(MAGMAMagma) [Lucas(4*n+2)/3: n in [0..30]]; // G. C. Greubel, Dec 17 2017

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In general, Sum_{k=0..n} binomial(2*n-k,k)j^(n-k) = (-1)^n*U(2n, I*sqrt(j)/2), Ii=sqrt(-1). - Paul Barry, Mar 13 2005

All positive solutions of the Diophantine equation x^2 + y^2 - 7*x*y = -5 are given by [x(n) = S(n, 7) - S(n-1, 7), y(n) = x(n-1)], for all integer numbers n, with the Chebyshev S-polynomials (A049310), with S(-1, 0) = 0, and S(-n, x) = - S(n-2, x), for n >= 2. x(n) = a(n), for n >= 0.

All proper and improper solutions of the generalized Pell equation X^2 - 45*Y^2 = +4 are given, up to a combined sign change in X and Y, in terms of x(n) = a(n) from the preceding comment, by X(n) = x(n) + x(n-1) = S(n-1, 7) - S(n-2, 7) and Y(n) = (x(n) - x(n-1))/3 = S(n-1, 7), for all integer numbers n. For positive integers X(n) = A056854(n) and Y(n) = A004187(n). X(-n) = X(n) and Y(-n) = - Y(n), for n >= 1.

a(n) = (-1)^n*U(2n, Ii*sqrt(5)/2), U(n, x) Chebyshev polynomial of second kind, Ii=sqrt(-1). - Paul Barry, Mar 13 2005

a(3) = L(4 * 3 + 2) / 3 = 843 / 3 = 281. - Indranil Ghosh, Feb 06 2017

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In general, sumSum_{k=0..n, } binomial(2*n-k,k)j^(n-k)} = (-1)^n*U(2n, I*sqrt(j)/2), I=sqrt(-1). - Paul Barry, Mar 13 2005

J.-C. Novelli, and J.-Y. Thibon, <a href="https://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv preprint arXiv:1403.5962 [math.CO], 2014.

Let q(n, x) = sum(Sum_{i=0, n, } x^(n-i)*binomial(2*n-i, i)); then q(n, 5)=a(n); a(n) = 7a(n-1) - a(n-2). - Benoit Cloitre, Nov 10 2002

a(n) = sum_Sum_{k=0..n} binomial(n+k, 2k)*5^k. - Paul Barry, Aug 30 2004

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From Wolfdieter Lang, Feb 09 2021: (Start)

All positive solutions of the Diophantine equation x^2 + y^2 - 7*x*y = -5 are given by [x(n) = S(n, 7) - S(n-1, 7), y(n) = x(n-1)], for all integer numbers n, with the Chebyshev S-polynomials (A049310), with S(-1, 0) = 0, and S(-n, x) = - S(n-2, x), for n >= 2. x(n) = a(n), for n >= 0.

This indefinite binary quadratic form has discriminant D = +45. There is only this family representing -5 properly with x and y positive, and there are no improper solutions.

All proper and improper solutions of the generalized Pell equation X^2 - 45*Y^2 = +4 are given, up to a combined sign change in X and Y, in terms of x(n) = a(n) from the preceding comment, by X(n) = x(n) + x(n-1) = S(n-1, 7) - S(n-2, 7) and Y(n) = (x(n) - x(n-1))/3 = S(n-1, 7), for all integer numbers n. For positive integers X(n) = A056854(n) and Y(n) = A004187(n). X(-n) = X(n) and Y(-n) = - Y(n), for n >= 1.

The two conjugated proper family of solutions are given by [X(3*n+1), Y(3*n+1)] and [X(3*n+2), Y(3*n+2)], and the one improper family by [X(3*n), Y(3*n)], for all integer numbers n.

This comment is inspired by a paper by Robert K. Moniot (private communication). See his Oct 04 2020 comment in A027941 related to the case of x^2 + y^2 - 3*x*y = -1 (special Markov solutions). (End)

a(n) = S(n, 7) - S(n-1, 7) with Chebyshev S polynomials S(n-1, 7) = A004187(n), for n >= 0. - Wolfdieter Lang, Feb 09 2021

Cf. A004187A002310, A002320, A049310, A049685, A056854.

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Row 7 of array A094954. First differences of A004187.

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