A030186 -id:A030186 - cp's OEIS frontend

A268630 a(n)^2 + a(n+1) is prime; lexicographically earliest sequence of nonnegative integers with this property and containing no duplicates.

0, 2, 1, 4, 3, 8, 7, 10, 9, 16, 13, 12, 5, 6, 11, 18, 23, 28, 25, 22, 15, 14, 27, 32, 37, 30, 19, 36, 31, 48, 29, 40, 21, 20, 33, 34, 45, 38, 39, 46, 63, 44, 43, 24, 17, 42, 47, 58, 49, 66, 35, 52, 73, 64, 57, 50, 51, 56, 55, 54, 41, 60, 59, 76, 67, 72, 53, 70, 69, 26, 75, 68, 79, 82, 99, 86, 61, 100, 91, 88, 85, 84, 65, 102, 83, 78, 89, 90, 71, 106, 81

Offset: 0

Eric Angelini and M. F. Hasler, Feb 09 2016

nonn

Conjectured to be a permutation of the nonnegative integers.
Terms are of alternating parity.
The sequence cannot have a fixed point other than a(0)=0 because for n>0, the terms are of parity opposite to that of their indices.
The number of distinct m-digit primes arising from the sequence appears to be bounded by the entries of A030186. The counts here for m=1 to 9 are 2,7,21,69,216,684,2162,6801,21623 compared to A030186's 2,7,22,71,228,733,2356,7573,24342. - Bill McEachen, Feb 15 2016

Zak Seidov, Table of n, a(n) for n = 0..50000
E. Angelini, A formula for a permutation, SeqFan list, Feb. 9, 2016.

Cf. A268494, A268495, A268496, A268497 for records and late birds.

s = {0, 2, 1, 4}; a = 4; Do[b = Mod[a, 2] + 3; While[MemberQ[s, b] || ! PrimeQ[a^2 + b], b = b + 2]; AppendTo[s, b]; a = b, {1000}]; s (* Zak Seidov, Feb 09 2016 *)

{u=[a=0]; for(n=1, 99, for(k=1, 9e9, setsearch(u,k)&&next; isprime(a*a+k)||next; print1(k","); u=setunion(u,[a=k]); break))}

A088132 a(n) equals the square of the n-th partial sum added to twice the n-th partial sum of the squares, divided by a(n-1), for all n>1, with a(0)=1, a(1)=3.

Original entry on oeis.org

1, 3, 12, 47, 185, 728, 2865, 11275, 44372, 174623, 687217, 2704496, 10643361, 41886227, 164840412, 648718287, 2552986921, 10047107272, 39539710801, 155605856283, 612376317860, 2409965560639, 9484256386273, 37324649227232

Offset: 0

Paul D. Hanna, Sep 19 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (4,0,-1).

Cf. A030186, A088131.

a:=[1,3,12];; for n in [4..30] do a[n]:=34a[n-1]-a[n-3]; od; a; # G. C. Greubel, Oct 26 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)/(1-4*x+x^3) )); // G. C. Greubel, Oct 26 2019

seq(coeff(series((1-x)/(1-4*x+x^3), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 26 2019

LinearRecurrence[{4,0,-1}, {1,3,12}, 30] (* or *) CoefficientList[Series[ (1-x)/(1-4x+x^3), {x,0,30}], x] (* Harvey P. Dale, Jun 24 2011 *)

{a(n)=if(n==0,1, if(n==1,3, (sum(k=0, n-1, a(k))^2 + 2*sum(k=0, n-1, a(k)^2))/a(n-1)))}
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Feb 20 2014

Vec( (1-x)/(1-4*x+x^3) + O(x^66) ) \\ Joerg Arndt, Feb 16 2014

def A088132_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-x)/(1-4*x+x^3)).list()
A088132_list(30) # G. C. Greubel, Oct 26 2019

a(n) = 4*a(n-1) - a(n-3) for n>3.
G.f.: (1-x)/(1-4*x+x^3).
G.f.: 1/(x - x^2*Sum_{n>=0} A030186(n)*x^n) - 1/x.

A353878 Number of tilings of a 3 X n rectangle using right trominoes, dominoes and 1 X 1 tiles.

