<Text-field style="Heading 1" layout="Heading 1">Examples</Text-field>

f[1]:= 255*x^77+13*x^21;

f[2]:= 1/(1-25*x);

f[3]:= 1/(1-3*x+x^2);

f[4]:=1/((1-2*x)^2*(x+1));

f[5]:=1/((1-2*x)^3);

f[6]:= 1/(1-10*x)^7/(1-7*x)^3/(1+x);

f[7]:= 1/((1-x+x^2)*(1-2*x^2)*(1-36*x*x));

f[8]:= 1/((1-2*x+x^2)*(1-2*x^2)*(1-36*x^2)^2*(1-17*x));

f[9]:=1/((1-2*x+x^2)*(1-2*x^2)*(1-6*x^2)*(1-9*x^2)^2);

f[10]:= 1/(-462*x^6-156*x^5+117*x^4-196*x^3-38*x^2-6*x+1);

f[11]:= (1+3*x-4*x^2+11*x^3)/(1+5*x+17*x^2-9*x^3+18*x^4+42*x^5);

f[12]:= (1+x+12*x^2+x^3+x^4+x^5+x^6-x^7+x^8-5*x^9+x^10)/ (1+12*x+23*x^2+34*x^3+45*x^4+67*x^6+56*x^5+78*x^7+89*x^8+170*x^17+100*x^9+111*x^10+122*x^11+133*x^12+144*x^13+155*x^14+166*x^15+177*x^16);

f[13]:= 1/((1+2*x+4*x^2+8*x^3+16*x^4)*(1-x+x^3-x^4+x^5-x^7+x^8));

# A094423 A rational function with nonnegative integer coefficients that is not N-rational. f[14]:= (x+5*x^2)/(1+x-5*x^2-125*x^3);

# Example from Gourdon/Salvy p.12 f[15]:= (x^9-87*x^8-4047*x^7+42186*x^6+205690*x^5+42186*x^4-4047*x^3-87*x^2+x)/ (x^11-89*x^10-4895*x^9+83215*x^8+582505*x^7-1514513*x^6-1514513*x^5+582505*x^4+83215*x^3-4895*x^2-89*x+1);

# A002727 Number of 3 X n binary matrices f[16]:= (x^6+x^4+2*x^3+x^2+1)/((1-x)^4*(1-x^2)^2*(1-x^3)^2);

# A006381 Number of 3 X n binary matrices under row and column permutations and column complementations. f[17]:= (x^14-2*x^13+3*x^12-2*x^11+5*x^10-4*x^9+7*x^8-4*x^7+7*x^6-4*x^5+5*x^4-2*x^3+3*x^2-2*x+1)/((x^6-1)*(x^2+1)^2*(x^2+x+1)*(x+1)^3*(x-1)^7);

# A005408 The odd numbers. f[18]:= (1+x)/(1-x)^2;

# A000032 Lucas-Numbers f[19]:= (2-x)/(1-x-x^2);

# Hofstadters MIU-system f[20]:= x^2 / (1-3*x+3*x^2-2*x^3);

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

# A001945 f[21]:= (x^5+2*x^4+x^3+2*x^2+x)/(x^6+x^5-x^4-3*x^3-x^2+x+1);

# A001608 Perrin sequence f[22]:= (3-x^2)/(1-x^2-x^3);

# A000124 Maximal number of pieces formed when slicing a pancake with n cuts. # Central polygonal numbers (the Lazy Caterer's sequence). f[23]:= (1-x+x^2)/(1-x)^3;

# A033547 Otto Haxel's guess for magic numbers of nuclear shells f[24]:= 2*x*(x^2-x+1)/(1-x)^4;

# A014206 Draw n+1 circles in the plane; sequence gives maximal number of regions into which the plane is divided f[25]:= 2*x*(x^2-x+1)/(1-x)^3;

# A002315 NSW numbers f[26]:= (1+x)/(1-6*x+x^2);

# A000129 Pell numbers f[27]:= x/(1-2*x-x^2);

# A001519 # F(2n+1) = bisection of Fibonacci sequence. # Number of ordered trees with n+1 edges and height at most 3 # (height=number of edges on a maximal path starting at the root). # Directed column-convex polyominoes of area n+1. # Nondecreasing Dyck paths of length 2n+2 f[28]:= (1-x)/(1-3*x+x^2);

