A137708 Secondary Lower Wythoff Sequence.

1, 2, 5, 6, 7, 8, 11, 12, 15, 16, 17, 18, 21, 22, 23, 24, 27, 28, 31, 32, 33, 34, 37, 38, 41, 42, 43, 44, 47, 48, 49, 50, 53, 54, 57, 58, 59, 60, 63, 64, 65, 66, 69, 70, 73, 74, 75, 76, 79, 80, 83, 84, 85, 86, 89, 90, 91, 92, 95, 96, 99, 100, 101, 102, 105, 106, 109, 110, 111

Offset: 1

Author

Clark Kimberling, Feb 07 2008

Keywords

• nonn
• easy

Comments

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

Examples

			The lower Wythoff sequence begins with 1,3,4,6,8,9,...
Double these: 2,6,8,12,16,18...; subtract 1: 1,5,7,11,15,17;
Then merge: 1,2,5,6,7,8,11,12,15,16,17,18,...
		

Crossrefs

Cf. A035513, A137709.

Programs

Mathematica

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

Python

from math import isqrt
def A137708(n): return ((m:=n+1>>1)+isqrt(5*m**2)&-2)-(n&1) # Chai Wah Wu, Aug 11 2022

Formula

a(2n) = 2*A000201(n) = a(2n-1)+1; A000201 is the lower Wythoff sequence.

Extensions

a(37)=59 corrected by Georg Fischer, Nov 24 2022