Original entry on oeis.org

1, 3, 44, 369, 3633, 34002, 323293, 3058623, 28982628, 274494621, 2600148629, 24628666626, 233286962601, 2209723174731, 20930806288252, 198259418947833, 1877940242218857, 17788105074906162, 168491350295593637, 1595972975308532199, 15117273008425964916

Offset: 0

Gerhard Kirchner, May 09 2022

Tiling algorithm see A351322.

			a(2)=44
The number of tilings (mirroring included) using r trominoes
      ___   ___        ___
r=1: |  _| | |_| r=2: |  _| r=0: 22 = A030186(3)
     |_|3| |___|      |_| |
     |___| |_2_|      |___|
      4*3 + 4*2   +    2*1   +   22 = 44
Legend:
   ___              ___      ___
  |_2_| stands for |___| or |_|_|
     _                _        _        _
   _|3|             _| |     _|_|     _|_|
  |___| stands for |_|_| or |___| or |_|_|

Index entries for linear recurrences with constant coefficients, signature (6,33,3,-40,15).

Cf. A030186, A127867, A165716, A351322, A352589, A353877, A353879.

Maxima
```
See A352589.
```

G.f.: (1-3*x-7*x^2+3*x^3-2*x^4) / (1-6*x-33*x^2-3*x^3+40*x^4-15*x^5).

a(n) = 6*a(n-1) + 33*a(n-2) + 3*a(n-3) - 40*a(n-4) + 15*a(n-5).

A030236 Cycle-path coverings of a family of digraphs.

Original entry on oeis.org

1, 2, 7, 18, 49, 136, 377, 1044, 2891, 8006, 22171, 61398, 170029, 470860, 1303949, 3611016, 9999959, 27692810, 76689487, 212375610, 588130153, 1628704336, 4510358465, 12490501212, 34589849507, 95789405774, 265268869027

Offset: 0

Ottavio D'Antona (dantona(AT)dsi.unimi.it) and Emanuele Munarini

G. C. Greubel, Table of n, a(n) for n = 0..1000
O. M. D'Antona and E. Munarini, The Cycle-path Indicator Polynomial of a Digraph, Advances in Applied Mathematics 25 (2000), 41-56.
Index entries for linear recurrences with constant coefficients, signature (3,-1,1).

Cf. A030186.

a:=[2,7,18];; for n in [4..40] do a[n]:=3*a[n-1]-a[n-2]+a[n-3]; od; Concatenation([1],a); # G. C. Greubel, Oct 27 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x+2*x^2-2*x^3)/(1-3*x+x^2-x^3) )); // G. C. Greubel, Oct 27 2019

seq(coeff(series((1-x+2*x^2-2*x^3)/(1-3*x+x^2-x^3), x, n+1), x, n), n = 0 .. 40); # G. C. Greubel, Oct 27 2019

LinearRecurrence[{3,-1,1}, {1,2,7,18}, 40] (* G. C. Greubel, Oct 27 2019 *)

makelist(sum(binomial(n+k+1,3*k+1)*2^k, k,0,n) + 2*sum(2^k* binomial(n+k-1,3*k+1), k,0,n-1), n,0,60); /* Emanuele Munarini, Dec 03 2012 */

my(x='x+O('x^40)); Vec((1-x+2*x^2-2*x^3)/(1-3*x+x^2-x^3)) \\ G. C. Greubel, Oct 27 2019

def A030236_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x+2*x^2-2*x^3)/(1-3*x+x^2-x^3) ).list()
A030236_list(40) # G. C. Greubel, Oct 27 2019

a(n+4) = 3*a(n+3) - a(n+2) + a(n+1), n >= 0.
a(n+3) = 2*a(n+2) + a(n+1) + 2*Sum_{k=0..n} a(k), n >= 0.
G.f.: (1-x+2*x^2-2*x^3)/(1-3*x+x^2-x^3).
a(n) = Sum_{k=0..n} binomial(n+k+1,3*k+1)*2^k + 2*Sum_{j=0..n-1}  binomial(n+j-1,3*j+1)*2^j. - Emanuele Munarini, Dec 03 2012

A088016 To obtain a(n+1), add the square of the n-th partial sum to the n-th partial sum of the squares, then divide this result by a(n), for all n >= 0, with a(0)=1.

Original entry on oeis.org

1, 1, 6, 17, 56, 179, 576, 1851, 5950, 19125, 61474, 197597, 635140, 2041543, 6562172, 21092919, 67799386, 217928905, 700493182, 2251609065, 7237391472, 23263290299, 74775653304, 240352858739, 772570939222, 2483290023101

Offset: 0

Paul D. Hanna, Sep 18 2003

nonn

			G.f.: A(x) = 1 + x + 6*x^2 + 17*x^3 + 56*x^4 + 179*x^5 + 576*x^6 + ...
where A(x) * (1 - 3*x - x^2 + x^3) = 1 - 2*x + 2*x^2 - x^3.
Illustration of the initial terms: set a(0) = a(1) = 1, then
a(2) = ((1+1)^2 + (1^2 + 1^2))/1 = 6;
a(3) = ((1+1+6)^2 + (1^2 + 1^2 + 6^2))/6 = 17;
a(4) = ((1+1+6+17)^2 + (1^2 + 1^2 + 6^2 + 17^2))/17 = 56;
a(5) = ((1+1+6+17+56)^2 + (1^2 + 1^2 + 6^2 + 17^2 + 56^2))/56 = 179; ...

