A028841 Iterated sum of digits of n is a Fibonacci number.

1, 2, 3, 5, 8, 10, 11, 12, 14, 17, 19, 20, 21, 23, 26, 28, 29, 30, 32, 35, 37, 38, 39, 41, 44, 46, 47, 48, 50, 53, 55, 56, 57, 59, 62, 64, 65, 66, 68, 71, 73, 74, 75, 77, 80, 82, 83, 84, 86, 89, 91, 92, 93, 95, 98, 100, 101, 102, 104, 107, 109, 110, 111, 113, 116, 118, 119

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• base

Comments

Intermediate iterations don't count. For example, with 85, we have 8 + 5 = 13, which is a Fibonacci number, but 1 + 3 = 4, which is not a Fibonacci numbers, so 85 is not in the sequence. - Alonso del Arte, Jan 20 2020

Examples

			98 -> 9 + 8 = 17 -> 1 + 7 = 8 is a Fibonacci number.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1001 [Offset adapted by _Georg Fischer_, Feb 28 2020]

Crossrefs

Cf. A010888, A028840, A028891.

Programs

Mathematica

With[{fibo = {1, 2, 3, 5, 8}}, Select[Range[120], MemberQ[fibo, NestWhile[Total[IntegerDigits[#]] &, #, # > 9 &]]&]] (* Harvey P. Dale, Apr 11 2013 *)

Scala

def fiboDRQ(n: Int): Boolean = List(1, 2, 3, 5, 8).contains(n % 9)
(1 to 100).filter(fiboDRQ) // Alonso del Arte, Jan 28 2020

Formula

Conjectures from Colin Barker, Feb 18 2020: (Start)

G.f.: x*(1 + x + x^2 + 2*x^3 + 3*x^4 + x^5) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4)).

a(n) = a(n-1) + a(n-5) - a(n-6) for n>6.

(End)

Extensions

More terms from Patrick De Geest, Jun 15 1999

Offset corrected to 1 by Alonso del Arte, Jan 28 2020 at Michel Marcus's suggestion