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(x,y) = (a(n),a(n+1)) are the solutions of (t(x)+t(y))/(1+xy) = t(4) = 10, where t(n) denotes the n-th triangular number t(n) = n*(n+1)/2.
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%I #22 Jan 08 2024 09:00:02

%S 4,79,1575,31420,626824,12505059,249474355,4976982040,99290166444,

%T 1980826346839,39517236770335,788363909059860,15727760944426864,

%U 313766854979477419,6259609338645121515,124878419917922952880

%N (x,y) = (a(n),a(n+1)) are the solutions of (t(x)+t(y))/(1+xy) = t(4) = 10, where t(n) denotes the n-th triangular number t(n) = n*(n+1)/2.

%H Paolo Xausa, <a href="/A065930/b065930.txt">Table of n, a(n) for n = 0..500</a>

%H J.-P. Ehrmann et al., <a href="http://forumgeom.fau.edu/POLYA/ProblemCenter/POLYA002.html">Problem POLYA002, Integer pairs (x,y) for which (x^2+y^2)/(1+pxy) is an integer</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (21,-21,1).

%F a(n) = 2t(m)a(n-1)-a(n-2)-1, a(0) = m, a(1) = m^3+m^2-1 with m = 4.

%F G.f.: (5x-4)/((1-20x+x^2)(x-1)).

%p g := (5*x-4)/(1-20*x+x^2)/(x-1): s := series(g, x, 40): for i from 0 to 30 do printf(`%d,`,coeff(s, x, i)) od: # _James A. Sellers_, Feb 11 2002

%t LinearRecurrence[{21,-21,1},{4,79,1575},25] (* _Paolo Xausa_, Jan 08 2024 *)

%Y Cf. A000217 (triangular numbers).

%K easy,nonn

%O 0,1

%A _Floor van Lamoen_, Nov 29 2001

%E More terms from _James A. Sellers_, Feb 11 2002