A072264 -id:A072264 - cp's OEIS frontend

A015523 a(n) = 3a(n-1) + 5a(n-2), with a(0)=0, a(1)=1.

0, 1, 3, 14, 57, 241, 1008, 4229, 17727, 74326, 311613, 1306469, 5477472, 22964761, 96281643, 403668734, 1692414417, 7095586921, 29748832848, 124724433149, 522917463687, 2192374556806, 9191710988853, 38537005750589

Offset: 0

Olivier Gérard

From Johannes W. Meijer, Aug 01 2010: (Start)
a(n) represents the number of n-move routes of a fairy chess piece starting in a given corner square (m = 1, 3, 7 and 9) on a 3 X 3 chessboard. This fairy chess piece behaves like a king on the eight side and corner squares but on the central square the king goes crazy and turns into a red king, see A179596.
For n >= 1, the sequence above corresponds to 24 red king vectors, i.e., A[5] vectors, with decimal values 27, 30, 51, 54, 57, 60, 90, 114, 120, 147, 150, 153, 156, 177, 180, 210, 216, 240, 282, 306, 312, 402, 408 and 432. These vectors lead for the side squares to A152187 and for the central square to A179606.
This sequence belongs to a family of sequences with g.f. 1/(1-3*x-k*x^2). Red king sequences that are members of this family are A007482 (k=2), A015521 (k=4), A015523 (k=5; this sequence), A083858 (k=6), A015524 (k=7) and A015525 (k=8). We observe that there is no red king sequence for k=3. Other members of this family are A049072 (k=-4), A057083 (k=-3), A000225 (k=-2), A001906 (k=-1), A000244 (k=0), A006190 (k=1), A030195 (k=3), A099012 (k=9), A015528 (k=10) and A015529 (k=11).
Inverse binomial transform of A052918 (with extra leading 0).
(End)
First differences in A197189. - Bruno Berselli, Oct 11 2011
Pisano period lengths:  1, 3, 4, 6, 4, 12, 3, 12, 12, 12, 120, 12, 12, 3, 4, 24, 288, 12, 72, 12, ... - R. J. Mathar, Aug 10 2012
This is the Lucas U(P=3, Q=-5) sequence, and hence for n >= 0, a(n+2)/a(n+1) equals the continued fraction 3 + 5/(3 + 5/(3 + 5/(3 + ... + 5/3))) with n 5's. - Greg Dresden, Oct 06 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,5).

Cf. A072263, A072264, A152187, A179606, A197189.

[ n eq 1 select 0 else n eq 2 select 1 else 3*Self(n-1)+5*Self(n-2): n in [1..30] ]; // Vincenzo Librandi, Aug 23 2011

Join[{a = 0, b = 1}, Table[c = 3 * b + 5 * a; a = b; b = c, {n, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 16 2011 *)
a[0] := 0; a[1] := 1; a[n_] := a[n] = 3a[n - 1] + 5a[n - 2]; Table[a[n], {n, 0, 49}] (* Alonso del Arte, Jan 16 2011 *)

x='x+O('x^30); concat([0], Vec(x/(1-3*x-5*x^2))) \\ G. C. Greubel, Jan 01 2018

[lucas_number1(n,3,-5) for n in range(0, 24)] # Zerinvary Lajos, Apr 22 2009

a(n) = 3*a(n-1) + 5*a(n-2).
From Paul Barry, Jul 20 2004: (Start)
a(n) = ((3/2 + sqrt(29)/2)^n - (3/2 - sqrt(29)/2)^n)/sqrt(29).
a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-k-1,k)*5^k*3^(n-2*k-1). (End)
G.f.: x/(1 - 3*x - 5*x^2). - R. J. Mathar, Nov 16 2007
From Johannes W. Meijer, Aug 01 2010: (Start)
Limit_{k->oo} a(n+k)/a(k) = (A072263(n) + a(n)*sqrt(29))/2.
Limit_{n->oo} A072263(n)/a(n) = sqrt(29). (End)
G.f.: G(0)*x/(2-3*x), where G(k) = 1 + 1/(1 - x*(29*k-9)/(x*(29*k+20) - 6/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 17 2013
E.g.f.: 2*exp(3*x/2)*sinh(sqrt(29)*x/2)/sqrt(29). - Stefano Spezia, Oct 06 2019

A179606 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 3x - 5x^2).

