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%I #10 Mar 31 2012 10:26:02
%S 5,185,1313,4925,13325,29585,57545,101813,167765,261545,390065,561005,
%T 782813,1064705,1416665,1849445,2374565,3004313,3751745,4630685,
%U 5655725,6842225,8206313,9764885,11535605,13536905,15787985,18308813,21120125
%N Smallest integral ladder whose ends slide over the respective distances 1 and m=2n+1 while slipping down along horizontal ground and vertical wall against which it leans.
%C Ladder has upper end at height M(8*M - 5) and lower end distance 4*M*m off the wall (or vice versa), where 2M=m^2 + 1. {The Pythagorean triple is M times (8*M-3, 8*M-5, 4*m)}.
%F a(n) = M(8*M - 3), where M=2n^2 + 2n + 1=A001844(n).
%F a(n) = (2n^2 + 2n + 1)*(16n^2 + 16n + 5).
%F G.f.: (5+160*x+438*x^2+160*x^3+5*x^4)/(1-x)^5. - Colin Barker, Jan 23 2012
%e The expressions associated with the first few entries are:
%e 5^2=3^2 + 4^2=(3+1)^2 + (4-1)^2.
%e 185^2=175^2 + 60^2=(175+1)^2 + (60-3)^2.
%e 1313^2=1287^2 + 260^2=(1287+1)^2 + (260-5)^2.
%e 4925^2=4875^2 + 700^2=(4875+1)^2 + (700-7)^2.
%e 13325^2=13243^2 +1476^2=(13243+1)^2 + (1476-9)^2.
%e Consider the case n=2. For a ladder L with upper end at height h off ground and lower end at distance s off wall, we have relations L^2=h^2 + s^2=(h-1)^2 + (s+5)^2.....(*), which boil down to X^2 - 26*Y^2=-1 using the parameters X=2k+7, Y=L/13, h=5k+18, s=k+1, so that triples (L, h, s) are generated from the recurrence V(i)=102*V(i-1) - V(i-2) + W, where vectors V(i)=[L(i) h(i) s(i)], W=[0 -50 250], with V(-1)=[13 -12 -5], V(0)=[13 13 0], yielding solutions (1313, 1288, 255);(133913, 131313, 26260);(13657813, 13392588, 2678515);(1392963013, 1365912613, 273182520);...all satisfying relation (*) above with the smallest solution L(1) being 1313=a(2).
%K nonn,easy
%O 0,1
%A _Lekraj Beedassy_, Apr 30 2004
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