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

1,1

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.

Links

Table of n, a(n) for n=1..60.

Formula

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

Example

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,...

Mathematica

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

Prog

(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

Crossrefs

Cf. A001950, A035513, A137707, A137708.

Sequence in context: A377430 A287323 A059985 * A275538 A090120 A129783

Adjacent sequences: A137706 A137707 A137708 * A137710 A137711 A137712

Keyword

nonn,easy

Author

Clark Kimberling, Feb 07 2008

Status

approved