A090017 -id:A090017 - cp's OEIS frontend

A057088 Scaled Chebyshev U-polynomials evaluated at i*sqrt(5)/2. Generalized Fibonacci sequence.

1, 5, 30, 175, 1025, 6000, 35125, 205625, 1203750, 7046875, 41253125, 241500000, 1413765625, 8276328125, 48450468750, 283633984375, 1660422265625, 9720281250000, 56903517578125, 333118994140625, 1950112558593750, 11416157763671875, 66831351611328125, 391237546875000000

Offset: 0

Wolfdieter Lang, Aug 11 2000

a(n) gives the length of the word obtained after n steps with the substitution rule 0->11111, 1->111110, starting from 0. The number of 1's and 0's of this word is 5*a(n-1) and 5*a(n-2), resp.
a(n) / a(n-1) converges to (5 + (3 * sqrt(5))) / 2 as n approaches infinity. (5 + (3 * sqrt(5))) / 2 can also be written as phi^2 + (2 * phi), phi^3 + phi, phi + sqrt(5) + 2, (3 * phi) + 1, (3 * phi^2) - 2, phi^4 - 1 and (5 + (3 * (L(n) / F(n)))) / 2, where L(n) is the n-th Lucas number and F(n) is the n-th Fibonacci number as n approaches infinity. - Ross La Haye, Aug 18 2003, on another version
Pisano period lengths: 1, 3, 3, 6, 1, 3, 24, 12, 9, 3, 10, 6, 56, 24, 3, 24,288, 9, 18, 6, ... - R. J. Mathar, Aug 10 2012

Indranil Ghosh, Table of n, a(n) for n = 0..1300
Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=5, q=5.
Tanya Khovanova, Recursive Sequences
W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eqs.(39) and (45),rhs, m=5.
Eric Weisstein's World of Mathematics, Horadam Sequence
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (5,5)

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

I:=[1, 5]; [n le 2 select I[n] else 5*Self(n-1) + 5*Self(n-2): n in [0..30]]; // G. C. Greubel, Jan 16 2018

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=5*a[n-1]+5*a[n-2]od: seq(a[n], n=1..33); # Zerinvary Lajos, Dec 14 2008

LinearRecurrence[{5,5}, {1,5}, 30] (* G. C. Greubel, Jan 16 2018 *)

x='x+O('x^30); Vec(1/(1 - 5*x - 5*x^2)) \\ G. C. Greubel, Jan 16 2018

[lucas_number1(n,5,-5) for n in range(1, 22)] # Zerinvary Lajos, Apr 24 2009

a(n) = 5*(a(n-1) + a(n-2)), a(-1)=0, a(0)=1.
a(n) = S(n, i*sqrt(5))*(-i*sqrt(5))^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.
G.f.: 1/(1 - 5*x - 5*x^2).
a(n) = (1/3)*Sum_{k=0..n} binomial(n, k)*Fibonacci(k)*3^k. - Benoit Cloitre, Oct 25 2003
a(n) = ((5 + 3*sqrt(5))/2)^n(1/2 + sqrt(5)/6) + (1/2 - sqrt(5)/6)((5 - 3*sqrt(5))/2)^n. - Paul Barry, Sep 22 2004
(a(n)) appears to be given by the floretion - 0.75'i - 0.5'j + 'k - 0.75i' + 0.5j' + 0.5k' + 1.75'ii' - 1.25'jj' + 1.75'kk' - 'ij' - 0.5'ji' - 0.75'jk' - 0.75'kj' - 1.25e ("jes"). - Creighton Dement, Nov 28 2004
a(n) = Sum_{k=0..n} 4^k*A063967(n,k). - Philippe Deléham, Nov 03 2006
G.f.: G(0)/(2-5*x), where G(k)= 1 + 1/(1 - x*(9*k-5)/(x*(9*k+4) - 2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 17 2013
From Ehren Metcalfe, Nov 18 2017: (Start)
With F(n) = A000045(n), L(n) = A000032(n), beta = (1-sqrt(5))/2:
a(2*n-1) = 5^n*F(4*n)/3 = (5^(n-1/2)*L(4*n) - 2*5^(n-1/2)*beta^(4*n))/3.
a(2*n) = 5^n*L(4*n+2)/3 = (5^(n+1/2)*F(4*n+2) + 2*5^n*beta^(4*n+2))/3.
a(n) = round 5^((n+1)/2)*F(2*(n+1))/3.
a(n) = round 5^(n/2)*L(2*(n+1))/3. (End)