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,1,-1).

Cf. A030186, A087640.

a:=[1,6,17];; for n in [4..40] do a[n]:=3*a[n-1]+a[n-2]-a[n-3]; od; Concatenation([1], a); # G. C. Greubel, Oct 27 2019

I:=[1,6,17]; [1] cat [n le 3 select I[n] else 3*Self(n-1) +Self(n-2) -Self(n-3): n in [1..30]]; // G. C. Greubel, Oct 27 2019

seq(coeff(series((1-2*x+2*x^2-x^3)/(1-3*x-x^2+x^3), x, n+1), x, n), n = 0 .. 40); # G. C. Greubel, Oct 27 2019

LinearRecurrence[{3,1,-1},{1,1,6,17},40] (* Harvey P. Dale, Nov 06 2012 *)

a(n)=(sum(k=0,n-1,a(k))^2+sum(k=0,n-1,a(k)^2))/a(n-1)

my(x='x+O('x^40)); Vec((1-2*x+2*x^2-x^3)/(1-3*x-x^2+x^3)) \\ G. C. Greubel, Oct 27 2019

def A088016_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-2*x+2*x^2-x^3)/(1-3*x-x^2+x^3)).list()
A088016_list(40) # G. C. Greubel, Oct 27 2019

a(n) = 3*a(n-1) + a(n-2) - a(n-3) for n>3.
G.f.: (1-2*x+2*x^2-x^3) / (1-3*x-x^2+x^3).
G.f.: A(x) = A030186(x) * (1-x+x^2), where A030186(x) = gf of A030186.

A125693 Riordan array ((1-x)/(1-3x), x(1-x)/(1-3*x)).

Original entry on oeis.org

1, 2, 1, 6, 4, 1, 18, 16, 6, 1, 54, 60, 30, 8, 1, 162, 216, 134, 48, 10, 1, 486, 756, 558, 248, 70, 12, 1, 1458, 2592, 2214, 1168, 410, 96, 14, 1, 4374, 8748, 8478, 5160, 2150, 628, 126, 16, 1, 13122, 29160, 31590, 21744, 10442, 3624, 910, 160, 18, 1

Offset: 0

Paul Barry, Nov 30 2006

Row sums are A001835(n+1). Diagonal sums are A030186. Inverse is A125694. Equal to product of A007318 and A073370.

			Triangle begins
    1;
    2,   1;
    6,   4,   1;
   18,  16,   6,  1;
   54,  60,  30,  8,  1;
  162, 216, 134, 48, 10, 1;

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

Flat(List([0..10], n-> List([0..n], k-> Sum([0..n], j->
(-1)^j*3^(n-k-j)*Binomial(k+1,j)*Binomial(n-j, n-k-j) )))); # G. C. Greubel, Oct 28 2019

T:= func< n,k | &+[(-1)^j*3^(n-k-j)*Binomial(k+1,j)*Binomial(n-j, n-k-j): j in [0..n]] >;
[T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Oct 28 2019

seq(seq( add( (-1)^j*3^(n-k-j)*binomial(k+1,j)*binomial(n-j, n-k-j), j=0..n), k=0..n), n=0..10); # G. C. Greubel, Oct 28 2019

T[0, 0]=1; T[1, 0]=2; T[1, 1]=1; T[n_, k_]/; 0<=k<=n:= T[n, k]= 3T[n-1, k] + T[n-1, k-1] - T[n-2, k-1]; T[, ]=0; Table[T[n, k], {n, 0, 9}, {k, 0, n}] (* Jean-François Alcover, Jun 13 2019 *)
T[n_, k_]:= Sum[(-1)^j*3^(n-k-j)*Binomial[k+1,j]*Binomial[n-j,n-k-j], {j, 0, n}]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 28 2019 *)

T(n,k) = sum(j=0,n, (-1)^j*3^(n-k-j)*binomial(k+1,j)*binomial(n-j, n-k-j));
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Oct 28 2019

[[sum( (-1)^j*3^(n-k-j)*binomial(k+1,j)*binomial(n-j, n-k-j) for j in (0..n) ) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Oct 28 2019

Number triangle T(n,k) = Sum_{j=0..k+1} C(k+1,j)*C(n-j,n-k-j)* (-1)^j * 3^(n-k-j).

