Data update

Task/Pell-numbers/Julia/pell-numbers-2.jl

@@ -0,0 +1,2 @@
+pell(n::Integer) = last(BigInt[2 1; 1 0] ^ n * BigInt[1, 0])
+pelllucas(n::Integer) = last(BigInt[2 1; 1 0] ^ n * BigInt[2, 2])

Task/Pell-numbers/PARI-GP/pell-numbers.parigp

@@ -0,0 +1,59 @@
+\\--- pell.gp ------------------------------------------------------------
+
+\\ 1) Define Pell numbers and Pell–Lucas numbers by simple recursion
+
+pell(n) =
+{
+  if (n==0, return(0));
+  if (n==1, return(1));
+  return(2*pell(n-1) + pell(n-2));
+}
+
+pellL(n) =
+{
+  if (n==0, return(2));
+  if (n==1, return(2));
+  return(2*pellL(n-1) + pellL(n-2));
+}
+
+\\ 2) Compute the first 10 Pell and Pell–Lucas numbers
+pns  = vector(10, i, pell(i-1));      \\ pell(0) .. pell(9)
+plns = vector(10, i, pellL(i-1));     \\ pellL(0) .. pellL(9)
+
+print("Pell numbers (n=0..9): ", pns);
+print("Pell–Lucas numbers (n=0..9): ", plns);
+
+\\ 3) Form the ratios (Pell–Lucas)/2 over Pell, skipping n=0
+\\    i.e. for n=1..9
+denom  = vector(9, i, pns[i+1]);      \\ pns[2..10]
+numr   = vector(9, i, plns[i+1]/2);   \\ plns[2..10]/2
+approx = vector(9, i, numr[i]/denom[i]);
+
+print("Ratios (pellL(n)/2) / pell(n) for n=1..9:");
+for (i=1,9, print(i, ": ", approx[i]));
+print("Numeric floats:");
+for (i=1,9, print(i, ": ", real(approx[i])));
+
+\\ 4) List (n, pell(n)) for n=0..100 and pick the first 10 where pell(n) is prime
+plist = vector(101, i, [i-1, pell(i-1)]);
+primespell = select(x->isprime(x[2]), plist);
+print("First 10 Pell(n) that are prime:");
+for (i=1,10, print(primespell[i]));
+
+\\ 5) Define PellS(n) = If[n==0,1, pell(2n) + pell(2n+1)]
+pellS(n) = if(n==0, 1, pell(2*n) + pell(2*n+1));
+print("PellS(n) for n=0..19:");
+print(vector(20, n, pellS(n)));
+
+\\ 6) Generate Pythagorean triples from Pell numbers:
+\\   short = sum_{k=1..2n} pell(k), long = short+1, hypo = pell(2n+1)
+pythag(n) =
+{
+  short = sum(k=1, 2*n, pell(k));
+  long  = short + 1;
+  hypo  = pell(2*n + 1);
+  return([short, long, hypo]);
+}
+
+print("Pythagorean triples for n=0..9:");
+for (n=0,9, print(n, ": ", pythag(n)));