A214992 Power ceiling-floor sequence of (golden ratio)^4.

Original entry on oeis.org

7, 47, 323, 2213, 15169, 103969, 712615, 4884335, 33477731, 229459781, 1572740737, 10779725377, 73885336903, 506417632943, 3471038093699, 23790849022949, 163064905066945, 1117663486445665, 7660579500052711

Offset: 0

Clark Kimberling, Nov 08 2012, Jan 24 2013

Let f = floor and c = ceiling.  For x > 1, define four sequences as functions of x, as follows:
p1(0) = f(x), p1(n) = f(x*p1(n-1));
p2(0) = f(x), p2(n) = c(x*p2(n-1)) if n is odd and p2(n) = f(x*p1(n-1)) if n is even;
p3(0) = c(x), p3(n) = f(x*p3(n-1)) if n is odd and p3(n) = c(x*p3(n-1)) if n is even;
p4(0) = c(x), p4(n) = c(x*p4(n-1)).
The present sequence is given by a(n) = p3(n).
Following the terminology at A214986, call the four sequences power floor, power floor-ceiling, power ceiling-floor, and power ceiling sequences.  In the table below, a sequence is identified with an A-numbered sequence if they appear to agree except possibly for initial terms.  Notation: S(t)=sqrt(t), r = (1+S(5))/2 = golden ratio, and Limit = limit of p3(n)/p2(n).
x ......p1..... p2..... p3..... p4.......Limit
r.......A000045 A000045 A000045 A000045..r
r^2.....A001519 A001654 A061646 A001906..-1+S(5)
r^3.....A024551 A001076 A015448 A049652..-1+S(5)
r^4.....A049685 A157335 A214992 A004187..-19+9*S(5)
r^5.....A214993 A049666 A015457 A214994...(-9+5*S(5))/2
r^6.....A007805 A156085 A214995 A049660..-151+68*S(5)
1+S(2)..A024537 A000129 A001333 A048739..S(2)
2+S(2)..A007052 A214996 A214997 A007070..(1+S(2))/2
1+S(3)..A057960 A002605 A028859 A077846..(1+S(3))/2
2+S(3)..A001835 A109437 A214998 A001353..-4+3*S(3)
S(5)....A214999 A215091 A218982 A218983..1.26879683...
2+S(5)..A024551 A001076 A015448 A049652..-1+S(5)
2+S(6)..A218984 A090017 A123347 A218985..S(3/2)
2+S(7)..A218986 A015530 A126473 A218987..(1+S(7))/3
2+S(8)..A218988 A057087 A086347 A218989..(1+S(2))/2
3+S(8)..A001653 A084158 A218990 A001109..-13+10*S(2)
3+S(10).A218991 A005668 A015451 A218992..-2+S(10)
...
Properties of p1, p2, p3, p4:
(1) If x > 2, the terms of p2 and p3 interlace: p2(0) < p3(0) < p2(1) < p3(1) < p2(2) < p3(2)... Also, p1(n) <= p2(n) <= p3(n) <= p4(n) <= p1(n+1) for all x>0 and n>=0.
(2) If x > 2, the limits L(x) = limit(p/x^n) exist for the four functions p(x), and L1(x) <= L2(x) <= L3(x) <= L4 (x). See the Mathematica programs for plots of the four functions; one of them also occurs in the Odlyzko and Wilf article, along with a discussion of the special case x = 3/2.
(3) Suppose that x = u + sqrt(v) where v is a nonsquare positive integer.  If u = f(x) or u = c(x), then p1, p2, p3, p4 are linear recurrence sequences.  Is this true for sequences p1, p2, p3, p4 obtained from x = (u + sqrt(v))^q for every positive integer q?
(4) Suppose that x is a Pisot-Vijayaraghavan number. Must p1, p2, p3, p4 then be linearly recurrent?  If x is also a quadratic irrational b + c*sqrt(d), must the four limits L(x) be in the field Q(sqrt(d))?
(5) The Odlyzko and Wilf article (page 239) raises three interesting questions about the power ceiling function; it appears that they remain open.

