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.

5, 185, 1313, 4925, 13325, 29585, 57545, 101813, 167765, 261545, 390065, 561005, 782813, 1064705, 1416665, 1849445, 2374565, 3004313, 3751745, 4630685, 5655725, 6842225, 8206313, 9764885, 11535605, 13536905, 15787985, 18308813, 21120125

Offset

0,1

Comments

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)}.

Links

Table of n, a(n) for n=0..28.

Formula

a(n) = M(8*M - 3), where M=2n^2 + 2n + 1=A001844(n).

a(n) = (2n^2 + 2n + 1)*(16n^2 + 16n + 5).

G.f.: (5+160*x+438*x^2+160*x^3+5*x^4)/(1-x)^5. - Colin Barker, Jan 23 2012

Example

The expressions associated with the first few entries are:
5^2=3^2 + 4^2=(3+1)^2 + (4-1)^2.
185^2=175^2 + 60^2=(175+1)^2 + (60-3)^2.
1313^2=1287^2 + 260^2=(1287+1)^2 + (260-5)^2.
4925^2=4875^2 + 700^2=(4875+1)^2 + (700-7)^2.
13325^2=13243^2 +1476^2=(13243+1)^2 + (1476-9)^2.
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).

Crossrefs

Sequence in context: A304277 A208403 A280797 * A189645 A228694 A268178

Adjacent sequences: A094078 A094079 A094080 * A094082 A094083 A094084

Keyword

nonn,easy

Author

Lekraj Beedassy, Apr 30 2004

Status

approved