T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - T(n-2,k-1), T(0,0)=1, T(1,0)=2, T(1,1)=1, T(n,k)=0 if k>n or if kPhilippe Deléham, Jan 08 2013

A159616 Expansion of (1-x)/(1-5x-2x^2+8*x^3).

Original entry on oeis.org

1, 4, 22, 110, 562, 2854, 14514, 73782, 375106, 1906982, 9694866, 49287446, 250571106, 1273871494, 6476200114, 32924174710, 167382301826, 850950257638, 4326122494162, 21993454571478, 111811915784610, 568437508112710

Offset: 0

R. J. Mathar, Apr 17 2009

Number of tilings of a 2 X n board with squares of 1 color and dominoes of 2 colors if n > 2. The number of tilings is 3 if n=1, and 17 if n=2.

a(n) = element(1,2) in A^n, where A is the 7 X 7 matrix defined by A(1,i) = A(7,i) = A(i,1) = A(i,7) = A(i,i) = A(i,7-i+1) = 1, and A(i,j) = 0 otherwise. - Lechoslaw Ratajczak, Jan 02 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. Katz, C. Stenson, Tiling a 2xn-board with squares and dominoes, J. Int. Seq. 12 (2009) # 09.2.2.
Index entries for linear recurrences with constant coefficients, signature (5,2,-8).

Cf. A030186, A102436, A159617.

a:=[1,4,22];; for n in [4..40] do a[n]:=5*a[n-1]+2*a[n-2]-8*a[n-3]; od; a; # G. C. Greubel, Oct 27 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x)/(1-5*x-2*x^2+8*x^3) )); // G. C. Greubel, Oct 27 2019

seq(coeff(series((1-x)/(1-5*x-2*x^2+8*x^3), x, n+1), x, n), n=0..40); # G. C. Greubel, Oct 27 2019~

CoefficientList[Series[(1-x)/(1-5*x-2*x^2+8*x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 11 2012 *)
LinearRecurrence[{5,2,-8}, {1,4,22}, 30] (* Harvey P. Dale, Dec 22 2013 *)

my(x='x+O('x^40)); Vec((1-x)/(1-5*x-2*x^2+8*x^3)) \\ G. C. Greubel, Oct 27 2019

def A159616_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-x)/(1-5*x-2*x^2+8*x^3)).list()
A159616_list(40) # G. C. Greubel, Oct 27 2019

G.f.: (1-x)/(1-5*x-2*x^2+8*x^3).

a(n) = 5*a(n-1) + 2*a(n-2) - 8*a(n-3) for n > 2 with a(0)=1, a(1)=4, a(2)=22. - Harvey P. Dale, Dec 22 2013

A159617 G.f.: (1-x)/(1-8x-8x^2+8*x^3).

Original entry on oeis.org

1, 7, 64, 560, 4936, 43456, 382656, 3369408, 29668864, 261244928, 2300355072, 20255449088, 178356473856, 1570492542976, 13828748541952, 121767076888576, 1072202663100416, 9441127931576320, 83132508142305280, 732011467286249472

Offset: 0

R. J. Mathar, Apr 17 2009

Number of tilings of a 2xn board with squares of 2 colors and dominoes of 2 colors if n>2. The number of tilings is 6 if n=1, and 56 if n=2.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. Katz, C. Stenson, Tiling a 2xn-board with squares and dominoes, J. Int. Seq. 12 (2009) # 09.2.2.
Index entries for linear recurrences with constant coefficients, signature (8,8,-8).

Cf. A030186, A102436, A159616.

CoefficientList[Series[(1 - x)/(1 - 8 x - 8 x^2 + 8 x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 11 2012 *)

Vec((1 - x) / (1 - 8*x - 8*x^2 + 8*x^3) + O(x^25)) \\ Colin Barker, Jul 05 2020

a(n) = 8*a(n-1) + 8*a(n-2) - 8*a(n-3) for n>2. - Colin Barker, Jul 05 2020

A253265 The number of tilings of 2 X n boards with squares of 2 colors and dominoes of 3 colors.

Original entry on oeis.org

1, 7, 82, 877, 9565, 103960, 1130701, 12296275, 133724242, 1454268793, 15815379409, 171994465072, 1870463946217, 20341557798991, 221217294787570, 2405769114915733, 26163076626035413, 284527128680078536, 3094272440210485525, 33650646877362841531, 365955505581792121138

Offset: 0

R. J. Mathar, Sep 30 2015

The numerator in Formula (3) in the JIS article should be 1-b*x, not 1-x.