			a(0) = ceiling(r) = 7, where r = ((1+sqrt(5))/2)^4 = 6.8...; a(1) = floor(7*r) = 47; a(2) = ceiling(47) = 323.

Clark Kimberling, Table of n, a(n) for n = 0..250
A. M. Odlyzko and H. S. Wilf, Functional iteration and the Josephus problem, Glasgow Math. J. 33, 235-240, 1991.
Index entries for linear recurrences with constant coefficients, signature (6,6,-1).
Index entries for sequences related to the Josephus Problem.

Cf. A001622, A214986, A049685, A157335, A004187.

(* Program 1.  A214992 and related sequences *)
x = GoldenRatio^4; z = 30; (* z = # terms in sequences *)
z1 = 100; (* z1 = # digits in approximations *)
f[x_] := Floor[x]; c[x_] := Ceiling[x];
p1[0] = f[x]; p2[0] = f[x]; p3[0] = c[x]; p4[0] = c[x];
p1[n_] := f[x*p1[n - 1]]
p2[n_] := If[Mod[n, 2] == 1, c[x*p2[n - 1]], f[x*p2[n - 1]]]
p3[n_] := If[Mod[n, 2] == 1, f[x*p3[n - 1]], c[x*p3[n - 1]]]
p4[n_] := c[x*p4[n - 1]]
Table[p1[n], {n, 0, z}]  (* A049685 *)
Table[p2[n], {n, 0, z}]  (* A157335 *)
Table[p3[n], {n, 0, z}]  (* A214992 *)
Table[p4[n], {n, 0, z}]  (* A004187 *)
Table[p4[n] - p1[n], {n, 0, z}]  (* A004187 *)
Table[p3[n] - p2[n], {n, 0, z}]  (* A098305 *)
(* Program 2.  Plot of power floor and power ceiling functions, p1(x) and p4(x) *)
f[x_] := f[x] = Floor[x]; c[x_] := c[x] = Ceiling[x];
p1[x_, 0] := f[x]; p1[x_, n_] := f[x*p1[x, n - 1]];
p4[x_, 0] := c[x]; p4[x_, n_] := c[x*p4[x, n - 1]];
Plot[Evaluate[{p1[x, 10]/x^10, p4[x, 10]/x^10}], {x, 2, 3}, PlotRange -> {0, 4}]
(* Program 3. Plot of power floor-ceiling and power ceiling-floor functions, p2(x) and p3(x) *)
f[x_] := f[x] = Floor[x]; c[x_] := c[x] = Ceiling[x];
p2[x_, 0] := f[x]; p3[x_, 0] := c[x];
p2[x_, n_] := If[Mod[n, 2] == 1, c[x*p2[x, n - 1]], f[x*p2[x, n - 1]]]
p3[x_, n_] := If[Mod[n, 2] == 1, f[x*p3[x, n - 1]], c[x*p3[x, n - 1]]]
Plot[Evaluate[{p2[x, 10]/x^10, p3[x, 10]/x^10}], {x, 2, 3}, PlotRange -> {0, 4}]

