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

%I A022413 #34 Aug 03 2025 21:04:33
%S A022413 1,5,8,10,13,16,18,21,23,26,29,31,34,37,39,42,44,47,50,52,55,57,60,63,
%T A022413 65,68,71,73,76,78,81,84,86,89,92,94,97,99,102,105,107,110,112,115,
%U A022413 118,120,123,126,128,131,133,136,139,141,144,146,149,152,154,157,160,162,165,167,170,173,175,178
%N A022413 Kim-sums: "Kimberling sums" K_n + K_2.
%C A022413 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
%C A022413 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
%D A022413 J. H. Conway, Posting to Math Fun Mailing List, Dec 02 1996.
%D A022413 M. LeBrun, Posting to Math-Fun Mailing List Jan 10 1997.
%H A022413 N. J. A. Sloane, <a href="/A022413/b022413.txt">Table of n, a(n) for n = 0..10000</a>
%H A022413 Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, and Antonio Vera-López, <a href="https://arxiv.org/abs/2503.19696">Fibonacci-like partitions and their associated piecewise-defined permutations</a>, arXiv:2503.19696 [math.CO], 2025. See p. 4.
%H A022413 J. H. Conway, Allan Wechsler, Marc LeBrun, Dan Hoey, N. J. A. Sloane, <a href="/A269725/a269725.txt">On Kimberling Sums and Para-Fibonacci Sequences</a>, Correspondence and Postings to Math-Fun Mailing List, Nov 1996 to Jan 1997
%F A022413 a(n) = 1 if n=0, otherwise a(n) = A000201(n)+n+3. - _N. J. A. Sloane_, Mar 07 2016
%t A022413 a[n_] := If[n == 0, 1, Floor[n GoldenRatio] + n + 3];
%t A022413 Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Oct 15 2023 *)
%o A022413 (Python)
%o A022413 from math import isqrt
%o A022413 def A022413(n): return (n+isqrt(5*n**2)>>1)+n+3 if n else 1 # _Chai Wah Wu_, Aug 29 2022
%Y A022413 Cf. A000201, A035513, A357097.
%Y A022413 The "Kim-sums" K_n + K_i for i = 2 through 12 are given in A022413, A022414, A022415, ..., A022423.
%K A022413 nonn
%O A022413 0,2
%A A022413 _Marc LeBrun_
%E A022413 Edited and extended by _N. J. A. Sloane_, Mar 07 2016