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

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.

Sequence in context: A039505 A166875 A282109 * A128546 A188200 A060875

Adjacent sequences: A307860 A307861 A307862 * A307864 A307865 A307866

Keyword

nonn,base

Author

Paolo P. Lava, May 02 2019

Status

approved