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A022414 Kim-sums: "Kimberling sums" K_n + K_3.

%I A022414 #27 Oct 15 2023 12:25:58
%S A022414 2,7,10,4,15,18,20,23,9,28,31,12,36,39,41,44,17,49,52,54,57,22,62,65,
%T A022414 25,70,73,75,78,30,83,86,33,91,94,96,99,38,104,107,109,112,43,117,120,
%U A022414 46,125,128,130,133,51,138,141,143,146,56,151,154,59,159,162,164,167
%N A022414 Kim-sums: "Kimberling sums" K_n + K_3.
%C A022414 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 a(n) = W(2,n). - _N. J. A. Sloane_, Mar 07 2016
%D A022414 J. H. Conway, Posting to Math Fun Mailing List, Dec 02 1996.
%D A022414 M. LeBrun, Posting to Math Fun Mailing List Jan 10 1997.
%H A022414 J. H. Conway, Allan Wechsler, Marc LeBrun, Dan Hoey, and 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
%p A022414 Ki := proc(n,i)
%p A022414         option remember;
%p A022414         local phi ;
%p A022414         phi := (1+sqrt(5))/2 ;
%p A022414         if i= 0 then
%p A022414                 n;
%p A022414         elif i=1 then
%p A022414                 floor((n+1)*phi) ;
%p A022414         else
%p A022414                 procname(n,i-1)+procname(n,i-2) ;
%p A022414         end if;
%p A022414 end proc:
%p A022414 Kisum := proc(n,m)
%p A022414         local ks,a,i;
%p A022414         ks := [seq( Ki(n,i)+Ki(m,i),i=0..5)] ;
%p A022414         for i from 0 to 2 do
%p A022414                 for a from 0 do
%p A022414                         if Ki(a,0) = ks[i+1] and Ki(a,1) = ks[i+2] then
%p A022414                                 return a;
%p A022414                         end if;
%p A022414                         if Ki(a,0) > ks[i+1] then
%p A022414                                 break;
%p A022414                         end if;
%p A022414                 end do:
%p A022414         end do:
%p A022414 end proc:
%p A022414 A022414 := proc(n)
%p A022414     if n = 0 then
%p A022414         2;
%p A022414     else
%p A022414             Kisum(n-1,2) ;
%p A022414     end if;
%p A022414 end proc:
%p A022414 seq(A022414(i),i=0..80) ; # _R. J. Mathar_, Sep 03 2016
%t A022414 Ki[n_, i_] := Ki[n, i] = Which[i == 0, n, i == 1, Floor[(n + 1)* GoldenRatio], True, Ki[n, i - 1] + Ki[n, i - 2]];
%t A022414 Kisum [n_, m_] := Module[{ks, a, i}, ks = Table[Ki[n, i] + Ki[m, i], {i, 0, 5}]; For[i = 0, i <= 2, i++, For[a = 0, True, a++, If[Ki[a, 0] == ks[[i + 1]] && Ki[a, 1] == ks[[i + 2]], Return@a]; If[Ki[a, 0] > ks[[i + 1]], Break[]]]]];
%t A022414 a[n_] := If[n == 0, 2, Kisum[n - 1, 2]];
%t A022414 Table[a[n], {n, 0, 80}] (* _Jean-François Alcover_, Oct 15 2023, after _R. J. Mathar_ *)
%Y A022414 Cf. A000201, A035513.
%Y A022414 The "Kim-sums" K_n + K_i for i = 2 through 12 are given in A022413, A022414, A022415, A022416, ..., A022423.
%K A022414 nonn,easy
%O A022414 0,1
%A A022414 _Marc LeBrun_