 *)
LinearRecurrence[{0,3,0,2},{1,1,3,5},40] (* Harvey P. Dale, Feb 10 2015 *)

A007455(n)=[n%2*2+3,1]*([3,1;2,0]^(n\2-1))[,1] \\ M. F. Hasler, Jun 19 2019

a(n) = 3*a(n-2) + 2*a(n-4).

G.f. = (1 + x + 2 x^3)/(1 - 3 x^2 - 2 x^4). - Harvey P. Dale, Feb 18 2011, edited by M. F. Hasler, Jun 19 2019

A083857 Square array T(n,k) of binomial transforms of generalized Fibonacci numbers, read by ascending antidiagonals, with n, k >= 0.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 0, 1, 3, 7, 0, 1, 3, 8, 15, 0, 1, 3, 9, 21, 31, 0, 1, 3, 10, 27, 55, 63, 0, 1, 3, 11, 33, 81, 144, 127, 0, 1, 3, 12, 39, 109, 243, 377, 255, 0, 1, 3, 13, 45, 139, 360, 729, 987, 511, 0, 1, 3, 14, 51, 171, 495, 1189, 2187, 2584, 1023, 0, 1, 3, 15, 57, 205, 648

Offset: 0

Paul Barry, May 06 2003

Row n >= 0 of the array gives the solution to the recurrence b(k) = 3*b(k-1) + (n-2) * a(k-2) for k >= 2 with a(0) = 0 and a(1) = 1. These are the binomial transforms of the rows of the generalized Fibonacci numbers A083856.

			Array T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
  0, 1, 3,  7, 15,  31,  63,  127,  255, ...
  0, 1, 3,  8, 21,  55, 144,  377,  987, ...
  0, 1, 3,  9, 27,  81, 243,  729, 2187, ...
  0, 1, 3, 10, 33, 109, 360, 1189, 3927, ...
  0, 1, 3, 11, 39, 139, 495, 1763, 6279, ...
  0, 1, 3, 12, 45, 171, 648, 2457, 9315, ...
  ...

OEIS, Transformations of integer sequences.

Rows include A000225 (n=0), A001906 (n=1), A000244 (n=2), A006190 (n=3), A007482 (n=4), A030195 (n=5), A015521 (n=6).

Cf. A083856, A083861 (binomial transform), A083862 (main diagonal).

T(n, k) = ((3 + sqrt(4*n + 1))/2)^k / sqrt(4*n + 1) - ((3 - sqrt(4*n + 1))/2)^k / sqrt(4*n + 1) for n, k >= 0.
O.g.f. of row n >= 0: -x/(-1 + 3*x + (n-2)*x^2) . - R. J. Mathar, Nov 23 2007
T(n,k) = Sum_{i = 0..k} binomial(k,i)*A083856(n,i). - Petros Hadjicostas, Dec 24 2019

Various sections edited by Petros Hadjicostas, Dec 24 2019

A180143 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + x^2)/(1 - 4x + x^2 + 2x^3).

Original entry on oeis.org

1, 4, 16, 58, 208, 742, 2644, 9418, 33544, 119470, 425500, 1515442, 5397328, 19222870, 68463268, 243835546, 868433176, 3092970622, 11015778220, 39233275906, 139731384160, 497660704294, 1772444881204, 6312656052202

Offset: 0

Johannes W. Meijer, Aug 13 2010

The a(n) represent 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 rook on the eight side and corner squares but on the central square the rook goes berserk and turns into a berserker, see A180140.

The sequence above corresponds to just one A[5] vectors with decimal value 16. This vector leads for the side squares to A180144 and for the central square to A000012.

