If x=a(n), y=a(n+1), z=a(n+2), s=a(n+3), t=a(n+4) then 81*x^5+z*(810*x^4-4860*y*x^3+12060*y^2*x^2-13710*y^3*x+5870*y^4)+t*(135*x^4+z*(1080*x^3- 5760*y*x^2+11040*y^2*x-7070*y^3)-810*y*x^3+s*(-720*x^3+3840*y*x^2+z*(-4710*x^2+20780*y*x- 22870*y^2)-7610*y^2*x+z^2*(33880*y-13670*x)-17190*z^3+5080*y^3)+z^2*(3720*x^2-15710*y*x+ 16370*y^2)+s^2*(1470*x^2+z*(8480*x-22870*y)-6770*y*x+17290*z^2+7940*y^2)+2070*y^2*x^2+z^3* (7180*x-16590*y)-2470*y^3*x+s^3*(-1690*x-7690*z+5060*y)+6390*z^4+1120*y^4+1280*s^4)-540*y* x^4+s*(-540*x^4+3240*y*x^3+z*(-4410*x^3+23340*y*x^2-43810*y^2*x+27280*y^3)-8190*y^2*x^2+z^2* (-15480*x^2+63840*y*x-64480*y^2)+9580*y^3*x+z^3*(66580*y-29870*x)-26010*z^4-4230*y^4)+z^2* (3420*x^3-17790*y*x^2+32270*y^2*x-19360*y^3)+s^2*(1440*x^3+z*(9720*x^2-41910*y*x+44520*y^2)- 7680*y*x^2+z^2*(28370*x-67360*y)+14920*y^2*x+34680*z^3-9660*y^3)+t^2*(90*x^3+z*(570*x^2-2570* y*x+2940*y^2)-480*y*x^2+s*(-360*x^2+1680*y*x+z*(5780*y-2020*x)-4310*z^2-2060*y^2)+z^2* (1640*x-4260*y)+s^2*(590*x+2880*z-1910*y)+970*y^2*x+2130*z^3-670*y^3-640*s^3)+1620*y^2* x^3+z^3*(8190*x^2-32370*y*x+31240*y^2)+t^3*(30*x^2+z*(160*x-490*y)-140*y*x+s*(-90*x-480*z+ 320*y)+360*z^2+180*y^2+160*s^2)-2580*y^3*x^2+s^3*(-2010*x^2+9080*y*x+z*(30160*y-11790*x)- 23120*z^2-10220*y^2)+z^4*(11720*x-24730*y)+s^4*(1790*x+7690*z-5010*y)+t^4*(5*x+30*z-20*y- 20*s)+2110*y^4*x+7834*z^5-698*y^5+t^5-1022*s^5 = 16*3^n; ===================================================================================================================== F1(i) = k[11]*F1(i-1) + k[12]*F2(i-1) +......+ k[1n]*Fn(i-1) F2(i) = k[21]*F1(i-1) + k[22]*F2(i-1) +......+ k[2n]*Fn(i-1) F3(i) = k[31]*F1(i-1) + k[32]*F2(i-1) +......+ k[3n]*Fn(i-1) ............................................................ Fn(i) = k[n1]*F1(i-1) + k[n2]*F2(i-1) +......+ k[nn]*Fn(i-1) with (k[22] - k[11])/k[21] = (k[32] - k[21])/k[31] = ... = (k[n2] - k[(n-1)1])/k[n1] = a[1], (k[23] - k[12])/k[21] = (k[33] - k[22])/k[31] = ... = (k[n3] - k[(n-1)2])/k[n1] = a[2], ..................................................................................... (k[2n] - k[1(n-1)])/k[21] = (k[3n] - k[2(n-1])/k[31] = ... = (k[nn] - k[(n-1)(n-1)])/k[n1] = a[n-1], k[1n]/k[21] = k[2n]/k[31] = ... = k[(n-1)n]/k[n1] = -a[n]. then: lim_{i->inf} F1(i)/F2(i) = F2(i)/F3(i) = ... = F[n-1](i)/Fn(i) = R, R - the root of the equation: x^n + a[1]*x^(n-1) + a[2]*x^(n-2) + ... + a[n-1]*x - a[n] = 0. Matrix: [a, b*m] [b, b*q+a] x^2+q*x-m=0. [a, c*m-q*b, b*m] [b, a+p*b, c*m] x^3+p*x^2+q*x-m=0 [c, b+p*c, a+p*b+q*c] [a, -c*t-b*q+d*m, c*m-b*t, b*m] [b, b*p+a, d*m-c*t, c*m] x^4+p*x^3+q*x^2+t*x-m=0 [c, c*p+b, c*q+b*p+a, d*m] [d, d*p+c, d*q+c*p+b, d*t+c*q+b*p+a] [a, -c*t-b*p+g*m-d*k, -b*t+d*m-c*k, c*m-b*k, b*m] [b, b*q+a, -c*t+g*m-d*k, d*m-c*k, c*m] [c, c*q+b, b*q+c*p+a, g*m-d*k, d*m] x^5+q*x^4+p*x^3+t*x^2+k*x-m=0 [d, d*q+c, c*q+d*p+b, d*t+b*q+c*p+a, g*m] [g, g*q+d, d*q+g*p+c, g*t+c*q+d*p+b, d*t+b*q+c*p+g*k+a] determinant - diophantine equation. Example_1: [2, 3, 5, 9, 17, 33, 65, 129] A000051 2*k1 + 3*k2 = 5, 5*k1 + 9*k2 = 17. >>> k1 = -2, k2 = 3. 2*k3 + 3*k4 = 9, 5*k3 + 9*k4 = 33. >>> k3 = -6, k4 = 7. [a, b*m] <<->> [-2 3] [b, b*q+a] [-6 7] >>> m = -1/2, q = -3/2. [a, -b/2] [b, a-3*b/2] determinant >> (b-2*a)*(b-a) diophantine equation: 2*a - b = 1, b - a = 2^n. 2*5 - 9 = 1, 129 - 65 = 2^6. Example_2: [2, 3, 5, 9, 17, 33, 65, 129] A000051 2*k1 + 3*k2 + 5*k3 = 9, 3*k1 + 5*k2 + 9*k3 = 17, 5*k1 + 9*k2 + 17*k3 = 33. >>> k1 = -2, k2 = 1, k3 = 2. 2*k1 + 3*k2 + 5*k3 = 17, 3*k1 + 5*k2 + 9*k3 = 33, 5*k1 + 9*k2 + 17*k3 = 65. >>> k1 = -4, k2 = 0, k3 = 5. 2*k1 + 3*k2 + 5*k3 = 33, 3*k1 + 5*k2 + 9*k3 = 65, 5*k1 + 9*k2 + 17*k3 = 129. >>> k1 = -10, k2 = 1, k3 = 10. [a, c*m-q*b, b*m] [-2 1 2] [b, a+p*b, c*m] <<->> [-4 0 5] >>> m = -1/2, p = -1/2, q = -1. [c, b+p*c, a+p*b+q*c] [-10 1 10] [a, b-c/2, -b/2] [b, a-b/2, -c/2] determinant >> (c-a)*(c-3*b+2*a)*(c-b-2*a) [c, b-c/2, a-c-b/2] diophantine equation: c - a = 3*2^n, 3*b - 2*a = c, 2*a + b - c = 2. Example_3: [1, 5, 14, 34, 124, 260, 1016, 2056] A083332 1*k1 + 5*k2 = 14, 14*k1 + 34*k2 = 124. >>> k1 = 4, k2 = 2. 