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Re: trouble with a Binet in a generalized Pell recursion


Testing Pell relations on the generalizations:

Clear[f, g, a, b, v1, v2, n, a0, b0, f1, g1]
f[0, a_] := 0; f[1, a_] := 1;
f[n_, a_] := f[n, a] = a*f[n - 1, a] + f[n - 2, a]
g[0, a_] := 1; g[1, a_] := 1;
g[n_, a_] := g[n, a] = a*g[n - 1, a] + g[n - 2, a]
Table[f[n, a], {n, 0, 10}, {a, 1, 11}]
Table[g[n, a], {n, 0, 10}, {a, 1, 11}]
(*Pell definitions*)
Table[2*f[n, a]*g[n, a] - f[2*n, a], {n, 0, 10}, {a, 1, 11}]
(* this one doesn't run well for some reason*)
Table[f[n, a] + f[n - 1, a] - g[n, a], {n, 0, 10}, {a, 1, 11}]
(* these are fine*)
Table[2*g[n, a]^2 - g[2*n, a] - (-1)^n, {n, 0, 10}, {a, 1, 11}]
Table[g[n, a]^2 - 2*f[n, a]^2 - (-1)^n, {n, 0, 10}, {a, 1, 11}]
Some new sequences result from these generalized relationships.
Reference:Charter 14, page 332 Elementary Number Theory, 5th edition,
David m. Burton,
ISBN 0071121749
Roger Bagula


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