%I A094081
%S A094081 5,185,1313,4925,13325,29585,57545,101813,167765,261545,390065,561005,
%T A094081 782813,1064705,1416665,1849445,2374565,3004313,3751745,4630685,5655725,
%U A094081 6842225,8206313,9764885,11535605,13536905,15787985,18308813,21120125
%N A094081 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 A094081 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 A094081 a(n)=M(8*M - 3), where M=2n^2 + 2n + 1=A001844(n).
%F A094081 (2n^2 - 2n + 1)*(16n^2 - 16n + 5).
%e A094081 The expressions associated with the first few entries are:
%e A094081 5^2=3^2 + 4^2=(3+1)^2 + (4-1)^2.
%e A094081 185^2=175^2 + 60^2=(175+1)^2 + (60-3)^2.
%e A094081 1313^2=1287^2 + 260^2=(1287+1)^2 + (260-5)^2.
%e A094081 4925^2=4875^2 + 700^2=(4875+1)^2 + (700-7)^2.
%e A094081 13325^2=13243^2 +1476^2=(13243+1)^2 + (1476-9)^2.
%e A094081 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).
%Y A094081 Sequence in context: A053363 A027572 A050239 this_sequence A096543 A138733 
               A015102
%Y A094081 Adjacent sequences: A094078 A094079 A094080 this_sequence A094082 A094083 
               A094084
%K A094081 nonn
%O A094081 0,1
%A A094081 Lekraj Beedassy (blekraj(AT)yahoo.com), Apr 30 2004