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a(1) = 1; a(n+1) = smallest Fibonacci number > a(n) with leading digit equal to the final digit of a(n).
1

%I #27 May 03 2016 23:14:41

%S 1,13,34,4181,10946,63245986,6557470319842,27777890035288,

%T 806515533049393,3416454622906707,7540113804746346429,

%U 927372692193078999176,6356306993006846248183,3311648143516982017180081,14028366653498915298923761,155576970220531065681649693,30960598847965113057878492344

%N a(1) = 1; a(n+1) = smallest Fibonacci number > a(n) with leading digit equal to the final digit of a(n).

%C Look at this array:

%C ...1

%C ...13

%C ....34

%C .....4181

%C ........10946

%C ............63245986

%C ...................6557470319842

%C .

%C .

%C .

%C ................................34507973060837282187130139035400899082304280

%C Since no positive Fibonacci number begins with 0, the sequence ends here.

%C - _N. J. A. Sloane_, May 02 2016

%H Chai Wah Wu, <a href="/A271750/b271750.txt">Table of n, a(n) for n = 1..25</a>

%e Since a(1) = 1, a(2) must start with 1 and so a(2) = 13;

%e since a(2) = 13, a(3) must start with 3, and so a(3) = 34;

%e since a(3) = 34, a(4) must start with 4 and the smallest Fibonacci number greater than 34 and beginning with 4 is 4181; ...

%t F = Fibonacci; L={2}; While[(d = Mod[F[k = Last@L], 10]) > 0, While[ IntegerDigits[ F[++k]][[1]] != d]; AppendTo[L, k]]; F@L (* _Giovanni Resta_, May 02 2016 *)

%Y Cf. A061448.

%K nonn,base,fini,full

%O 1,2

%A _José de Jesús Camacho Medina_, Apr 13 2016