A004253 -id:A004253 - cp's OEIS frontend

A110307 Expansion of (1+2x)/((1+x+x^2)(1+5*x+x^2)).

1, -4, 17, -80, 384, -1842, 8827, -42292, 202631, -970862, 4651680, -22287540, 106786021, -511642564, 2451426797, -11745491420, 56276030304, -269634660102, 1291897270207, -6189851690932, 29657361184451, -142096954231322, 680827409972160, -3262040095629480

Offset: 0

Creighton Dement, Jul 19 2005

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-6,-7,-6,-1).

Cf. A002320, A004253, A049347, A099837, A110308, A110309, A110310, A110311.

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+2*x)/((1+x+x^2)*(1+5*x+x^2)) )); // G. C. Greubel, Jan 03 2023

seriestolist(series((1+2*x)/((x^2+x+1)*(x^2+5*x+1)), x=0,25));

LinearRecurrence[{-6,-7,-6,-1}, {1,-4,17,-80}, 41] (* G. C. Greubel, Jan 03 2023 *)

Vec((1+2*x)/((1+x+x^2)*(1+5*x+x^2)) + O(x^25)) \\ Colin Barker, Apr 30 2019

def U(n,x): return chebyshev_U(n, x)
def A110307(n): return (1/4)*(3*U(n,-5/2) +U(n-1,-5/2) +U(n,-1/2) -U(n-1,-1/2))
[A110307(n) for n in range(41)] # G. C. Greubel, Jan 03 2023

a(n+2) = - 5*a(n+1) - a(n) - A099837(n+1).
a(n) + a(n+1) + a(n+2) = A002320(n).
a(n) = -6*a(n-1) - 7*a(n-2) - 6*a(n-3) - a(n-4) for n>3. - Colin Barker, Apr 30 2019
a(n) = (1/4)*(3*U(n,-5/2) + U(n-1,-5/2) + U(n,-1/2) - U(n-1,-1/2)), where U(n, x) = ChebyshevU(n, x). - G. C. Greubel, Jan 03 2023

A110308 Expansion of -x(2+x)/((1+x+x^2)(1+5*x+x^2)).

Original entry on oeis.org

0, -2, 11, -52, 247, -1182, 5664, -27140, 130037, -623044, 2985181, -14302860, 68529120, -328342742, 1573184591, -7537580212, 36114716467, -173036002122, 829065294144, -3972290468600, 19032387048857, -91189644775684, 436915836829561, -2093389539372120

Offset: 0

Creighton Dement, Jul 19 2005

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-6,-7,-6,-1).

Cf. A004253, A049347, A099837, A110307, A110309, A110310, A110311.

R:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( -x*(2+x)/((1+x+x^2)*(1+5*x+x^2)) )); // G. C. Greubel, Jan 03 2023

seriestolist(series(-x*(2+x)/((x^2+x+1)*(x^2+5*x+1)), x=0,25));

LinearRecurrence[{-6,-7,-6,-1}, {0,-2,11,-52}, 40] (* G. C. Greubel, Jan 03 2023 *)

concat(0, Vec(-x*(2+x)/((1+x+x^2)*(1+5*x+x^2)) + O(x^25))) \\ Colin Barker, Apr 30 2019

def U(n, x): return chebyshev_U(n,x)
def A110308(n): return (1/4)*(2*U(n, -5/2) +U(n-1, -5/2) -2*U(n, -1/2) -U(n-1, -1/2))
[A110308(n) for n in range(41)] # G. C. Greubel, Jan 03 2023

a(n+2) = - 5*a(n+1) - a(n) - A099837(n+2).
a(n) = -6*a(n-1) - 7*a(n-2) - 6*a(n-3) - a(n-4) for n>3. - Colin Barker, Apr 30 2019
a(n) = (1/4)*(2*U(n, -5/2) + U(n-1, -5/2) - 2*U(n, -1/2) - U(n-1, -1/2)), where U(n, x) = ChebyshevU(n, x). - G. C. Greubel, Jan 03 2023

A110309 Expansion of (1+3x+x^2)/((1+x+x^2)(1+5*x+x^2)).