a(n) = floor(r*a(n-1)) if n is odd and a(n) = ceiling(r*a(n-1)) if n is even, where a(0) = ceiling(r), r = (golden ratio)^4 = (7 + sqrt(5))/2.
a(n) = 6*a(n-1) + 6*a(n-2) - a(n-3).
G.f.: (7 + 5*x - x^2)/((1 + x)*(1 - 7*x + x^2)).
a(n) = (10*(-2)^n+(10+3*sqrt(5))*(7-3*sqrt(5))^(n+2)+(10-3*sqrt(5))*(7+3*sqrt(5))^(n+2))/(90*2^n). - Bruno Berselli, Nov 14 2012
a(n) = 7*A157335(n) + 5*A157335(n-1) - A157335(n-2). - R. J. Mathar, Feb 05 2020
E.g.f.: exp(-x)*(5 + 2*exp(9*x/2)*(155*cosh(3*sqrt(5)*x/2) + 69*sqrt(5)*sinh(3*sqrt(5)*x/2)))/45. - Stefano Spezia, Oct 28 2024

A057089 Scaled Chebyshev U-polynomials evaluated at i*sqrt(6)/2. Generalized Fibonacci sequence.

Original entry on oeis.org

1, 6, 42, 288, 1980, 13608, 93528, 642816, 4418064, 30365280, 208700064, 1434392064, 9858552768, 67757668992, 465697330560, 3200729997312, 21998563967232, 151195763787264, 1039165966526976, 7142170381885440

Offset: 0

Wolfdieter Lang, Aug 11 2000

a(n) gives the length of the word obtained after n steps with the substitution rule 0->1^6, 1->(1^6)0, starting from 0. The number of 1's and 0's of this word is 6*a(n-1) and 6*a(n-2), resp.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=6, q=6.
Tanya Khovanova, Recursive Sequences
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eqs.(39) and (45),rhs, m=6.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (6,6).

Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015537, A015440, A015441, A015443, A015444, A015445, A015447, A015548, A030195, A053404, A057087, A057088, A083858, A085939, A090017, A091914, A099012, A135030, A135032, A180222, A180226, A180250.

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

Join[{a=0,b=1},Table[c=6*b+6*a;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 16 2011 *)
LinearRecurrence[{6,6},{1,6},40] (* Harvey P. Dale, Nov 05 2011 *)

x='x+O('x^30); Vec(1/(1-6*x-6*x^2)) \\ G. C. Greubel, Jan 24 2018

[lucas_number1(n,6,-6) for n in range(1, 21)] # Zerinvary Lajos, Apr 24 2009

a(n) = 6*a(n-1) + 6*a(n-2); a(0)=1, a(1)=6.
a(n) = S(n, i*sqrt(6))*(-i*sqrt(6))^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.
G.f.: 1/(1-6*x-6*x^2).
a(n) = Sum_{k=0..n} 5^k*A063967(n,k). - Philippe Deléham, Nov 03 2006

A015537 Expansion of x/(1 - 5x - 4x^2).

Original entry on oeis.org

0, 1, 5, 29, 165, 941, 5365, 30589, 174405, 994381, 5669525, 32325149, 184303845, 1050819821, 5991314485, 34159851709, 194764516485, 1110461989261, 6331368012245, 36098688018269, 205818912140325, 1173489312774701, 6690722212434805

Offset: 0

Olivier Gérard

First differences give A122690(n) = {1, 4, 24, 136, 776, 4424, 25224, ...}. Partial sums of a(n) are {0, 1, 6, 35, 200, ...} = (A123270(n) - 1)/8. - Alexander Adamchuk, Nov 03 2006
For n >= 2, a(n) equals the permanent of the (n-1) X (n-1) tridiagonal matrix with 5's along the main diagonal, and 2's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 19 2011
Pisano period lengths:  1, 1, 8, 1, 4, 8, 48, 1, 24, 4, 40, 8, 42, 48, 8, 2, 72, 24, 360, 4, ... - R. J. Mathar, Aug 10 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Lucyna Trojnar-Spelina and Iwona Włoch, On Generalized Pell and Pell-Lucas Numbers, Iranian Journal of Science and Technology, Transactions A: Science (2019), 1-7.
Index entries for linear recurrences with constant coefficients, signature (5,4).

Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015443, A015447, A030195, A053404, A057087, A083858, A085939, A090017, A091914, A099012, A122690, A123270, A180222, A180226.

a:=[0,1];; for n in [3..30] do a[n]:=5*a[n-1]+4*a[n-2]; od; a; # G. C. Greubel, Dec 26 2019

[n le 2 select n-1 else 5*Self(n-1)+4*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 12 2012

seq( simplify((2/I)^(n-1)*ChebyshevU(n-1, 5*I/4)), n=0..20); # G. C. Greubel, Dec 26 2019

LinearRecurrence[{5,4}, {0,1}, 30] (* Vincenzo Librandi, Nov 12 2012 *)
Table[2^(n-1)*Fibonacci[n, 5/2], {n, 0, 30}] (* G. C. Greubel, Dec 26 2019 *)

x='x+O('x^30); concat([0], Vec(x/(1-5*x-4*x^2))) \\ G. C. Greubel, Jan 01 2018

[lucas_number1(n,5,-4) for n in range(0, 22)] # Zerinvary Lajos, Apr 24 2009

a(n) = 5*a(n-1) + 4*a(n-2).
a(n) = Sum_{k=0..floor((n-1)/2)} C(n-k-1, k)*4^k*5^(n-2*k-1). - Paul Barry, Apr 23 2005
a(n) = Sum_{k=0..(n-1)} A122690(k). - Alexander Adamchuk, Nov 03 2006
a(n) = 2^(n-1)*Fibonacci(n, 5/2) = (2/i)^(n-1)*ChebyshevU(n-1, 5*i/4). - G. C. Greubel, Dec 26 2019

A057091 Scaled Chebyshev U-polynomials evaluated at i*sqrt(2). Generalized Fibonacci sequence.

Original entry on oeis.org

1, 8, 72, 640, 5696, 50688, 451072, 4014080, 35721216, 317882368, 2828828672, 25173688320, 224020135936, 1993550594048, 17740565839872, 157872931471360, 1404907978489856, 12502247279689728, 111257242065436672, 990075914761011200, 8810665254611582976

Offset: 0

Wolfdieter Lang, Aug 11 2000

a(n) gives the length of the word obtained after n steps with the substitution rule 0->1^8, 1->(1^8)0, starting from 0. The number of 1's and 0's of this word is 8*a(n-1) and 8*a(n-2), resp.

Colin Barker, Table of n, a(n) for n = 0..1000
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=8, q=8.
Tanya Khovanova, Recursive Sequences
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eqs.(39) and (45),rhs, m=8.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (8,8).

I:=[1,8]; [n le 2 select I[n] else 8*Self(n-1) + 8*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 24 2018

LinearRecurrence[{8,8}, {1,8}, 50] (* G. C. Greubel, Jan 24 2018 *)

Vec(1/(1-8*x-8*x^2) + O(x^30)) \\ Colin Barker, Jun 14 2015

[lucas_number1(n,8,-8) for n in range(0, 20)] # Zerinvary Lajos, Apr 25 2009

a(n) = 8*(a(n-1) + a(n-2)), a(-1)=0, a(0)=1.
a(n) = S(n, i*2*sqrt(2))*(-i*2*sqrt(2))^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.
G.f.: 1/(1 - 8*x - 8*x^2).
a(n) = Sum_{k=0..n} 7^k*A063967(n,k). - Philippe Deléham, Nov 03 2006
a(n) = 2^n*A090017(n+1). - R. J. Mathar, Mar 08 2021

A277767 T(n,k)=Number of nXk 0..2 arrays with every element equal to some element at offset (-1,-1) (-1,0) (-1,1) (0,-1) (0,1) or (1,0) both plus 1 mod 3 and minus 1 mod 3, with new values introduced in order 0..2.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 4, 2, 0, 0, 18, 17, 14, 0, 0, 80, 204, 330, 56, 0, 0, 356, 1989, 9741, 3666, 284, 0, 0, 1584, 21141, 275018, 270291, 46289, 1304, 0, 0, 7048, 220549, 7824415, 20049229, 8971150, 560809, 6248, 0, 0, 31360, 2292380, 221983169, 1487830718

