A372729 a(n) is the smallest k such that the first n digits of Fibonacci(k) are the reverse of its last n digits.

0, 10, 317, 1235, 28898, 120742, 1411753, 201095722, 306312948, 12306316582, 32679761048, 806327047899, 9600042921304, 172192972068022

Offset: 1

Author

Jon E. Schoenfield, May 11 2024

Keywords

• nonn
• base
• hard
• more

Comments

(Inspired by D. S. McNeil's comment at A045504 that typically a Fibonacci number can be ruled out as a possible palindrome by checking only a few digits at the start and end of the number.)

Examples

			a(1) = 0 (as Fibonacci(0) = 0 is the smallest Fibonacci number).
a(2) = 10 (Fibonacci(10) = 55 is the only 2-digit Fibonacci number that is palindromic, and almost certainly the only multidigit palindromic Fibonacci number; see A045504).
a(3) = 317 because Fibonacci(317) (a 66-digit number) is the smallest Fibonacci number whose first 3 digits (793) are the reverse of its last 3 digits (397).
The table below lists the first 8 terms and the corresponding Fibonacci numbers (abbreviated, for n > 2):
.
  n       a(n)    Fibonacci(a(n))
  -  ---------  -------------------
  1          0           0
  2         10          55
  3        317       793...397
  4       1235      5626...6265
  5      28898     94480...08449
  6     120742    172255...552271
  7    1411753   3789665...5669873
  8  201095722  11367389...98376311
		

Links

• Kevin Ryde, C Code

Crossrefs

Cf. A000045, A002113, A045504.

Programs

C

/* See links. */

Extensions

a(9)-a(14) from Kevin Ryde, Aug 22 2024