Original entry on oeis.org

1, -3, 12, -57, 275, -1320, 6325, -30303, 145188, -695637, 3332999, -15969360, 76513801, -366599643, 1756484412, -8415822417, 40322627675, -193197315960, 925663952125, -4435122444663, 21249948271188, -101814618911277, 487823146285199, -2337301112514720

Offset: 0

Creighton Dement, Jul 19 2005

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-6,-7,-6,-1).

Cf. A004253, A049347, A109265, A110307, A110308, A110310, A110311.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+3*x+x^2)/((1+x+x^2)*(1+5*x+x^2)) )); // G. C. Greubel, Jan 03 2023

seriestolist(series((1+3*x+x^2)/((x^2+5*x+1)*(x^2+x+1)), x=0,25));

LinearRecurrence[{-6,-7,-6,-1}, {1,-3,12,-57}, 40] (* G. C. Greubel, Jan 03 2023 *)

Vec((1+3*x+x^2)/((1+x+x^2)*(1+5*x+x^2)) + O(x^25)) \\ Colin Barker, Apr 30 2019

def A110309(n): return (1/2)*(chebyshev_U(n,-5/2)+chebyshev_U(n,-1/2))
[A110309(n) for n in range(41)] # G. C. Greubel, Jan 03 2023

a(n+2) = - 5*a(n+1) - a(n) + (-1)^n*A109265(n+3).
a(n) = -6*a(n-1) - 7*a(n-2) - 6*a(n-3) - a(n-4) for n>3. - Colin Barker, Apr 30 2019
a(n) = (1/2)*(ChebyshevU(n, -5/2) + ChebyshevU(n, -1/2)). - G. C. Greubel, Jan 03 2023

A110310 Expansion of (1-x+x^2)/((x^2+x+1)(x^2+5x+1)).

Original entry on oeis.org

1, -7, 36, -173, 827, -3960, 18973, -90907, 435564, -2086913, 9998999, -47908080, 229541401, -1099798927, 5269453236, -25247467253, 120967883027, -579591947880, 2776991856373, -13305367333987, 63749844813564, -305443856733833, 1463469438855599, -7011903337544160

Offset: 0

Creighton Dement, Jul 19 2005

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-6,-7,-6,-1).

Cf. A004253, A049347, A109265, A110307, A110308, A110309, A110311.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x+x^2)/((1+x+x^2)*(1+5*x+x^2)) )); // G. C. Greubel, Jan 02 2023

seriestolist(series((1-x+x^2)/((x^2+x+1)*(x^2+5*x+1)), x=0,25));

LinearRecurrence[{-6,-7,-6,-1}, {1,-7,36,-173}, 40] (* G. C. Greubel, Jan 02 2023 *)

Vec((1-x+x^2)/((1+x+x^2)*(1+5*x+x^2)) + O(x^25)) \\ Colin Barker, Apr 30 2019

def U(n,x): return chebyshev_U(n,x)
def A110310(n): return (1/2)*(3*U(n, -5/2) - U(n, -1/2))
[A110310(n) for n in range(41)] # G. C. Greubel, Jan 02 2023

a(n+2) = - 5*a(n+1) - a(n) - (-1)^n*A109265(n+3).
a(n) = -6*a(n-1) - 7*a(n-2) - 6*a(n-3) - a(n-4) for n>3. - Colin Barker, Apr 30 2019
a(n) = (1/2)*(3*ChevyshevU(n, -5/2) - ChebyshevU(n, -1/2)). - G. C. Greubel, Jan 02 2023

A110311 Expansion of 1/((1+x+x^2)(1+5x+x^2)).

Original entry on oeis.org

1, -6, 29, -138, 660, -3162, 15151, -72594, 347819, -1666500, 7984680, -38256900, 183299821, -878242206, 4207911209, -20161313838, 96598657980, -462831976062, 2217561222331, -10624974135594, 50907309455639, -243911573142600, 1168650556257360, -5599341208144200

Offset: 0

Creighton Dement, Jul 19 2005

In reference to the program code, A004254(n+1) = 1ibaseiseq[A*B](n).