Offset: 1

R. H. Hardin, Oct 29 2016

Table starts
.0......0..........0.............0.................0...................0
.0......1..........4............18................80.................356
.0......2.........17...........204..............1989...............21141
.0.....14........330..........9741............275018.............7824415
.0.....56.......3666........270291..........20049229..........1487830718
.0....284......46289.......8971150........1762881313........343944986355
.0...1304.....560809.....280603511......145416104585......74591651561541
.0...6248....6883464....8946059000....12253138042478...16513537201433122
.0..29408...84161576..283436060320..1025207978301185.3631417278822015869
.0.139472.1030163755.8998418743638.85977721285058269

			Some solutions for n=4 k=4
..0..1..2..0. .0..1..2..0. .0..1..2..0. .0..1..2..0. .0..1..2..0
..2..1..1..1. .2..0..0..1. .2..2..0..1. .2..0..1..1. .2..2..1..1
..2..0..2..0. .0..1..0..2. .1..0..0..1. .0..1..2..0. .1..0..2..0
..1..2..1..0. .2..2..1..2. .2..1..2..2. .1..2..1..1. .2..0..1..2

R. H. Hardin, Table of n, a(n) for n = 1..112

Row 2 is A090017(n-1).

Empirical for column k:
k=2: a(n) = 4*a(n-1) +6*a(n-2) -12*a(n-3)
k=3: [order 11]
k=4: [order 44] for n>45
Empirical for row n:
n=2: a(n) = 4*a(n-1) +2*a(n-2)
n=3: [order 18]
n=4: [order 98]

A180141 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + x - 2x^2)/(1 - 3x - 6*x^2).

Original entry on oeis.org

1, 4, 16, 72, 312, 1368, 5976, 26136, 114264, 499608, 2184408, 9550872, 41759064, 182582424, 798301656, 3490399512, 15261008472, 66725422488, 291742318296, 1275579489816, 5577192379224, 24385054076568, 106618316505048

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.
On a 3 X 3 chessboard there are 2^9 = 512 ways to go berserk on the central square (we assume here that a berserker might behave like a rook). The berserker is represented by the A[5] vector in the fifth row of the adjacency matrix A, see the Maple program. For the corner squares the 512 berserkers lead to 42 berserker sequences, see the cross-references for some examples.
The sequence above corresponds to just one A[5] vectors with decimal value 495. This vector leads for the side squares to 4*A154964 (for n >= 1 with a(0) = 1) and for the central square to 2*A180141 (for n >= 1 with a(0)=1).
This sequence belongs to a family of sequences with g.f. (1 + x + k*x^2)/(1 - 3*x + (k-4)*x^2), see A123620.

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

Cf. A180140 (side squares) and A180147 (central square).

Cf. Berserker sequences corner squares [numerical value A[5]]: 4*A055099 [0, with leading 1 added], A180143 [16], 4*A001353 [17, n>=1 and a(0)=1], A123620 [3], 2*A018916 [19, with leading 1 added], A000302 [15], 4*A179606 [111, with leading 1 added], A089979 [343], 4*A001076 [95, n>=1 and a(0)=1], A180145 [191], A180141 [495, this sequence], 4*A090017 [383, n>=1 and a(0)=1].

with(LinearAlgebra): nmax:=22; m:=1; A[5]:= [1,1,1,1,0,1,1,1,1]: 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);

LinearRecurrence[{3, 6}, {1, 4, 16}, 23] (* Jean-François Alcover, Jan 18 2025 *)

G.f.: (1 + x - 2*x^2)/(1 - 3*x - 6*x^2).
a(n) = 4*a(n-1) - 2*a(n-3) with a(0)=2, a(1)=8 and a(2)=31.
a(n) = 3*a(n-1) + 6*a(n-2) for n >= 3 with a(0)=1, a(1)=4 and a(2)=16.
a(n) = (6+2*A)*A^(-n-1)/33 + (6+2*B)*B^(-n-1)/33 with A=(-3-sqrt(33))/12 and B=(-3+sqrt(33))/12 for n >= 1 with a(0)=1.