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

Cf. A180141 (corner squares), A180140 (side squares), A180147 (central square).

with(LinearAlgebra): nmax:=23; m:=1; A[5]:=[0,0,0,0,1,0,0,0,0]: A:= Matrix([[0,1,1,1,0,0,1,0,0], [1,0,1,0,1,0,0,1,0], [1,1,0,0,0,1,0,0,1], [1,0,0,0,1,1,1,0,0], A[5], [0,0,1,1,1,0,0,0,1], [1,0,0,1,0,0,0,1,1], [0,1,0,0,1,0,1,0,1], [0,0,1,0,0,1,1,1,0]]): 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);

G.f.: (1+x^2)/(1 - 4*x + x^2 + 2*x^3).
a(n) = 4*a(n-1) - 1*a(n-2) - 2*a(n-3) with a(0)=1, a(1)=4 and a(2)=16.
a(n) = -1/2 + (9+12*A)*A^(-n-1)/34 + (9+12*B)*B^(-n-1)/34 with A=(-3+sqrt(17))/4 and B=(-3-sqrt(17))/4.
Lim_{k->infinity} a(n+k)/a(k) = (-1)^(n)*(2)^(n+1)/((2*A007482(n) - 3*A007482(n-1)) - A007482(n-1)*sqrt(17)) for n >= 1.

A189800 a(n) = 6a(n-1) + 8a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 6, 44, 312, 2224, 15840, 112832, 803712, 5724928, 40779264, 290475008, 2069084160, 14738305024, 104982503424, 747801460736, 5326668791808, 37942424436736, 270267896954880, 1925146777223168, 13713023838978048, 97679317251653632, 695780094221746176

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,8).

Sequences of the form a(n) = c*a(n-1) + d*a(n-2), with a(0)=0, a(1)=1:
c/d...1.......2.......3.......4.......5.......6.......7.......8.......9......10
.A000045,A001045,A006130,A006131,A015440,A015441,A015442,A015443,A015445,A015446
.A000129,A002605,A015518,A063727,A002532,A083099,A015519,A003683,A002534,A083102
.A006190,A007482,A030195,A015521,A015523,A083858,A015524,A015525,A099012,A015528
.A001076,A090017,A015530,A057087,A015531,A085939,A015532,A180222,A015533,A180226
.A052918,A015535,A015536,A015537,A057088,A015540,A015541,A015544,A015545,A180250
.A005668,A135030,A090018,A135032,A015551,A057089,A015552,A189800,A189801,A190005
.A054413,A015555,A015559,A015561,A015562,A015564,A057090,A015565,A015566,A015568
.A041025,A190331,A015574,A190510,A015575,A190560,A015576,A057091,A015577,A190953
.A099371,A015579,A181353,A015580,A015581,A153191,A015583,A015584,A057092,A015585
A041041,A191014,A015588,A190954,A190955,A190956,A015589,A190957,A015591,A057093

I:=[0,1]; [n le 2 select I[n] else 6*Self(n-1)+8*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 14 2011

LinearRecurrence[{6, 8}, {0, 1}, 50]
CoefficientList[Series[-(x/(-1+6 x+8 x^2)),{x,0,50}],x] (* Harvey P. Dale, Jul 26 2011 *)

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

G.f.: x/(1 - 2*x*(3+4*x)). - Harvey P. Dale, Jul 26 2011

A015541 Expansion of x/(1 - 5x - 7x^2).

Original entry on oeis.org

0, 1, 5, 32, 195, 1199, 7360, 45193, 277485, 1703776, 10461275, 64232807, 394392960, 2421594449, 14868722965, 91294775968, 560554940595, 3441838134751, 21133075257920, 129758243232857, 796722742969725, 4891921417478624, 30036666288181195

Offset: 0

Olivier Gérard

Pisano period lengths: 1, 3, 8, 6, 8, 24, 6, 6, 24, 24, 5, 24, 12, 6, 8, 12, 16, 24, 120, 24, ... - R. J. Mathar, Aug 10 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,7).

Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015537, A015443, A015447, A030195, A053404, A057087, A057088, A083858, A085939, A090017, A091914, A099012, A180222, A180226.