Superseeker finds: a(n) + a(n+1) + a(n+2) = (-1)^n*A004254(n+3).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-6,-7,-6,-1).

Cf. A004253, A004254, A110307, A110308, A110309, A110310.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/((1+x+x^2)*(1+5*x+x^2)) )); // G. C. Greubel, Jan 02 2023

seriestolist(series(1/((x^2+5*x+1)*(x^2+x+1)), x=0,25));

LinearRecurrence[{-6,-7,-6,-1}, {1,-6,29,-138}, 40] (* G. C. Greubel, Jan 02 2023 *)

Vec(1/((1+x+x^2)*(1+5*x+x^2)) + O(x^25)) \\ Colin Barker, May 14 2019

def U(n,x): return chebyshev_U(n,x)
def A110311(n): return (1/4)*(5*U(n, -5/2) + U(n-1, -5/2) - U(n, -1/2) - U(n-1, -1/2))
[A110311(n) for n in range(41)] # G. C. Greubel, Jan 02 2023

a(n+2) = - 5*a(n+1) - a(n) + ((-1)^n)*A109265(n+1)/2.
a(n) = -6*a(n-1) - 7*a(n-2) - 6*a(n-3) - a(n-4) for n>3. - Colin Barker, May 14 2019
a(n) = (1/4)*(5*U(n, -5/2) + U(n-1, -5/2) - U(n, -1/2) - U(n-1, -1/2)), where U(n, x) = ChebyshevU(n, x). - G. C. Greubel, Jan 02 2023

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

Original entry on oeis.org

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.

A160695 Integers m such that 3m+1 and 7m+1 are both perfect squares.

Original entry on oeis.org

0, 5, 120, 2760, 63365, 1454640, 33393360, 766592645, 17598237480, 403992869400, 9274237758725, 212903475581280, 4887505700610720, 112199727638465285, 2575706229984090840, 59129043561995624040, 1357392295695915262085, 31160893757444055403920

Offset: 1

Paul Weisenhorn, May 24 2009

The ansatz 3*a(n)+1=A^2, 7*a(n)+1=B^2 is equivalent to the Pell equation x^2-21*y^2=1 (see A077232 for d=21), with x=(21*a(n)+5)/2 and y=A*B/2.
The associated A are in A004253, the B in A030221.
Bisection of A089927. - R. J. Mathar, Jul 10 2009

Michael De Vlieger, Table of n, a(n) for n = 1..736
Francesca Arici and Jens Kaad, Gysin sequences and SU(2)-symmetries of C*-algebras, arXiv:2012.11186 [math.OA], 2020.
Index entries for linear recurrences with constant coefficients, signature (24,-24,1).

Cf. A004253, A004254, A030221, A089927, A160682, A245031, A334673.

j:=0: for n from 0 to 1000000 do a:=sqrt(3*n+1): b:=sqrt(7*n+1):
if (trunc(a)=a) and (trunc(b)=b) then j:=j+1: print(j,n,a,b): end if:
end do:

LinearRecurrence[{24,-24,1},{0,5,120},30] (* Harvey P. Dale, Dec 17 2013 *)

a(n) = 24*a(n-1) - 24*a(n-2) + a(n-3).
a(n) = (A004253(n)^2 - 1)/3 = (A030221(n)^2 - 1)/7.
a(n) = ((5+w)/2*((23+5*w)/2)^(n-1) + (5-w)/2*((23-5*w)/2)^(n-1) - 5)/21; where w=sqrt(21). [Corrected by Kevin Ryde, Sep 11 2020]
G.f.: 5*x^2/((1-x)*(x^2-23*x+1)). - R. J. Mathar, Jul 10 2009
From Francesca Arici, Sep 12 2020: (Start)
a(n) = 23*a(n-1) - a(n-2) + 5.
a(n) = A004254(n)* A004254(n+1). (End)
a(n) = 5*A334673(n-1). - Hugo Pfoertner, Apr 07 2021

Edited and extended by R. J. Mathar, Jul 10 2009

Name edited by Michel Marcus, Sep 12 2020

A237254 Values of x in the solutions to x^2 - 5xy + y^2 + 5 = 0, where 0 < x < y.

