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 A205845 [s(k)-s(j)]/3, where the pairs (k,j) are given by A205842 and A205843, and s(k) denotes the (k+1)-st Fibonacci number. 5

%I

%S 1,2,1,4,6,11,7,18,14,7,29,28,27,47,41,77,76,75,48,125,124,123,96,48,

%T 203,199,192,185,328,322,281,532,528,521,514,329,861,857,850,843,658,

%U 329,1393,1392,1391,1364,1316,1268,2254,2248,2207,1926,3648

%N [s(k)-s(j)]/3, where the pairs (k,j) are given by A205842 and A205843, and s(k) denotes the (k+1)-st Fibonacci number.

%C For a guide to related sequences, see A205840.

%C The first six terms match these differences:

%C s(4)-s(2) = 5-2 = 3 = 3*1

%C s(5)-s(2) = 8-2 = 6 = 3*2

%C s(5)-s(4) = 8-5 = 3 = 3*1

%C s(6)-s(1) = 13-1 = 12 = 3*4

%C s(7)-s(3) = 21-3 = 18 = 3*6

%C s(8)-s(1) = 34-1 = 33 + 3*11

%C (See the program at A205842.)

%e The first six terms match these differences:

%e s(4)-s(2) = 5-2 = 3 = 3*1

%e s(5)-s(2) = 8-2 = 6 = 3*2

%e s(5)-s(4) = 8-5 = 3 = 3*1

%e s(6)-s(1) = 13-1 = 12 = 3*4

%e s(7)-s(3) = 21-3 = 18 = 3*6

%e s(8)-s(1) = 34-1 = 33 + 3*11

%t s[n_] := s[n] = Fibonacci[n + 1]; z1 = 400; z2 = 60;

%t f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];

%t Table[s[n], {n, 1, 30}]

%t u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]

%t Table[u[m], {m, 1, z1}] (* A204922 *)

%t v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]

%t w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]

%t d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]

%t c = 3; t = d[c] (* A205841 *)

%t k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]

%t j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2

%t Table[k[n], {n, 1, z2}] (* A205842 *)

%t Table[j[n], {n, 1, z2}] (* A205843 *)

%t Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A205844 *)

%t Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205845 *)

%Y Cf. A204890, A205842, A205845.

%K nonn

%O 1,2

%A _Clark Kimberling_, Feb 01 2012

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