[n le 2 select n-1 else 5*Self(n-1) + 7*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 13 2012

Join[{a=0,b=1},Table[c=5*b+7*a;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 16 2011 *)
LinearRecurrence[{5, 7}, {0, 1}, 30] (* Vincenzo Librandi, Nov 13 2012 *)

x='x+O('x^30); concat([0], Vec(x/(1-5*x-7*x^2))) \\ G. C. Greubel, Jan 24 2018

[lucas_number1(n,5,-7) for n in range(0, 21)] # Zerinvary Lajos, Apr 24 2009

a(n) = 5*a(n-1) + 7*a(n-2).

A015544 Lucas sequence U(5,-8): a(n+1) = 5a(n) + 8a(n-1), a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 5, 33, 205, 1289, 8085, 50737, 318365, 1997721, 12535525, 78659393, 493581165, 3097180969, 19434554165, 121950218577, 765227526205, 4801739379641, 30130517107845, 189066500576353, 1186376639744525, 7444415203333449, 46713089134623445

Offset: 0

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,8).

Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015537, A015441, A015443, A015447, A030195, A053404, A057087, A057088, A083858, A085939, A090017, A091914, A099012, A180222, A180226, A015555 (binomial transform).

[n le 2 select n-1 else 5*Self(n-1) + 8*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 13 2012

a[n_]:=(MatrixPower[{{1,2},{1,-6}},n].{{1},{1}})[[2,1]]; Table[Abs[a[n]],{n,-1,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{5, 8}, {0, 1}, 30] (* Vincenzo Librandi, Nov 13 2012 *)

A015544(n)=imag((2+quadgen(57))^n) \\ M. F. Hasler, Mar 06 2009

x='x+O('x^30); concat([0], Vec(x/(1 - 5*x - 8*x^2))) \\ G. C. Greubel, Jan 01 2018

[lucas_number1(n,5,-8) for n in range(0, 21)] # Zerinvary Lajos, Apr 24 2009

a(n) = 5*a(n-1) + 8*a(n-2).

G.f.: x/(1 - 5*x - 8*x^2). - M. F. Hasler, Mar 06 2009

More precise definition by M. F. Hasler, Mar 06 2009

A112906 A skew generalized Pascal triangle.

Original entry on oeis.org

1, 0, 3, 0, 1, 10, 0, 0, 6, 33, 0, 0, 1, 29, 109, 0, 0, 0, 9, 126, 360, 0, 0, 0, 1, 57, 516, 1189, 0, 0, 0, 0, 12, 306, 2034, 3927, 0, 0, 0, 0, 1, 94, 1491, 7807, 12970, 0, 0, 0, 0, 0, 15, 600, 6813, 29382, 42837, 0, 0, 0, 0, 0, 1, 140, 3385, 29737, 108923, 141481, 0, 0, 0, 0, 0, 0

Offset: 0

Paul Barry, Oct 05 2005

Main diagonal is A006190. Row sums are A007482. Column sums are A001076(n+1). Compare with [0,1/3,-1/3,0,0,..] DELTA [3,1/3,-1/3,0,0,...] where DELTA is the operator defined in A084938. A skewed version of the Riordan array (1/(1-3x-x^2),x/(1-3x-x^2)).

			Triangle begins
1;
0, 3;
0, 1, 10;
0, 0, 6, 33;
0, 0, 1, 29, 109;
0, 0, 0, 9, 126, 360,
0, 0, 0, 1, 57, 516, 1189;
0, 0, 0, 0, 12, 306, 2034, 3927;
0, 0, 0, 0, 1, 94, 1491, 7809, 12970;

Cf. A112899.

G.f.: 1/(1-3xy(1+x/3)-x^2*y^2); T(n, k)=sum{j=0..floor((2k-n)/2), C(k-j, n-k)C(2k-n, j)3^(2k-2j-n)}; T(n, k) = 3*T(n-1, k-1)+T(n-2, k-1)+T(n-2, k-2).

A152268 Expansion of g.f. x/(1-7x+8x^2).