Original entry on oeis.org

1, 2, 3, 9, 14, 43, 67, 206, 321, 987, 1538, 4729, 7369, 22658, 35307, 108561, 169166, 520147, 810523, 2492174, 3883449, 11940723, 18606722, 57211441, 89150161, 274116482, 427144083, 1313370969, 2046570254, 6292738363, 9805707187, 30150320846, 46981965681

Offset: 1

Colin Barker, Feb 05 2014

The corresponding values of y are given by a(n+2).

Also the solutions to 21x^2-20 is a perfect square. - Jaimal Ichharam, Jul 13 2014

			9 is in the sequence because (x, y) = (9, 43) is a solution to x^2 - 5xy + y^2 + 5 = 0.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-1).

Cf. A004253, A237255.

A237254 := proc(n)
    coeftayl( -x*(x-1)*(x^2+3*x+1) / (x^4-5*x^2+1), x=0, n);
end proc:
seq(A237254(n), n=1..40); # Wesley Ivan Hurt, Jul 14 2014

Rest[CoefficientList[Series[- x (x - 1) (x^2 + 3 x + 1)/(x^4 - 5 x^2 + 1), {x, 0, 40}], x]] (* Vincenzo Librandi, Jul 01 2014 *)
LinearRecurrence[{0,5,0,-1},{1,2,3,9},40] (* Harvey P. Dale, Aug 24 2024 *)

Vec(-x*(x-1)*(x^2+3*x+1)/(x^4-5*x^2+1) + O(x^100))

a(n) = 5*a(n-2)-a(n-4).

G.f.: -x*(x-1)*(x^2+3*x+1) / (x^4-5*x^2+1).

A125078 Fifth in an infinite set of generalized Pascal's triangles, with trigonometric properties.

Original entry on oeis.org

1, 1, 4, 1, 5, 19, 1, 9, 24, 91, 1, 10, 63, 115, 436, 1, 14, 73, 397, 551, 2089, 1, 15, 132, 470, 2358, 2640, 10009, 1, 19, 147, 1043, 2828, 13482, 12649, 47956, 1, 20, 226, 1190, 7441, 16310, 75061, 60605, 229771

Offset: 1

Gary W. Adamson, Nov 18 2006

The triangle is the fifth in an infinite set of generalized Pascal's triangles constrained by two properties: row sums = powers of N and upward sloping diagonals solve for N + 2*Cos 2Pi/Q. Row sums are powers of 5. Right border (1, 4, 19, 91, 436...) = A004253. Next to right border (1, 5, 24, 115...) = A004254.

			First few rows of the triangle are:
1;
1, 4;
1, 5, 19;
1, 9, 24, 91;
1, 10, 63, 115, 436;
1, 14, 73, 397, 551, 2089;
1, 15, 132, 470, 2358, 2640, 10009;
...
The upward sloping diagonal (1, 14, 63, 91) is derived from the characteristic polynomial x^3 - 14x^2 + 63x - 91 and relates to the Heptagon (Q=7) since a root = 6.24697960...= 5 + 2*Cos 2Pi/7. The corresponding matrix is [4, 1, 0; 1, 5, 1; 0, 1, 5]. The next upward sloping diagonal (1, 15, 73, 115) relates to the Octagon (Q=8) since a root = 6.41421356... = 5 + 2*Cos 2Pi/8. The corresponding matrix is [5, 1, 0; 1, 5, 1; 0, 1, 5].

Cf. A125076, A125078, A004253, A004254.

Upward sloping diagonals are derived from interleaved characteristic polynomials of two types of matrices, relating to odd and even polygons. Matrices with an eigenvalue 5 + 2*Cos 2Pi/Q, Q is odd, are of the form: all 1's in the super and subdiagonals and 4,5,5,5... in the main diagonal. Matrices (Q is even) are of the form: all 1's in the super and subdiagonals and 5,5,5... in the main diagonal.

A136210 Numerators in continued fraction [0; 1, 3, 1, 3, 1, 3, ...].

