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Positive Fibonacci numbers that are also triangular numbers.
5

%I #23 Oct 13 2024 07:09:16

%S 1,3,21,55

%N Positive Fibonacci numbers that are also triangular numbers.

%C Vern Hoggatt conjectured and Luo Ming confirmed that these are all the terms. - _Tomohiro Yamada_, Sep 23 2017

%D J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 55, p. 20, Ellipses, Paris 2008.

%D R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section D26.

%D D. Wells, Curious and interesting numbers, Penguin Books, p. 108

%H Luo Ming, <a href="http://www.fq.math.ca/Scanned/27-2/ming.pdf">On triangular Fibonacci numbers</a>, Fibonacci Quart. 27 (1989), 98--108.

%t Union[Select[Fibonacci[Range[20]],OddQ[Sqrt[8#+1]]&]] (* _Harvey P. Dale_, Jan 17 2015 *)

%Y Cf. A000045, A000217.

%K nonn,fini,full

%O 1,2

%A _Felice Russo_