1,3

The Mathematica coding Robert G. Wilson v used is the Binet's Fibonacci number formula as suggested by David W. Wilson and further increase in speed by Benoit Cloitre's use of logarithms.

R. Knott, Fibonacci Numbers and the Golden Section

Eric Weisstein's World of Mathematics, Fibonacci numbers

n>5 is in the sequence if a=(1+sqrt(5))/2 b=1/sqrt(5) and n==floor(b*(a^n)/10^(floor((log(b) +n*log(a))/log(10))-floor(log(n)/log(10))) ) - Benoit Cloitre (benoit7848c(AT)orange.fr), Feb 27 2002

a(3)=43 since 43rd Fibonacci number starts with 43 -> {43}3494437.

Fibonacci(53) is 53316291173, which begins with 53, so 53 is a term in the sequence.

a = N[ Log[10, Sqrt[5]/5], 24]; b = N [Log[10, GoldenRatio], 24]; Do[ If[ IntegerPart[10^FractionalPart[a + n*b]*10^Floor[ Log[10, n]]] == n, Print[n]], {n, 225000000}] (from Robert G. Wilson v (rgwv(AT)rgwv.com), May 09 2005)

confirmed with fQ[n_] := (FromDigits[ Take[ IntegerDigits[ Fibonacci[n]], Floor[ Log[10, n] + 1]]] == n)

(PARI) To obtain terms > 5: a=(1+sqrt(5))/2; b=1/sqrt(5); for(n=1, 3500, if(n==floor(b*(a^n)/10^( floor(log(b *(a^n))/log(10))-floor(log(n)/log(10)))), print1(n, ", "))) - Benoit Cloitre Feb 27 2002

Cf. A000045, A052000, A000350, A050816.

Adjacent sequences: A038543 A038544 A038545 this_sequence A038547 A038548 A038549

Sequence in context: A102244 A132487 A067927 this_sequence A022891 A106940 A106941

nonn,base,nice

Jeff Burch (gburch(AT)erols.com)

Term a(6) from Patrick De Geest (pdg(AT)worldofnumbers.com), Oct 15 1999.

a(7) from Benoit Cloitre, Feb 27 2002

a(8), a(9), a(10) & a(11) from Robert G. Wilson v (rgwv(AT)rgwv.com), May 09 2005

a(12) from Robert G. Wilson v (rgwv(AT)rgwv.com), May 11 2005

More terms from Robert Gerbicz (robert.gerbicz(AT)gmail.com), Aug 22 2006