G. C. Greubel, Table of n, a(n) for n = 0..950
M. Katz, C. Stenson, Tiling a (2 x n)-board with squares and dominoes, JIS 12 (2009) 09.2.2, Table 1, a=2, b=3.
Index entries for linear recurrences with constant coefficients, signature (10,12,-27).

Cf. A030186 (pieces of a single color), A102436.

a:=[1,7,82];; for n in [4..30] do a[n]:=10*a[n-1]+12*a[n-2] -27*a[n-3]; od; a; # G. C. Greubel, Oct 28 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-3*x)/(1-10*x-12*x^2+27*x^3) )); // G. C. Greubel, Oct 28 2019

seq(coeff(series((1-3*x)/(1-10*x-12*x^2+27*x^3), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 28 2019

CoefficientList[Series[(1-3x)/(1-10x-12x^2+27x^3), {x, 0, 20}], x] (* Michael De Vlieger, Sep 30 2015 *)
LinearRecurrence[{10,12,-27},{1,7,82},30] (* Harvey P. Dale, Dec 30 2015 *)

my(x='x+O('x^30)); Vec((1-3*x)/(1-10*x-12*x^2+27*x^3)) \\ G. C. Greubel, Oct 28 2019

def A253265_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-3*x)/(1-10*x-12*x^2+27*x^3)).list()
A253265_list(30) # G. C. Greubel, Oct 28 2019

G.f.: ( 1-3*x ) / ( 1 - 10*x - 12*x^2 + 27*x^3 ).

A260033 Number of configurations of the general monomer-dimer model for a 2 X 2n square lattice.

Original entry on oeis.org

1, 7, 71, 733, 7573, 78243, 808395, 8352217, 86293865, 891575391, 9211624463, 95173135221, 983314691581, 10159461285307, 104966044432531, 1084493574452273, 11204826469232593, 115766602184825143, 1196083332322900695, 12357755266727364237, 127678491209925526885

Offset: 0

N. J. A. Sloane, Jul 19 2015

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. Allegra, Exact solution of the 2d dimer model: Corner free energy, correlation functions and combinatorics, arXiv:1410.4131 [cond-mat.stat-mech], 2014. See Table 5.
Index entries for linear recurrences with constant coefficients, signature (11,-7,1).

Bisection (even part) of A030186.

a:=[1,7,71];; for n in [4..30] do a[n]:=11*a[n-1]-7*a[n-2]+a[n-3]; od; a; # G. C. Greubel, Oct 27 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-4*x+x^2)/(1-11*x+7*x^2-x^3) )); // G. C. Greubel, Oct 27 2019

seq(coeff(series((1-4*x+x^2)/(1-11*x+7*x^2-x^3), x, n+1), x, n), n = 0 .. 30); # G. C. Greubel, Oct 27 2019

LinearRecurrence[{11,-7,1}, {1,7,71}, 30] (* G. C. Greubel, Oct 27 2019 *)

my(x='x+O('x^30)); Vec((1-4*x+x^2)/(1-11*x+7*x^2-x^3)) \\ G. C. Greubel, Oct 27 2019

def A260033_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-4*x+x^2)/(1-11*x+7*x^2-x^3)).list()
A260033_list(30) # G. C. Greubel, Oct 27 2019

G.f.: (1-4*x+x^2)/(1-11*x+7*x^2-x^3). - Alois P. Heinz, Mar 07 2016

a(0), a(5)-a(20) from Alois P. Heinz, Mar 07 2016

cp's OEIS Frontend

A268630 a(n)^2 + a(n+1) is prime; lexicographically earliest sequence of nonnegative integers with this property and containing no duplicates.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A088132 a(n) equals the square of the n-th partial sum added to twice the n-th partial sum of the squares, divided by a(n-1), for all n>1, with a(0)=1, a(1)=3.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

A353878 Number of tilings of a 3 X n rectangle using right trominoes, dominoes and 1 X 1 tiles.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maxima

Formula

A030236 Cycle-path coverings of a family of digraphs.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

Maxima

PARI

Sage

Formula

A088016 To obtain a(n+1), add the square of the n-th partial sum to the n-th partial sum of the squares, then divide this result by a(n), for all n >= 0, with a(0)=1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

A125693 Riordan array ((1-x)/(1-3*x), x*(1-x)/(1-3*x)).

Views

Author

Keywords

Comments

Examples

Links

Programs

GAP

Magma

Maple

Mathematica

A125693 Riordan array ((1-x)/(1-3x), x(1-x)/(1-3*x)).

A159616 Expansion of (1-x)/(1-5x-2x^2+8*x^3).

A159617 G.f.: (1-x)/(1-8x-8x^2+8*x^3).