Original entry on oeis.org

1, 3, 4, 15, 19, 72, 91, 345, 436, 1653, 2089, 7920, 10009, 37947, 47956, 181815, 229771, 871128, 1100899, 4173825, 5274724, 19997997, 25272721, 95816160, 121088881, 459082803, 580171684, 2199597855, 2779769539, 10538906472

Offset: 1

Gary W. Adamson, Dec 21 2007

A136210(n)/A136211(n) tends to 0.7912878474... = (sqrt(21) - 3)/2 = continued fraction [0; 1, 3, 1, 3, 1, 3, ...] = the inradius of a right triangle with hypotenuse 5, legs 2 and sqrt(21).

This is a strong divisibility sequence, that is, GCD(a(n),a(m)) = a(GCD(n,m)) for all natural numbers n and m. - Peter Bala, May 14 2014

			a(4) = 15 = 3*a(3) + a(2) = 3*4 + 3.
a(5) = 19 = a(4) + a(3) = 15 + 4.
T^3 = [19, 72; 24, 91], where [19, 72] = [a(5), a(6)]. [24, 91] = [A136211(5), A136211(6)].
G.f. = x + 3*x^2 + 4*x^3 + 15*x^4 + 19*x^5 + 72*x^6 + 91*x^7 + 345*x^8 + ...

J. L. Ramirez, F. Sirvent, A q-Analogue of the Bi-Periodic Fibonacci Sequence, J. Int. Seq. 19 (2016) # 16.4.6, t_n at a=3, b=1.
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-1).

Cf. A004253, A004254, A136211.

a = {1, 3}; Do[If[EvenQ[n], AppendTo[a, 3*a[[ -1]] + a[[ -2]]], AppendTo[a, a[[ -1]] + a[[ -2]]]], {n, 3, 30}]; a (* Stefan Steinerberger, Dec 31 2007 *)
a[n_] := FromContinuedFraction[ Join[{0}, 3 - 2*Array[Mod[#, 2]&, n]]] // Numerator; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, May 15 2014 *)

{a(n) = (-1)^((n+1) * (n<0)) * polcoeff( x * (1 + 3*x - x^2) / (1 - 5*x^2 + x^4) + x * O(x^abs(n)), abs(n))}; /* Michael Somos, May 15 2014 */

a(0) = 0, a(1) = 1, a(2n) = 3*a(2n-1) + a(2n-2); a(2n-1) = a(2n-2) + a(2n-3). Given the 2 X 2 matrix [1, 3; 1, 4] = T, [a(2n-1), a(2n)] = top row of T^n.
g.f.: x*(1+3*x-x^2)/(1-5*x^2+x^4). - Colin Barker, Jan 04 2012
a(-n) = -(-1)^n * a(n). a(2*n - 1) = A004253(n). a(2*n) = 3 * A004254(n). - Michael Somos, May 15 2014
a(n+1) - a(n-1) = a(n) * (2 - (-1)^n) for all n in Z. - Michael Somos, May 15 2014

More terms from Stefan Steinerberger, Dec 31 2007

cp's OEIS Frontend

A110307 Expansion of (1+2*x)/((1+x+x^2)*(1+5*x+x^2)).

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A110308 Expansion of -x*(2+x)/((1+x+x^2)*(1+5*x+x^2)).

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A110309 Expansion of (1+3*x+x^2)/((1+x+x^2)*(1+5*x+x^2)).

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A110310 Expansion of (1-x+x^2)/((x^2+x+1)*(x^2+5*x+1)).

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A110311 Expansion of 1/((1+x+x^2)*(1+5*x+x^2)).

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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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A160695 Integers m such that 3*m+1 and 7*m+1 are both perfect squares.

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A110307 Expansion of (1+2x)/((1+x+x^2)(1+5*x+x^2)).

A110308 Expansion of -x(2+x)/((1+x+x^2)(1+5*x+x^2)).

A110309 Expansion of (1+3x+x^2)/((1+x+x^2)(1+5*x+x^2)).

A110310 Expansion of (1-x+x^2)/((x^2+x+1)(x^2+5x+1)).

A110311 Expansion of 1/((1+x+x^2)(1+5x+x^2)).

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

A160695 Integers m such that 3m+1 and 7m+1 are both perfect squares.