A180226 a(n) = 4a(n-1) + 10a(n-2), with a(1)=0 and a(2)=1.

Original entry on oeis.org

0, 1, 4, 26, 144, 836, 4784, 27496, 157824, 906256, 5203264, 29875616, 171535104, 984896576, 5654937344, 32468715136, 186424233984, 1070384087296, 6145778689024, 35286955629056, 202605609406464, 1163291993916416, 6679224069730304, 38349816218085376

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 16 2011

Nathaniel Johnston, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (4, 10).

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

I:=[0,1]; [n le 2 select I[n] else 4*Self(n-1) + 10*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 16 2018

Join[{a=0,b=1},Table[c=4*b+10*a;a=b;b=c,{n,100}]]
LinearRecurrence[{4,10}, {0,1}, 30] (* G. C. Greubel, Jan 16 2018 *)

x='x+O('x^30); concat([0], Vec(x^2/(1-4*x-10*x^2))) \\ G. C. Greubel, Jan 16 2018

a(n) = ((2+sqrt(14))^(n-1) - (2-sqrt(14))^(n-1))/(2*sqrt(14)). - Rolf Pleisch, May 14 2011

G.f.: x^2/(1-4*x-10*x^2).

A099089 Riordan array (1, 2+x).

Original entry on oeis.org

1, 0, 2, 0, 1, 4, 0, 0, 4, 8, 0, 0, 1, 12, 16, 0, 0, 0, 6, 32, 32, 0, 0, 0, 1, 24, 80, 64, 0, 0, 0, 0, 8, 80, 192, 128, 0, 0, 0, 0, 1, 40, 240, 448, 256, 0, 0, 0, 0, 0, 10, 160, 672, 1024, 512, 0, 0, 0, 0, 0, 1, 60, 560, 1792, 2304, 1024, 0, 0, 0, 0, 0, 0, 12, 280, 1792, 4608, 5120, 2048

Offset: 0

Paul Barry, Sep 25 2004

Row sums are A000129. Diagonal sums are A008346. The Riordan array (1, s+tx) defines T(n,k) = binomial(k,n-k)*s^k*(t/s)^(n-k). The row sums satisfy a(n) = s*a(n-1) + t*a(n-2) and the diagonal sums satisfy a(n) = s*a(n-2) + t*a(n-3).
Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1/2, -1/2, 0, 0, 0, 0, ...] DELTA [2, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 10 2008
As an upper right triangle (in the example), table rows give number of points, edges, faces, cubes, 4D hypercubes etc. in hypercubes of increasing dimension by column. - Henry Bottomley, Apr 14 2000. More precisely, the (i,j)-th entry is the number of j-dimensional subspaces of an i-dimensional hypercube (see the Coxeter reference). - Christof Weber, May 08 2009

			Triangle begins:
  1;
  0,  2;
  0,  1,  4;
  0,  0,  4,  8;
  0,  0,  1, 12, 16;
  0,  0,  0,  6, 32, 32;
  0,  0,  0,  1, 24, 80, 64;
The entries can also be interpreted as the antidiagonal reading of the following array:
  1,    2,    4,    8,   16,   32,   64,  128,  256,  512, 1024,... A000079
  0,    1,    4,   12,   32,   80,  192,  448, 1024, 2304, 5120,... A001787
  0,    0,    1,    6,   24,   80,  240,  672, 1792, 4608,11520,... A001788
  0,    0,    0,    1,    8,   40,  160,  560, 1792, 5376,15360,... A001789
  0,    0,    0,    0,    1,   10,   60,  280, 1120, 4032,13440,...
  0,    0,    0,    0,    0,    1,   12,   84,  448, 2016, 8064,...
  0,    0,    0,    0,    0,    0,    1,   14,  112,  672, 3360,...
  0,    0,    0,    0,    0,    0,    0,    1,   16,  144,  960,...
  0,    0,    0,    0,    0,    0,    0,    0,    1,   18,  180,...
  0,    0,    0,    0,    0,    0,    0,    0,    0,    1,   20,...
  0,    0,    0,    0,    0,    0,    0,    0,    0,    0,    1,...

