A166074 a(n) = n^2 - [largest Fibonacci number <= n^2].

0, 1, 1, 3, 4, 2, 15, 9, 26, 11, 32, 0, 25, 52, 81, 23, 56, 91, 128, 23, 64, 107, 152, 199, 15, 66, 119, 174, 231, 290, 351, 37, 102, 169, 238, 309, 382, 457, 534, 3, 84, 167, 252, 339, 428, 519, 612, 707, 804, 903, 17, 120, 225, 332, 441, 552, 665, 780, 897, 1016, 1137

Offset

1,4

Comments

The only numbers n where a(n)=0 are 1 and 12: a(1) = 1*1 - 1 = 0 and a(12) = 12*12 - 144 = 0. Cohn (1964) proved that 1 and 144 are the only Fibonacci numbers which are perfect squares. In general, for a positive integer k, how many Fibonacci numbers exist such that k = n^2 - (largest Fibonacci number <= n^2)? The only proved answer is 2 for k=0.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

J. H. E. Cohn, On square Fibonacci numbers, Journal London Math.Soc., 39 (1964), pp. 537-540.

Mathematica

a[n_]:=Block[{k=1}, While[Fibonacci[k]<=n^2, k++]; Return[n^2 - Fibonacci[k - 1]]]; Table[a[n], {n, 80}] (* Vincenzo Librandi, Jan 09 2019 *)
With[{fibs=Fibonacci[Range[40]]}, Table[n^2-Select[Nearest[fibs, n^2, 2], #<= n^2&], {n, 70}]][[All, 1]] (* Harvey P. Dale, Feb 05 2019 *)

Crossrefs

Cf. A000045.

Sequence in context: A159672 A059114 A246322 * A225475 A259334 A210488

Adjacent sequences: A166071 A166072 A166073 * A166075 A166076 A166077

Keyword

easy,nonn

Author

Ctibor O. Zizka, Oct 06 2009

Extensions

a(7) corrected and more terms appended by R. J. Mathar, Oct 08 2009

Status

approved