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Number of sequences F such that F(k) = F(k-1) + F(k-2), F(1), F(2) are positive integers, and there exists some integer x>2 such that F(x) = n.

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`%I #19 Sep 25 2023 11:40:36
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`%S 0,1,3,4,7,7,10,12,13,14,18,17,22,22,23,25,28,29,31,32,36,35,40,38,41,
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`%T 44,44,47,51,48,53,53,56,59,59,60,64,65,66,66,71,71,74,75,77,78,83,81,
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`%U 84,86,87,88,94,91,97,96,97,101,102,103,107,106,110,109,112
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`%N Number of sequences F such that F(k) = F(k-1) + F(k-2), F(1), F(2) are positive integers, and there exists some integer x>2 such that F(x) = n.
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`%C This is the number of Fibonacci-style sequences seeded with positive integers that contain each n, after the seeds.
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`%H Giovanni Resta, <a href="/A261368/b261368.txt">Table of n, a(n) for n = 1..10000</a>
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`%e For n=4, a(4) = 4 as the sequences <1,3,4,7...>, <2,1,3,4...>, <2,2,4,6...>, and <3,1,4,5...> each contain 4 outside of the initial 2 numbers.
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`%t a[n_] := Sum[Block[{s, L={x, y}}, While[(s = Total@L ) < n, L = Rest@ Append[L, s]]; If[s == n, 1, 0]], {x, n-1}, {y,n-x}]; Array[a, 65] (* _Giovanni Resta_, Aug 17 2015 *)
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`%o (PARI) isok(x, y, n) = {ny = 0; while (ny <= n, ny = x + y; if (ny == n, return (1)); x = y; y = ny;); return (0);}
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`%o a(n) = {nb = 0; for (j=1, n-1, for (k=1, n-j, if (isok(j, k, n), nb++););); nb;} \\ _Michel Marcus_, Aug 17 2015
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`%K nonn
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`%O 1,3
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`%A _Ben Johnsrude_, Aug 16 2015
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`%E More terms from _Michel Marcus_, Aug 17 2015
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