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A307863
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Numbers x = concat(a,b) such that b and a are the first two terms for a Fibonacci-like sequence containing x itself.
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1
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17, 21, 25, 42, 63, 84, 105, 123, 126, 147, 168, 189, 197, 246, 295, 369, 492, 787, 1033, 1115, 1141, 1248, 1279, 1997, 2066, 2230, 2282, 2496, 2995, 3099, 3345, 3423, 3744, 4460, 4564, 4992, 5411, 5575, 5705, 6690, 6846, 7987, 10112, 10483, 10822, 11059, 11107
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OFFSET
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1,1
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COMMENTS
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Similar to A130792 but here the sums start b + a = c, a + c = d, etc.
First six terms are also the first six Inrepfigit numbers (A128546).
Being x = concat(a,b), the problem is to find an index y such that x = b*F(y) + a*F(y+1), where F(y) is a Fibonacci number (see file with values of x, b, a, y, for 1< x <10^6, in Links). All the listed numbers admit only one unique concatenation that, through the addition process, leads to themselves. Is there any number that admits more than one single concatenation?
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LINKS
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EXAMPLE
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123 can be split into 1 and 23 and the Fibonacci-like sequence: 23, 1, 24, 25, 49, 74, 123, ... contains 123 itself.
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MAPLE
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P:=proc(n) local j, t, v; v:=array(1..100);
for j from 1 to length(n)-1 do v[1]:=n mod 10^j; v[2]:=trunc(n/10^j);
v[3]:=v[1]+v[2]; t:=3; while v[t]<n do t:=t+1; v[t]:=v[t-2]+v[t-1]; od;
if v[t]=n then RETURN(n); break; fi; od; end: seq(P(i), i=1..11107); # Paolo P. Lava, May 02 2019
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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