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%I #30 Mar 07 2023 10:23:54
%S 1,2,3,5,8,13,22,36,58,95,154,249,403,652,1056,1709,2766,4475,7241,
%T 11717,18959,30676,49635,80311,129947,210258,340205,550464,890670,
%U 1441135,2331806,3772941,6104748,9877690,15982438,25860128,41842566,67702694,109545261,177247955
%N a(n) = floor(sqrt(F(n+2)^2 + F(n)^2)), where F(n) = A000045(n).
%C a(n) is the length of the short side (rounded down) of the parallelogram appearing in the dissection fallacy using the square F(n+3) X F(n+3) (see the links and references). Let the actual length of the short side be L(n) and the one of the long side LL(n), then L(n) = LL(n-1). See the Ngaokrajang link for an illustration. Also floor(LL(n)*L(n)) = A014742(n), n >= 1 (proof by _Wolfdieter Lang_ given there).
%C Note that F(n+2)^2 + F(n)^2 = 3*F(n+1)^2 - 2*(-1)^n = A069921(n). It appears that for n > 1, a(n) = floor(sqrt(3)*F(n+1)). - _Robert Israel_, Jul 16 2014
%D T. Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley & Sons, 2001, ch. 6, pp. 100-108.
%H Kival Ngaokrajang, <a href="/A245271/a245271.pdf">Illustration of initial terms</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DissectionFallacy.html">Dissection Fallacy</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Missing_square_puzzle">Missing square puzzle</a>.
%F a(n) = floor(sqrt(F(n+2)^2 + F(n)^2))), n >= 0, with F(n) = A000045(n), and F(n+2)^2 + F(n)^2 = A069921(n).
%p A245271 := n -> floor(sqrt(3*combinat:-fibonacci(n+1)^2 - 2*(-1)^n)):
%p seq(A245271(n), n=0..100); # _Robert Israel_, Jul 16 2014
%t Table[Floor[Sqrt[Fibonacci[n + 2]^2 + Fibonacci[n]^2]], {n, 0, 50}] (* _Wesley Ivan Hurt_, Jul 17 2014 *)
%o (PARI)a(n) = floor(sqrt((fibonacci(n+2)^2 + fibonacci(n)^2)))
%o for (n=0,50,print1(a(n),", "))
%Y Cf. A000045, A014742, A069921.
%K nonn,easy
%O 0,2
%A _Kival Ngaokrajang_, Jul 15 2014
%E A069921 added to Crossrefs and to the _Robert Israel_ comment by _Wolfdieter Lang_, Jul 17 2014
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