H. S. M. Coxeter, Regular Polytopes, Dover Publications, New York (1973), p. 122.

Eric W. Weisstein's Mathworld, Hypercube.

Cf. A053118, A008312, A062715, A038207.

Number triangle T(n,k) = binomial(k, n-k)*2^k*(1/2)^(n-k); columns have g.f. (2*x+x^2)^k.
G.f.: 1/(1-2y*x-y*x^2). - Philippe Deléham, Nov 20 2011
Sum_ {k=0..n} T(n,k)*x^k = A000007(n), A000129(n+1), A090017(n+1), A090018(n), A190510(n+1), A190955(n+1) for x = 0,1,2,3,4,5 respectively. - Philippe Deléham, Nov 20 2011
T(n,k) = 2*T(n-1,k-1) + T(n-2,k-1), T(0,0) = 1, T(1,0) = T(2,0) = 0, T(1,1) = 2, T(2,1) = 1, T(2,2) = 4, T(n,k) = 0 if k > n or if k < 0. - Philippe Deléham, Oct 30 2013

A123347 Number of words of length n over the alphabet {1,2,3,4,5} such that 1 is not followed by an odd letter.

Original entry on oeis.org

1, 5, 22, 98, 436, 1940, 8632, 38408, 170896, 760400, 3383392, 15054368, 66984256, 298045760, 1326151552, 5900697728, 26255094016, 116821771520, 519797274112, 2312832639488, 10290925106176, 45789365703680, 203739313027072, 906535983515648, 4033622560116736

Offset: 0

N. J. A. Sloane, Oct 10 2006

Appears to be Kekulé numbers for certain benzenoids (see the Cyvin-Gutman book for details).

			a(2) = 22 because all 25 words of length 2 are included except 11, 13 and 15.

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 78).

Andrew Howroyd, Table of n, a(n) for n = 0..500
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, example 17.
Index entries for linear recurrences with constant coefficients, signature (4, 2).

Cf. A138395.

I:=[1, 5]; [n le 2 select I[n] else 4*Self(n-1) + 2*Self(n-2): n in [1..30]]; // G. C. Greubel, Nov 29 2018

seq(coeff(series((1+x)/(1-4*x-2*x^2),x,n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Nov 27 2018

LinearRecurrence[{4, 2}, {1, 5}, 30] (* Amiram Eldar, Nov 26 2018 *)

Vec((1 + x)/(1 - 4*x - 2*x^2) + O(x^30)) \\ Andrew Howroyd, Nov 25 2018

s=((1+x)/(1-4*x-2*x^2)).series(x, 50); s.coefficients(x, sparse=False) # G. C. Greubel, Nov 29 2018

From Klaus Brockhaus, Oct 03 2009: (Start)
Inverse binomial transform of A138395.
a(n) = ((2+sqrt(6))^(n+1) + (2-sqrt(6))^(n+1))/4.
a(n) = 4*a(n-1) + 2*a(n-2) for n > 1.
G.f.: (1 + x)/(1 - 4*x - 2*x^2).
(End)
a(n) = A090017(n+1)+A090017(n). - R. J. Mathar, Aug 04 2019

Edited and new name by Armend Shabani and Andrew Howroyd, Nov 25 2018

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A180141 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + x - 2*x^2)/(1 - 3*x - 6*x^2).

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A180141 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + x - 2x^2)/(1 - 3x - 6*x^2).

A180226 a(n) = 4a(n-1) + 10a(n-2), with a(1)=0 and a(2)=1.