Original entry on oeis.org

1, 4, 17, 71, 298, 1249, 5237, 21956, 92053, 385939, 1618082, 6783941, 28442233, 119246404, 499950377, 2096083151, 8788001338, 36844419769, 154473265997, 647641896836, 2715292020493, 11384085545659, 47728716739442

Offset: 0

Johannes W. Meijer, Jul 28 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 king on the eight side and corner squares but on the central square the king goes crazy and turns into a red king, see A179596.
The sequence above corresponds to 24 red king vectors, i.e., A[5] vectors, with decimal values 27, 30, 51, 54, 57, 60, 90, 114, 120, 147, 150, 153, 156, 177, 180, 210, 216, 240, 282, 306, 312, 402, 408 and 432. These vectors lead for the corner squares to A015523 and for the side squares to A152187.
This sequence belongs to a family of sequences with g.f. (1 + (k-4)*x)/(1 - 3*x - k*x^2). Red king sequences that are members of this family are A007483 (k= 2), A015521 (k=4), A179606 (k=5; this sequence), A154964 (k=6), A179603 (k=7) and A179599 (k=8). We observe that there is no red king sequence for k=3. Other members of this family are A006190 (k=1), A133494 (k=0) and A168616 (k=-2).
Inverse binomial transform of A052918.
The sequence b(n+1) = 6*a(n), n >= 0 with b(0)=1, is a berserker sequence, see A180147. The b(n) sequence corresponds to 16 A[5] vectors with decimal values between 111 and 492. These vectors lead for the corner squares to sequence c(n+1)=4*A179606(n), n >= 0 with c(0)=1, and for the side squares to A180140. - Johannes W. Meijer, Aug 14 2010
Equals the INVERT transform of A063782: (1, 3, 10, 32, 104, ...). Example: a(3) = 71 = (1, 1, 4, 7) dot (32, 10, 3, 1) = (32 + 10 + 12 + 17). - Gary W. Adamson, Aug 14 2010

Indranil Ghosh, Table of n, a(n) for n = 0..1603
Index entries for linear recurrences with constant coefficients, signature (3,5).

Cf. A179597 (central square).

Cf. A072263, A072264, A152187, A197189.

with(LinearAlgebra): nmax:=22; m:=5; A[1]:= [0,1,0,1,1,0,0,0,0]: A[2]:= [1,0,1,1,1,1,0,0,0]: A[3]:= [0,1,0,0,1,1,0,0,0]: A[4]:= [1,1,0,0,1,0,1,1,0]: A[5]:= [0,0,0,1,1,1,0,0,1]: A[6]:= [0,1,1,0,1,0,0,1,1]: A[7]:= [0,0,0,1,1,0,0,1,0]: A[8]:= [0,0,0,1,1,1,1,0,1]: A[9]:= [0,0,0,0,1,1,0,1,0]: 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);

CoefficientList[Series[(1+x)/(1-3*x-5*x^2), {x, 0, 22}],x] (* or *) LinearRecurrence[{3,5,0},{1,4},23] (* Indranil Ghosh, Mar 05 2017 *)

print(Vec((1 + x)/(1- 3*x - 5*x^2) + O(x^23))); \\ Indranil Ghosh, Mar 05 2017

G.f.: (1+x)/(1 - 3*x - 5*x^2).
a(n) = A015523(n) + A015523(n+1).
a(n) = 3*a(n-1) + 5*a(n-2) with a(0) = 1 and a(1) = 4.
a(n) = ((29 + 7*sqrt(29))*A^(-n-1) + (29-7*sqrt(29))*B^(-n-1))/290 with A = (-3+sqrt(29))/10 and B = (-3-sqrt(29))/10
Limit_{k->oo} a(n+k)/a(k) = (-1)^(n+1)*A000351(n)*A130196(n)/(A015523(n)*sqrt(29) - A072263(n)) for n >= 1.

