A226857 Numbers that are both the sum of two Fibonacci numbers and the product of two Fibonacci numbers.

0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 15, 16, 21, 24, 26, 34, 39, 42, 55, 63, 68, 89, 102, 110, 144, 165, 178, 233, 267, 288, 377, 432, 466, 610, 699, 754, 987, 1131, 1220, 1597, 1830, 1974, 2584, 2961, 3194, 4181, 4791, 5168, 6765, 7752, 8362, 10946, 12543, 13530

Offset: 1

Author

Alonso del Arte, Jun 19 2013

Keywords

• nonn
• easy

Comments

All Fibonacci numbers are in the sequence. The only prime numbers in this sequence are prime Fibonacci numbers.

Examples

			5 + 21 = 2 * 13 = 26, therefore 26 is in the sequence.
8 + 21 = 1 * 34 = 34, therefore 34 is in the sequence.
5 + 34 = 3 * 13 = 39, therefore 39 is in the sequence.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A059389, A049997.

Programs

Mathematica

t = Fibonacci[Range[0, 25]]; t1 = Select[Union[Flatten[Table[a + b, {a, t}, {b, t}]]], # <= t[[-1]] &]; t2 = Select[Union[Flatten[Table[a*b, {a, t}, {b, t}]]], # <= t[[-1]] &]; Intersection[t1, t2] (* T. D. Noe, Jul 03 2013 *)

Formula

Conjecture: a(n) = a(n-3)+a(n-6) for n>12. - Colin Barker, Nov 09 2014

Empirical g.f.: -x^2*(x^10 +x^9 +x^8 +2*x^7 +3*x^6 +3*x^5 +3*x^4 +3*x^3 +3*x^2 +2*x +1) / (x^6 +x^3 -1). - Colin Barker, Nov 09 2014