# A038577 Number of self-avoiding walks of length n from origin in strip Z X {0,1} f[29]:= (1+2*x-x^3-x^4+x^7)/(1-x)^2/(1+x)^2/(1-x-x^2);

# A079983 Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=3, r=3, I={-2,1,2} f[30]:= -(x^2-1)*(x^12+2*x^9-x^6-2*x^3+1)/(x^20+x^19-x^18+x^17+2*x^16-x^15-4*x^14-4*x^13+5*x^12-3*x^11-3*x^10+3*x^9+3*x^8+4*x^7-4*x^6+x^5-x^3-x^2-x+1);

# A007598 F(n)^2, where F() = Fibonacci numbers f[31]:= (1-x)/((1+x)*(1-3*x+x^2));

# A011783 F(2n-1) where F() = Fibonacci numbers f[32]:= (1-2*x)/(1-3*x+x^2);

# A001906 F(2n) = bisection of Fibonacci sequence f[33]:= x/(1-3*x+x^2);

# A001654 F(n)F(n+1), where F() = Fibonacci numbers f[34]:= x/(1-2*x-2*x^2+x^3);

# A103142 Generalized Pell numbers f[35]:= 1/(1-2*x-x^2-x^3-x^4);

# A099496 f[36]:= (1+x)/(1+3*x+x^2);

# A001333 Numerators of continued fraction convergents to sqrt(2) f[37]:= (1-x)/(1-2*x-x^2);

# A001653 f[38]:= (1-x)/(1-6*x+x^2);

# A001109 f[39]:= x/(1-6*x+x^2);

# A094706 Convolution of Pell(n) and 2^n f[40]:= x/((1-2*x-x^2)*(1-2*x));

# A000079 Powers of 2: a(n) = 2^n f[41]:= 1/(1-2*x);

# A001110 Numbers that are both triangular and square f[42]:= (1+x)/((1-x)*(1-34*x+x^2));

# A001108 a(n)-th triangular number is a square f[43]:= x*(1+x)/(1-7*x+7*x^2-x^3);

# A002203 Companion Pell numbers f[44]:= (2-2*x)/(1-2*x-x^2);

# A052955 f[45]:= (1+x-x^2)/((1-x)*(1-2*x^2));

# A088014 E.g.f.: cosh(sqrt(2)x)(1+exp(x)) f[46]:= (2-3*x-3*x^2+2*x^2)/(1-2*x-3*x^2+4*x^3+2*x^4);

# A097075 Counts closed walks of length n at a vertex of a triangle, to which a loop has been added at one of the other vertices f[47]:= (1-x-x^2)/(1-x-3*x^2-x^3);

# A097076 Counts walks of length n between two vertices of a triangle, when a loop has been added at the third vertex. f[48]:= x/(1-x-3*x^2-x^3);

# A051927 Number of independent sets of vertices in graph K_2 X C_n (n > 2) f[49]:= (3-2*x-3*x^2)/((1-2*x-x^2)*(1+x));

# A005409 Number of polynomials of height n f[50]:= x*(1-2*x+2*x^2+x^3)/(1-3*x+x^2+x^3);

# A046090 Consider all Pythagorean triples (X,X+1,Z) ordered by increasing Z; sequence gives X+1 values. f[51]:= (1-3*x)/((1-6*x+x^2)*(1-x));

# A001541 Chebyshev polynomials of the first kind evaluated at 3 f[52]:= (1-3*x)/(1-6*x+x^2);

# A055997 Numbers n such that n(n-1)/2 is a square f[53]:= (1-5*x+2*x^2)/((1-x)*(1-6*x+x^2));

# A097165 f[54]:= (1-3*x)/((1-x)*(1-4*x)*(1-5*x));

# A057009 Number of conjugacy classes of subgroups of index 3 in free group of rank f[55]:= x*(1-4*x)/((1-2*x)*(1-3*x)*(1-6*x));

# A002061 Central polygonal numbers: n^2 - n + 1. f[56]:= (1-2*x+3*x^2)/(1-x)^3;

# A004006 3-dimensional analog of centered polygonal numbers. f[57]:= x*(x^2-x+1)/(1-x)^4;

# A050531 Number of multigraphs with loops on 3 nodes with n edges. f[58]:= (x^6+x^4+2*x^3+x^2+1)/((x^3-1)^2*(x^2-1)^2*(x-1)^2);

