A245271 a(n) = floor(sqrt(F(n+2)^2 + F(n)^2)), where F(n) = A000045(n).

1, 2, 3, 5, 8, 13, 22, 36, 58, 95, 154, 249, 403, 652, 1056, 1709, 2766, 4475, 7241, 11717, 18959, 30676, 49635, 80311, 129947, 210258, 340205, 550464, 890670, 1441135, 2331806, 3772941, 6104748, 9877690, 15982438, 25860128, 41842566, 67702694, 109545261, 177247955

Offset: 0

Author

Kival Ngaokrajang, Jul 15 2014

Keywords

• nonn
• easy

Comments

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).

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

References

• T. Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley & Sons, 2001, ch. 6, pp. 100-108.

Links

• Kival Ngaokrajang, Illustration of initial terms.
• Eric Weisstein's World of Mathematics, Dissection Fallacy.
• Wikipedia, Missing square puzzle.

Crossrefs

Cf. A000045, A000196, A014742, A069921.

Programs

Maple

A245271 := n -> floor(sqrt(3*combinat:-fibonacci(n+1)^2 - 2*(-1)^n)):
seq(A245271(n), n=0..100); # Robert Israel, Jul 16 2014

Mathematica

Table[Floor[Sqrt[Fibonacci[n + 2]^2 + Fibonacci[n]^2]], {n, 0, 50}] (* Wesley Ivan Hurt, Jul 17 2014 *)

PARI

a(n) = sqrtint(fibonacci(n+2)^2 + fibonacci(n)^2)
for (n=0,50,print1(a(n),", "))

Formula

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).

a(n) = A000196(A069921(n)). - Jason Yuen, Nov 10 2024

Extensions

A069921 added to Crossrefs and to the Robert Israel comment by Wolfdieter Lang, Jul 17 2014