A152187 a(n) = 3a(n-1) + 5a(n-2), with a(0)=1, a(1)=5.

Original entry on oeis.org

1, 5, 20, 85, 355, 1490, 6245, 26185, 109780, 460265, 1929695, 8090410, 33919705, 142211165, 596232020, 2499751885, 10480415755, 43940006690, 184222098845, 772366329985, 3238209484180, 13576460102465, 56920427728295

Offset: 0

Philippe Deléham, Nov 28 2008

Unsigned version of A152185.
From Johannes W. Meijer, Aug 01 2010: (Start)
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 and 8) on a 3 X 3 chessboard. This fairy chess piece behaves like a king on the eight side and corner squares but on the central square the king goes crazy and turns into a red king, see A179596.
The sequence above corresponds to 24 red king vectors, i.e., A[5] vectors, with decimal values 27, 30, 51, 54, 57, 60, 90, 114, 120, 147, 150, 153, 156, 177, 180, 210, 216, 240, 282, 306, 312, 402, 408 and 432. These vectors lead for the corner squares to A015523 and for the central square to A179606.
This sequence belongs to a family of sequences with g.f. (1+2*x)/(1 - 3*x - k*x^2). Red king sequences that are members of this family are A007483 (k=2), A108981 (k=4), A152187 (k=5; this sequence), A154964 (k=6), A179602 (k=7) and A179598 (k=8). We observe that there is no red king sequence for k=3. Other members of this family are A036563 (k=-2), A054486 (k=-1), A084244 (k=0), A108300 (k=1) and A000351 (k=10).
Inverse binomial transform of A015449 (without the first leading 1).
(End)

Index entries for linear recurrences with constant coefficients, signature (3, 5).

Cf. A015523, A072263, A072264, A179606, A197189.

LinearRecurrence[{3,5},{1,5},40] (* Harvey P. Dale, May 03 2013 *)

G.f.: (1+2*x)/(1 - 3*x - 5*x^2).
Lim_{k->infinity} a(n+k)/a(k) = (A072263(n) + A015523(n)*sqrt(29))/2. - Johannes W. Meijer, Aug 01 2010
G.f.: G(0)*(1+2*x)/(2-3*x), where G(k) = 1 + 1/(1 - x*(29*k-9)/(x*(29*k+20) - 6/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 17 2013

A197189 a(n) = 3a(n-1) + 5a(n-2), with a(0)=1, a(1)=2.

Original entry on oeis.org

1, 2, 11, 43, 184, 767, 3221, 13498, 56599, 237287, 994856, 4171003, 17487289, 73316882, 307387091, 1288745683, 5403172504, 22653245927, 94975600301, 398193030538, 1669457093119, 6999336432047, 29345294761736, 123032566445443, 515824173145009, 2162635351662242

Offset: 0

Bruno Berselli, Oct 11 2011

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,5).

Cf. for type of recurrence: A015523, A072263, A072264, A152187, A179606 and also A180140.

[n le 2 select n else 3*Self(n-1)+5*Self(n-2): n in [1..26]];

a = {1, 2}; Do[AppendTo[a, 3 a[[-1]] + 5 a[[-2]]], {24}]; a (* Bruno Berselli, Dec 26 2012 *)

v=vector(26); v[1]=1; v[2]=2; for(i=3, #v, v[i]=3*v[i-1]+5*v[i-2]); v

G.f.: (1-x)/(1-3*x-5*x^2).
a(n) = ((29+sqrt(29))*(3+sqrt(29))^n+(29-sqrt(29))*(3-sqrt(29))^n)/(58*2^n).
a(n) = A015523(n+1)-A015523(n).
G.f.: G(0)*(1-x)/(2-3*x), where G(k)= 1 + 1/(1 - x*(29*k-9)/(x*(29*k+20) - 6/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 17 2013

A072263 a(n) = 3a(n-1) + 5a(n-2), with a(0)=2, a(1)=3.

