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A205850 [s(k)-s(j)]/4, where the pairs (k,j) are given by A205847 and A205848, and s(k) denotes the (k+1)-st Fibonacci number.

%I A205850 #9 Oct 25 2024 04:53:37
%S A205850 1,3,2,5,4,2,8,13,22,21,19,17,34,58,57,55,53,36,94,93,91,89,72,36,152,
%T A205850 144,246,233,399,398,396,394,377,341,305,644,610,1045,1044,1042,1040,
%U A205850 1023,987,951,646,1691,1690,1688,1686,1669,1633,1597,1292,646
%N A205850 [s(k)-s(j)]/4, where the pairs (k,j) are given by A205847 and A205848, and s(k) denotes the (k+1)-st Fibonacci number.
%C A205850 For a guide to related sequences, see A205840.
%e A205850 The first six terms match these differences:
%e A205850 s(4)-s(1) = 5-1 = 4 = 4*1
%e A205850 s(6)-s(1) = 13-1 = 12= 4*3
%e A205850 s(6)-s(4) = 13-5 = 8 = 4*2
%e A205850 s(7)-s(1) = 21-1 = 20 = 4*5
%e A205850 s(7)-s(4) = 21-5 = 16 = 4*4
%e A205850 s(7)-s(6) = 21-13 = 8 = 4*2
%t A205850 s[n_] := s[n] = Fibonacci[n + 1]; z1 = 400; z2 = 60;
%t A205850 f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
%t A205850 Table[s[n], {n, 1, 30}]
%t A205850 u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
%t A205850 Table[u[m], {m, 1, z1}]  (* A204922 *)
%t A205850 v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
%t A205850 w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
%t A205850 d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]
%t A205850 c = 4; t = d[c]    (* A205846 *)
%t A205850 k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
%t A205850 j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
%t A205850 Table[k[n], {n, 1, z2}]    (* A205847 *)
%t A205850 Table[j[n], {n, 1, z2}]    (* A205848 *)
%t A205850 Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A205849 *)
%t A205850 Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205850 *)
%Y A205850 Cf. A204892, A205845, A205849.
%K A205850 nonn
%O A205850 1,2
%A A205850 _Clark Kimberling_, Feb 02 2012