1 / 89 one fraction whose decimal IS the Fibonacci sequence — and 89 = F₁₁
Divide 1 by 89 and read the digits: 0.011235955… — the Fibonacci numbers, laid end to end, each shifted one place and added where they overlap (which is why 8+ carries turn …8,13,21… into …9,5,5…). The reason is exact: ∑ F(n)/10ⁿ = 1/89, because 89 = 10² − 10 − 1 encodes the Fibonacci recurrence in base 10. And 89 = F(11) — the eleventh Fibonacci number, the other 11 in David's puzzle. The whole infinite sequence, folded into a single fraction.
The demo shows the Fibonacci numbers summed at decreasing place values converging on 1/89 — the sequence inside the fraction: live demo
HISTORY & CREDIT a Fibonacci curiosity · 89 = F(11)
“1/89 is just a repeating decimal.” — its digits are the Fibonacci sequence added with carries, because 89 = 10²−10−1 encodes ‘sum the last two’ in base ten. cited
the identity · ∑n≥1 F(n)/10ⁿ⁺¹ = 1/89. why 89 · 89 = 10² − 10 − 1 — the characteristic x²−x−1 at x=10. the 11 · 89 = F(11) — the eleventh Fibonacci, the second 11 in the puzzle.
The infinite sequence, summed at shrinking scales, is a single fraction — and its denominator is the eleventh Fibonacci. 89 = F(11)
RECOMMEND FOR I-13 1/89 and 89 = F(11), on the compiler
On the canonical compiler, 89 = F(11) and 1/89 = 0.0112359550… — the Fibonacci digits, and the eleventh Fibonacci as the denominator:
$ i13 run g_oneover89.i13 # 1/89, and 89 = F(11)
RUN OK · 3308 step(s) · peak stack 4 · call depth 11
n89 = 89 -- = F(11), the ~~11~~ referent
recip = 0.011235955056179775 -- 0.0 1 1 2 3 5 (9..) : the Fibonacci sequence
is_89 = 1
Recommend: 1/89 is the whole Fibonacci sequence folded into one fraction — its digits are the numbers added with carries, because 89 = 10²−10−1 is the recurrence in base ten — and 89 = F(11) pays off the second 11 in the puzzle. i13 grounds both. Not a keeper (a base-ten curiosity, not a mechanism), but the dart that ties the referent 11 to the deepest fact: the recurrence is encoded in the denominator.