1*k1 + 5*k2 = 34, 14*k1 + 34*k2 = 260. >>> k1 = 4, k2 = 6. [a, b*m] <<->> [4 2] [b, b*q+a] [4 6] >>> m = 1/2, q = 1/2. [a, b/2] [b, b/2+a] determinant >> b^2 - a*b - 2*a^2 >> a(2n+1)^2-a(2n+1)*a(2n)-2*a(2n)^2=9*2^(4*n+1) Example_4: [1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9] A000931 [b-(c-a*p)*p, a*m-(c-a*p)*q, (c-a*p)*m] [a+b-c, a, c-a] [ c-a*p, b, a*m ] >>> [c-a, b, a] [ a, c, b+a*q ] [a, c, b] c^3-a*c^2+(-b^2-3*a*b)*c+b^3+a*b^2+2*a^2*b+a^3=1. Example_5: [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233] A000045 a*F(2n) + b*F(2n+k) = F(2n+2k), a*F(2n+1) + b*F(2n+1+k) = F(2n+2k+1) b^2 - a*b - a^2 = +-1, b^2 - 3*a*b + a^2 = +-1, b^2 - 4*a*b - a^2 = +-4, b^2 - 7*a*b + a^2 = +-9, b^2 - 11*a*b - a^2 = +-25, etc. F(n+k)^2 - L(k)*F(n)*F(n+k) + (-1)^k*F(n)^2 = (-1)^n*F(k)^2. Example_6: [1, 1, 1, 1, 1, 3, 13, 43, 113, 253, 509, 969, 1849, 3719, 8009, 18027] A247584 81*x^5+z*(810*x^4-4860*y*x^3+12060*y^2*x^2-13710*y^3*x+5870*y^4)+t*(135*x^4+z*(1080*x^3- 5760*y*x^2+11040*y^2*x-7070*y^3)-810*y*x^3+s*(-720*x^3+3840*y*x^2+z*(-4710*x^2+20780*y*x- 22870*y^2)-7610*y^2*x+z^2*(33880*y-13670*x)-17190*z^3+5080*y^3)+z^2*(3720*x^2-15710*y*x+ 16370*y^2)+s^2*(1470*x^2+z*(8480*x-22870*y)-6770*y*x+17290*z^2+7940*y^2)+2070*y^2*x^2+z^3* (7180*x-16590*y)-2470*y^3*x+s^3*(-1690*x-7690*z+5060*y)+6390*z^4+1120*y^4+1280*s^4)-540*y* x^4+s*(-540*x^4+3240*y*x^3+z*(-4410*x^3+23340*y*x^2-43810*y^2*x+27280*y^3)-8190*y^2*x^2+z^2* (-15480*x^2+63840*y*x-64480*y^2)+9580*y^3*x+z^3*(66580*y-29870*x)-26010*z^4-4230*y^4)+z^2* (3420*x^3-17790*y*x^2+32270*y^2*x-19360*y^3)+s^2*(1440*x^3+z*(9720*x^2-41910*y*x+44520*y^2)- 7680*y*x^2+z^2*(28370*x-67360*y)+14920*y^2*x+34680*z^3-9660*y^3)+t^2*(90*x^3+z*(570*x^2-2570* y*x+2940*y^2)-480*y*x^2+s*(-360*x^2+1680*y*x+z*(5780*y-2020*x)-4310*z^2-2060*y^2)+z^2* (1640*x-4260*y)+s^2*(590*x+2880*z-1910*y)+970*y^2*x+2130*z^3-670*y^3-640*s^3)+1620*y^2* x^3+z^3*(8190*x^2-32370*y*x+31240*y^2)+t^3*(30*x^2+z*(160*x-490*y)-140*y*x+s*(-90*x-480*z+ 320*y)+360*z^2+180*y^2+160*s^2)-2580*y^3*x^2+s^3*(-2010*x^2+9080*y*x+z*(30160*y-11790*x)- 23120*z^2-10220*y^2)+z^4*(11720*x-24730*y)+s^4*(1790*x+7690*z-5010*y)+t^4*(5*x+30*z-20*y- 20*s)+2110*y^4*x+7834*z^5-698*y^5+t^5-1022*s^5 = 16*3^n, if x=a(n), y=a(n+1), z=a(n+2), s=a(n+3), t=a(n+4).