Original entry on oeis.org

2, 3, 19, 72, 311, 1293, 5434, 22767, 95471, 400248, 1678099, 7035537, 29497106, 123669003, 518492539, 2173822632, 9113930591, 38210904933, 160202367754, 671661627927, 2815996722551, 11806298307288, 49498878534619, 207528127140297

Offset: 0

Miklos Kristof, Jul 08 2002

Inverse binomial transform of A087130. - Johannes W. Meijer, Aug 01 2010
Pisano period lengths: 1, 3, 4, 6, 4, 12, 3, 12, 12, 12, 120, 12, 12, 3, 4, 24, 288, 12, 72, 12... - R. J. Mathar, Aug 10 2012
This is the Lucas sequence V(3,-5). - Bruno Berselli, Jan 09 2013

			a(5)=5*b(4)+b(6): 1293=5*57+1008.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Wikipedia, Lucas sequence: Specific names.
Index entries for linear recurrences with constant coefficients, signature (3,5).

Cf. A072264, A152187, A197189.

Appears in A179606 and A015523. - Johannes W. Meijer, Aug 01 2010

a:=[2,3];; for n in [3..40] do a[n]:=3*a[n-1]+5*a[n-2]; od; a; # G. C. Greubel, Jan 14 2020

I:=[2,3]; [n le 2 select I[n] else 3*Self(n-1) +5*Self(n-2): n in [1..40]]; // G. C. Greubel, Jan 14 2020

seq(coeff(series((2-3*x)/(1-3*x-5*x^2), x, n+1), x, n), n = 0..40); # G. C. Greubel, Jan 14 2020

LinearRecurrence[{3,5},{2,3},40] (* Harvey P. Dale, Nov 23 2018 *)

my(x='x+O('x^40)); Vec((2-3*x)/(1-3*x-5*x^2)) \\ G. C. Greubel, Jan 14 2020

[lucas_number2(n,3,-5) for n in range(0, 16)] # Zerinvary Lajos, Apr 30 2009

a(n) = 2*A015523(n+1) - 3*A015523(n).
a(n) = ((3 + sqrt(29))/2)^n + ((3 - sqrt(29))/2)^n.
G.f.: (2-3*x)/(1-3*x-5*x^2). - R. J. Mathar, Feb 06 2010
From Johannes W. Meijer, Aug 01 2010: (Start)
Limit_{k -> Infinity} a(n+k)/a(k) = (A072263(n) + A015523(n)*sqrt(29))/2.
Limit_{n -> Infinity} (A072263(n)/A015523(n)) = sqrt(29). (End)
G.f.: G(0), where G(k)= 1 + 1/(1 - x*(29*k-9)/(x*(29*k+20) - 6/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 17 2013
a(n) = [x^n] ( (1 + 3*x + sqrt(1 + 6*x + 29*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 23 2015
a(n) = 5^((n-1)/2)*( 2*sqrt(5)*Fibonacci(n+1, 3/sqrt(5)) - 3*Fibonacci(n, 3/sqrt(5)) ). - G. C. Greubel, Jan 14 2020

Offset changed and terms added by Johannes W. Meijer, Jul 19 2010

A206800 Riordan array (1/(1-3x+x^2), x(1-x)/(1-3*x+x^2)).

Original entry on oeis.org

1, 3, 1, 8, 5, 1, 21, 19, 7, 1, 55, 65, 34, 9, 1, 144, 210, 141, 53, 11, 1, 377, 654, 534, 257, 76, 13, 1, 987, 1985, 1905, 1111, 421, 103, 15, 1, 2584, 5911, 6512, 4447, 2041, 641, 134, 17, 1, 6765, 17345, 21557, 16837, 9038, 3440, 925, 169, 19, 1

Offset: 0

Philippe Deléham, Feb 12 2012

			Triangle begins :
1
3, 1
8, 5, 1
21, 19, 7, 1
55, 65, 34, 9, 1
144, 210, 141, 53, 11, 1
377, 654, 534, 257, 76, 13, 1
987, 1985, 1905, 1111, 421, 103, 15, 1
2584, 5911, 6512, 4447, 2041, 641, 134, 17, 1
6765, 17345, 21557, 16837, 9038, 3440, 925, 169, 19, 1
Triangle (0,3,-1/3,1/3,0,0,0,0,0,...) DELTA (1,0,-1/3,1/3,0,0,0,0,...) begins :
1
0, 1
0, 3, 1
0, 8, 5, 1
0, 21, 19, 7, 1
0, 55, 65, 34, 9, 1...

