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A205840 [s(k)-s(j)]/2, where the pairs (k,j) are given by A205837 and A205838.

%I A205840 #19 Oct 25 2024 04:52:31
%S A205840 1,2,1,3,6,5,4,10,9,8,4,16,13,27,26,25,21,17,44,43,42,38,34,17,71,68,
%T A205840 55,116,115,114,110,106,89,72,188,187,186,182,178,161,144,72,304,301,
%U A205840 288,233,493,492,491,487,483,466,449,377,305,798,797,796,792,788
%N A205840 [s(k)-s(j)]/2, where the pairs (k,j) are given by A205837 and A205838.
%C A205840 Let s(n)=F(n+1), where F=A000045 (Fibonacci numbers), so that s=(1,2,3,5,8,13,21,...).  If c is a positive integer, there are infinitely many pairs (k,j) such that c divides s(k)-s(j).  The set of differences s(k)-s(j) is ordered as a sequence at A204922.  Guide to related sequences:
%C A205840 c....k..........j..........s(k)-s(j)....[s(k)-s(j)]/c
%C A205840 2....A205837....A205838....A205839......A205840
%C A205840 3....A205842....A205843....A205844......A205845
%C A205840 4....A205847....A205848....A205849......A205850
%C A205840 5....A205852....A205853....A205854......A205855
%C A205840 6....A205857....A205858....A205859......A205860
%C A205840 7....A205862....A205863....A205864......A205865
%C A205840 8....A205867....A205868....A205869......A205870
%C A205840 9....A205872....A205873....A205874......A205875
%C A205840 10...A205877....A205878....A205879......A205880
%e A205840 The first six terms match these differences:
%e A205840 s(3)-s(1) = 3-1 = 2 = 2*1
%e A205840 s(4)-s(1) = 5-1 = 4 = 2*2
%e A205840 s(4)-s(3) = 5-3 = 2 = 2*1
%e A205840 s(5)-s(2) = 8-2 = 6 = 2*3
%e A205840 s(6)-s(1) = 13-1 = 12 = 2*6
%e A205840 s(6)-s(3) = 13-3 = 10 = 2*5
%t A205840 s[n_] := s[n] = Fibonacci[n + 1]; z1 = 400; z2 = 60;
%t A205840 f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
%t A205840 Table[s[n], {n, 1, 30}]
%t A205840 u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
%t A205840 Table[u[m], {m, 1, z1}]   (* A204922 *)
%t A205840 v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
%t A205840 w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
%t A205840 d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]
%t A205840 c = 2; t = d[c]    (* A205556 *)
%t A205840 k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
%t A205840 j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
%t A205840 Table[k[n], {n, 1, z2}]    (* A205837 *)
%t A205840 Table[j[n], {n, 1, z2}]    (* A205838 *)
%t A205840 Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A205839 *)
%t A205840 Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205840 *)
%Y A205840 Cf. A204890, A205587, A205839.
%K A205840 nonn
%O A205840 1,2
%A A205840 _Clark Kimberling_, Feb 01 2012