A022413 Kim-sums: "Kimberling sums" K_n + K_2.

1, 5, 8, 10, 13, 16, 18, 21, 23, 26, 29, 31, 34, 37, 39, 42, 44, 47, 50, 52, 55, 57, 60, 63, 65, 68, 71, 73, 76, 78, 81, 84, 86, 89, 92, 94, 97, 99, 102, 105, 107, 110, 112, 115, 118, 120, 123, 126, 128, 131, 133, 136, 139, 141, 144, 146, 149, 152, 154, 157, 160, 162, 165, 167, 170, 173, 175, 178

Offset: 0

Author

Marc LeBrun

Keywords

• nonn

Comments

Let W(i,j) denote the index of that row of the extended Wythoff array (see A035513) that contains the sequence formed by the sum of rows i and j. Then the "Kim-sum" or "Kimberling-sum" K_n + K_i is W(i-1,n). - N. J. A. Sloane, Mar 08 2016

The n-th Kimberling sequence K_n is defined (cf Links) by K_n(i) = K_n(i-1) + K_n(i-2), with initial values K_n(0) = n, K_n(1) = floor((n+1)*tau). - M. F. Hasler, Sep 02 2016

References

• J. H. Conway, Posting to Math Fun Mailing List, Dec 02 1996.
• M. LeBrun, Posting to Math-Fun Mailing List Jan 10 1997.

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..10000
• Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, and Antonio Vera-López, Fibonacci-like partitions and their associated piecewise-defined permutations, arXiv:2503.19696 [math.CO], 2025. See p. 4.
• J. H. Conway, Allan Wechsler, Marc LeBrun, Dan Hoey, N. J. A. Sloane, On Kimberling Sums and Para-Fibonacci Sequences, Correspondence and Postings to Math-Fun Mailing List, Nov 1996 to Jan 1997

Crossrefs

Cf. A000201, A035513, A357097.

The "Kim-sums" K_n + K_i for i = 2 through 12 are given in A022413, A022414, A022415, ..., A022423.

Programs

Mathematica

a[n_] := If[n == 0, 1, Floor[n GoldenRatio] + n + 3];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Oct 15 2023 *)

Python

from math import isqrt
def A022413(n): return (n+isqrt(5*n**2)>>1)+n+3 if n else 1 # Chai Wah Wu, Aug 29 2022

Formula

a(n) = 1 if n=0, otherwise a(n) = A000201(n)+n+3. - N. J. A. Sloane, Mar 07 2016

Extensions

Edited and extended by N. J. A. Sloane, Mar 07 2016