share/solve_rec/solve_rec.dem

   1 "This is a demo file for the solve_rec package."$
   2
   3 "Order 1 linear recurrences."$
   4
   5 solve_rec(a[n]=n/(n+1)*a[n-1], a[n]);
   6 solve_rec(a[n]=n/(n+1)*a[n-1], a[n], a[1]=1);
   7 solve_rec((n+1)*(2*n+1)*a[n+1]-(n+3)*(2*n-1)*a[n], a[n]);
   8 solve_rec(a[n]=n*a[n-1]/(n-2)+1, a[n]);
   9 solve_rec(a[n+1]=n*a[n]/(n-3) + 1/(n-5), a[n]);
  10
  11 "Linear recurrences with constant coefficients."$
  12
  13 solve_rec(a[n]=a[n-1]-a[n-2]+(n^2-1)/4^n, a[n]);
  14 solve_rec(a[n]=a[n-1]-a[n-2]+(n^2-1)/4^n, a[n], a[1]=1);
  15 solve_rec(a[n]=a[n-1]-a[n-2]+(n^2-1)/4^n, a[n], a[1]=1, a[2]=1);
  16 solve_rec(a[n]=2*a[n-1]-a[n-2]+n, a[n]);
  17 solve_rec(a[n]=3*a[n-1]-3*a[n-2]+a[n-3]+n+n*2^n, a[n]);
  18 solve_rec(a[n+4]-a[n+3]-13*a[n+2]+a[n+1]+12*a[n]+n^2+2^(2*n+1)-1, a[n]);
  19 solve_rec(a[n]=a[n+1]-a[n-1]-1, a[n]);
  20
  21 "Linear recurrences with rational coefficients - using hyper."$
  22
  23 solve_rec((n+4)*s[n+2] + s[n+1] - (n+1)*s[n], s[n]);
  24 solve_rec(9*(n+2)*s[n+2] - 3*(n+4)*s[n+1] - 2*(n+3)*s[n], s[n]);
  25 solve_rec(a[x+2]-2*(x+5)*(x-4)/((x+6)*(x-3))*a[x+1]+(x+4)*(x-5)/((x+6)*(x-3))*a[x], a[x]);
  26 solve_rec(2*x*(x+1)*y[x] - (x^2+3*x-2)*y[x+1] + (x-1)*y[x+2], y[x]);
  27 solve_rec(2*x*(x+1)*y[x] - (x^2+3*x-2)*y[x+1] + (x-1)*y[x+2], y[x], y[1]=1, y[3]=3);
  28 solve_rec((-n-5)*s[n+3]+(6*n+22)*s[n+2]+(-9*n-17)*s[n+1]+4*n*s[n],s[n]);
  29 solve_rec((n+3)*s[n+2]-3*(n+2)*s[n+1]+2*(n+1)*s[n],s[n]);
  30 solve_rec((x^6+5*x^5+5*x^4-5*x^3-6*x^2)*z[x+2]+(2*x^4+7*x^3+9*x^2+6*x)*z[x+1]+(x^2+4*x+3)*z[x], z[x]);
  31 solve_rec(n*(n+1)*a[n+2]-2*n*(n+k+1)*a[n+1]+(n+k)*(n+k+1)*a[n], a[n]);
  32 solve_rec(a[n+2]-(2*n+1)*a[n+1]+(n^2-u)*a[n], a[n]), hyper_factor_solve=true, assume_pos=true;
  33 solve_rec((n^2-n-2)*f[n+2]+(-n^3-3*n^2+2*n+8)*f[n+1]+(2*n^3+4*n^2-2*n-4)*f[n], f[n]);
  34 solve_rec((n^5+8*n^4+23*n^3+29*n^2+16*n+4)*f[n+2]+(-n^6-9*n^5-32*n^4-61*n^3-65*n^2-36*n-8)*f[n+1]+(-n^6-8*n^5-21*n^4-23*n^3-9*n^2)*f[n], f[n]);
  35 solve_rec((5*n^3+17*n^2+16*n+4)*f[n+2]+(-15*n^3-46*n^2-43*n-12)*f[n+1]+(-20*n^3-38*n^2-14*n)*f[n], f[n]), product_use_gamma=false;
  36 solve_rec((2*n^6+18*n^5+57*n^4+74*n^3+32*n^2-3*n)*f[n+3]+(-2*n^7-24*n^6-115*n^5-285*n^4-386*n^3-275*n^2-95*n-18)*f[n+2]+(2*n^6+26*n^5+123*n^4+280*n^3+326*n^2+181*n+36)*f[n+1]+(2*n^7+24*n^6+111*n^5+255*n^4+308*n^3+187*n^2+45*n)*f[n], f[n]);
  37 solve_rec((n^3-1)*f[n+2]+(-n^5-2*n^4-3*n^3-2*n^2-n+2)*f[n+1]+(n^5+3*n^4+4*n^3+3*n^2+3*n)*f[n], f[n]);
  38
  39 "Linear recurrences with rational coefficients - using ratsol."$
  40
  41 solve_rec_rat(a[x+2]-2*(x+5)*(x-4)/((x+6)*(x-3))*a[x+1]+(x+4)*(x-5)/((x+6)*(x-3))*a[x], a[x]);
  42 solve_rec_rat(a[x+2]-2*((x-99)*(x+101))/((x-98)*(x+102))*a[x+1]+((x-100)*(x+100))/((x-98)*(x+102))*a[x], a[x]);
  43 solve_rec((x+4)*a[x+3] + (x+3)*a[x+2] - x*a[x+1] + (x^2-1)*a[x]=(x+2)/(x+1), a[x]);
  44
  45 "Ricatti type equations."$
  46
  47 solve_rec(y[x]*y[x-1] + y[x]/(x+1) = 0, y[x]);
  48 solve_rec(y[x+1]*y[x] = y[x]-y[x+1]+2, y[x]);
  49 solve_rec(y[x+1]*y[x] = y[x]-y[x+1]+2, y[x], y[1]=5);
  50 ratsimp(%);
  51 solve_rec(x*y[x+1]*y[x] - y[x+1]/(x+2) + y[x]/(x-1) = 0, y[x], y[3]=5);
  52 ratsimp(minfactorial(factcomb(%)));