Subtriangle of the triangle given by (0, 3, -1/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Antidiagonal sums are A072264(n).

Cf. A000045, A001519, A001906, A001870,

T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1).
G.f.: 1/(1-(y+3)*x+(y+1)*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^n* A015587(n+1), (-1)^n*A190953(n+1), (-1)^n*A015566(n+1), (-1)*A189800(n+1), (-1)^n*A015541(n+1), (-1)^n*A085939(n+1), (-1)^n*A015523(n+1), (-1)^n*A063727(n), (-1)^n*A006130(n), A077957(n), A000045(n+1), A000079(n), A001906(n+1), A007070(n), A116415(n), A084326(n+1), A190974(n+1), A190978(n+1), A190984(n+1), A190990(n+1), A190872(n) for x = -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively.

A111365 a(n) = 5a(n-1) + 3a(n-2) where a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, 8, 43, 239, 1324, 7337, 40657, 225296, 1248451, 6918143, 38336068, 212434769, 1177182049, 6523214552, 36147618907, 200307738191, 1109981547676, 6150830952953, 34084099407793, 188872989897824, 1046617247712499

Offset: 0

Parthasarathy Nambi, Nov 07 2005

nonn

			a(2) = 5*a(1) + 3*a(0) = 5*1 + 3*1 = 8 which is the third term in the sequence.

Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", Wiley, 2001

Index entries for linear recurrences with constant coefficients, signature (5, 3).

Cf. A000045, A072264.

Transpose[NestList[Flatten[{Rest[#],ListCorrelate[{3,5},#]}]&, {1,1},40]][[1]]  (* Harvey P. Dale, Mar 23 2011 *)

a(n)=A015536(n+1)-4*A015536(n). G.f.: (1-4x)/(1-5x-3x^2). [From R. J. Mathar, Jul 08 2009]

More terms from Robert G. Wilson v, Nov 10 2005

cp's OEIS Frontend

A015523 a(n) = 3*a(n-1) + 5*a(n-2), with a(0)=0, a(1)=1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A179606 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 3*x - 5*x^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A152187 a(n) = 3*a(n-1) + 5*a(n-2), with a(0)=1, a(1)=5.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A197189 a(n) = 3*a(n-1) + 5*a(n-2), with a(0)=1, a(1)=2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A072263 a(n) = 3*a(n-1) + 5*a(n-2), with a(0)=2, a(1)=3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A206800 Riordan array (1/(1-3*x+x^2), x*(1-x)/(1-3*x+x^2)).

Views

Author

Keywords

Examples

References

Crossrefs

Formula

A111365 a(n) = 5*a(n-1) + 3*a(n-2) where a(0) = a(1) = 1.

Views

Author

Keywords

Examples

References

Links

Crossrefs

A015523 a(n) = 3a(n-1) + 5a(n-2), with a(0)=0, a(1)=1.

A179606 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 3x - 5x^2).

A152187 a(n) = 3a(n-1) + 5a(n-2), with a(0)=1, a(1)=5.

A197189 a(n) = 3a(n-1) + 5a(n-2), with a(0)=1, a(1)=2.

A072263 a(n) = 3a(n-1) + 5a(n-2), with a(0)=2, a(1)=3.

A206800 Riordan array (1/(1-3x+x^2), x(1-x)/(1-3*x+x^2)).

A111365 a(n) = 5a(n-1) + 3a(n-2) where a(0) = a(1) = 1.