A307863 - OEIS

A307863

Numbers x = concat(a,b) such that b and a are the first two terms for a Fibonacci-like sequence containing x itself.

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

1,1

COMMENTS

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?

LINKS

Giovanni Resta, Table of n, a(n) for n = 1..900

Paolo P. Lava, Values of x, b, a, y, for 1< x <10^6

EXAMPLE

123 can be split into 1 and 23 and the Fibonacci-like sequence: 23, 1, 24, 25, 49, 74, 123, ... contains 123 itself.

MAPLE

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

CROSSREFS

Cf. A128546, A130792, A289868.