A137709 Secondary upper Wythoff sequence.

3, 4, 9, 10, 13, 14, 19, 20, 25, 26, 29, 30, 35, 36, 39, 40, 45, 46, 51, 52, 55, 56, 61, 62, 67, 68, 71, 72, 77, 78, 81, 82, 87, 88, 93, 94, 97, 98, 103, 104, 107, 108, 113, 114, 119, 120, 123, 124, 129, 130, 135, 136, 139, 140, 145, 146, 149, 150, 155, 156

Offset: 1

Author

Clark Kimberling, Feb 07 2008

Keywords

• nonn
• easy

Comments

This sequence is the ordered union of even numbered columns of the Wythoff secondary array, A137707. See A137707 for complementary equations of which this sequence is a solution.

Examples

			The upper Wythoff sequence begins with 2,5,7,10,13,15,18,...
Double these: 4,10,14,20,26,30,36,...; subtract 1: 3,9,13,19,25,29,35,...;
Then merge: 3,4,9,10,13,14,19,20,25,26,29,30,35,36,...
		

Crossrefs

Cf. A001950, A035513, A137707, A137708.

Programs

Mathematica

Union[Flatten[{#, # - 1}]] &[2 Array[Floor[#*GoldenRatio^2] &, 20]] (* Peter J. C. Moses, May 12 2025 *)

Python

from math import isqrt
def A137709(n): return (((m:=n+1>>1)+isqrt(5*m**2)>>1)+m<<1)-(n&1) # Chai Wah Wu, May 30 2025

Formula

a(2n) = 2*A001950(n) = a(2n-1)+1; A001950 is the upper Wythoff sequence.