Original entry on oeis.org

0, 1, 7, 41, 231, 1289, 7175, 39913, 221991, 1234633, 6866503, 38188457, 212387175, 1181202569, 6569320583, 36535623529, 203194800039, 1130078612041, 6284991883975, 34954314291497, 194400264968679, 1081167340448777

Offset: 0

Roger L. Bagula, Dec 01 2008

Binomial transform of 0, 1, 5, 23, 105, ... (A107839 with an additional initial term 0) and second binomial transform of 0, 1, 3, 11, 39, ... (A007482 with an additional initial term 0). - Klaus Purath, Sep 09 2024

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

[lucas_number1(n,7,8) for n in range(0, 22)] # Zerinvary Lajos, Apr 23 2009

From R. J. Mathar, Dec 04 2008: (Start)
a(n) = 7*a(n-1) - 8*a(n-2).
G.f.: x/(1-7*x+8*x^2). (End)

A368156 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 2x, p(n,x) = up(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 + x^2.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 3, 10, 14, 12, 5, 20, 41, 44, 29, 8, 40, 98, 148, 131, 70, 13, 76, 224, 408, 497, 376, 169, 21, 142, 482, 1044, 1542, 1588, 1052, 408, 34, 260, 1003, 2492, 4351, 5456, 4894, 2888, 985, 55, 470, 2026, 5684, 11359, 16790, 18400, 14672, 7813

Offset: 1

Clark Kimberling, Jan 20 2024

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

			First eight rows:
   1
   1    2
   2    4    5
   3   10   14    12
   5   20   41    44    29
   8   40   98   148   131    70
  13   76  224   408   497   376   169
  21  142  482  1044  1542  1588  1052  408
Row 4 represents the polynomial p(4,x) = 3 + 10*x + 14*x^2 + 12*x^3, so (T(4,k)) = (3,10,14,12), k=0..3.

Rigoberto Flórez, Robinson A. Higuita, and Antara Mikherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers 18 (2018) 1-28.

Cf. A000045 (column 1); A000129, (p(n,n-1)); A007482 (row sums), (p(n,1)); A077925 (alternating row sums), (p(n,-1)); A057088, (p(n,2)); A015523, (p(n,-2)); A015568, (p(n,3)); A180250, (p(n,-3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150, A368151, A368152, A368153, A368154, A368155.

p[1, x_] := 1; p[2, x_] := 1 + 2 x; u[x_] := p[2, x]; v[x_] := 1 + 2x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 1 + 2*x, u = p(2,x), and v = 1 + x^2.

p(n,x) = k*(b^n - c^n), where k = -1/sqrt(5 + 4*x + 8*x^2), b = (1/2)*(2*x + 1 - 1/k), c = (1/2)*(2*x + 1 + 1/k).

cp's OEIS Frontend

A180142 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + x - x^2)/(1 - 3*x - 3*x^2).

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A007455 Number of subsequences of [ 1,...,n ] in which each odd number has an even neighbor.

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A083857 Square array T(n,k) of binomial transforms of generalized Fibonacci numbers, read by ascending antidiagonals, with n, k >= 0.

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A180143 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + x^2)/(1 - 4*x + x^2 + 2*x^3).

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A189800 a(n) = 6*a(n-1) + 8*a(n-2), with a(0)=0, a(1)=1.

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A015541 Expansion of x/(1 - 5*x - 7*x^2).

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A015544 Lucas sequence U(5,-8): a(n+1) = 5*a(n) + 8*a(n-1), a(0)=0, a(1)=1.

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A180142 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + x - x^2)/(1 - 3x - 3x^2).

A180143 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + x^2)/(1 - 4x + x^2 + 2x^3).

A189800 a(n) = 6a(n-1) + 8a(n-2), with a(0)=0, a(1)=1.

A015541 Expansion of x/(1 - 5x - 7x^2).

A015544 Lucas sequence U(5,-8): a(n+1) = 5a(n) + 8a(n-1), a(0)=0, a(1)=1.

A152268 Expansion of g.f. x/(1-7x+8x^2).

A368156 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 2x, p(n,x) = up(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 + x^2.