# A027083 f[59]:= (2*x^2*(1+x+x^2))/((1-x)*(1-x-x^2-x^3));

# A000217 Triangular numbers f[60]:= x/(1-x)^3;

# A000125 Cake numbers: maximal number of pieces resulting from n planar cuts through a cube (or cake): C(n+1,3)+n+1. f[61]:= (1-2*x+2*x^2)/(1-x)^4;

# A016028 For n>2, maximal number of edges in critical strongly connected digraphs on n-1 vertices. f[62]:= (1-x+x^4)/(1-x)^5;

# A000096 For n >= 1, a(n) = maximal number of pieces that can be obtained by cutting an annulus with n cuts. f[63]:= x*(2-x)/(1-x)^3;

# A098574 f[64]:= (1-x)/(1-2*x+x^2-x^7);

# A005689 f[65]:= (1+x^2+x^3+x^4+x^5)/(1-2*x+x^2-x^6);

# A000601 Molien series for 4-dimensional representation of S_3 f[66]:= 1/((1-x)^2*(1-x^2)*(1-x^3));

# A002524 Number of permutations of length n within distance 2. f[67]:= (1+2*x^2-x^4)/(1-2*x-2*x^3+x^5);

# A059929 Fib(n)*Fib(n+2). f[68]:= (2*x-x^2)/((1+x)*(1-3*x+x^2));

# A021913 Decimal expansion of 1/909. f[69]:= (x^2+x^3)/(1-x^4);

# A046127 Maximal number of regions into which space can be divided by n spheres. f[70]:= (2*x-4*x^2+4*x^3)/(1-x)^4;

# Attention: The following two examples are very nasty. # Lots of computation time will be needed to cope them!

# Example from Gourdon/Salvy p.12 (a mistake there is corrected here) # Sloane A005341 # see also: http://mathworld.wolfram.com/LookandSaySequence.html # but there is also a small difference at x^40: # Test: sort(-reciprocal(op(2, factor(denom(f[71])))), x); f[71]:=simplify( (1+x-x^2-x^3-x^4+x^5-5*x^7+6*x^9+8*x^10-10*x^12-5*x^13+x^14+4*x^15+x^16+4*x^17+7*x^18+9*x^19-4*x^20-22*x^21-39*x^22+4*x^23+52*x^24+38*x^25+17*x^26-68*x^27-28*x^28+22*x^29-12*x^30+13*x^31-37*x^32+45*x^33+54*x^34-12*x^35-34*x^36-82*x^37+17*x^38+89*x^39+13*x^40-34*x^42-89*x^43+73*x^44+x^45+26*x^46+31*x^47-128*x^48-14*x^49+49*x^50+56*x^51+74*x^52-99*x^53-20*x^54-43*x^55+33*x^56+47*x^57-41*x^58+18*x^59+50*x^60-10*x^61-13*x^62-9*x^63-17*x^64+38*x^65-42*x^66+37*x^67+8*x^68-4*x^69-29*x^70-19*x^71+28*x^72+30*x^73-22*x^74-18*x^76+12*x^77)/(1-x-x^2-x^3+x^4+3*x^5-x^7-2*x^8+3*x^13+3*x^14-2*x^15-5*x^16-8*x^17+7*x^18+x^19+8*x^20-5*x^22+8*x^23-12*x^24-4*x^25-x^26+18*x^28-4*x^29+2*x^30-13*x^31-7*x^32+19*x^33-14*x^34+14*x^35-6*x^36-4*x^37+13*x^38-9*x^39-7*x^40+4*x^41-8*x^42+7*x^43+5*x^44+7*x^45-12*x^46+17*x^47-22*x^48+8*x^49-7*x^50+16*x^51-6*x^52-7*x^53-6*x^54+3*x^55+19*x^56-5*x^57-5*x^58-14*x^59+8*x^60+2*x^61+7*x^62-5*x^63+x^64-8*x^65+14*x^66-11*x^67+16*x^68-18*x^69+9*x^70-9*x^71+6*x^72)-2*x^5-2*x^6);

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# Roots are very close together # The smallest p such that the c-condition for the denominator is fulfilled is 1031. f[72]:= (1+x-x^2-10*x^3)/(7*x^4-9*x^3-2*x^2+1);