%I #20 Nov 21 2018 03:18:07

%S 1,2,3,2,2,2,3,2,4,2,1,2,2,2,3,2,3,2,3,2,4,2,3,2,2,2,3,2,3,2,3,2,4,2,

%T 3,2,2,2,3,2,3,2,3,2,4,2,3,2,3,2,3,2,3,2,3,2,4,2,3,2,2,2,3,2,3,2,3,2,

%U 4,2,3,2,2,2,3,2,3,2,3,2,4,2,4,2,2,2,3,2,3,2,3,2,4,2,3,2,2,2,3,2,3,2,3,2,4

%N Minimal number of squares needed to sum to Fibonacci(n+1).

%C Since every positive integer is the sum of four squares, no term is greater than 4. Also, since any positive integer not of the form 4^k(8m+7) is the sum 3 or fewer squares, the next occurrences of a(n)=4 are at n = 45, 57, 69, 81, 83, 93, .... - _John W. Layman_, Mar 30 2005

%D Hardy and Wright, An Introduction to the Theory of Numbers, Fourth Ed., Oxford, Section 20.10.

%H Hans Havermann, <a href="/A103266/b103266.txt">Table of n, a(n) for n = 1..1400</a> (terms 1..465 from Antti Karttunen)

%F a(n) = A002828(A000045(n+1)).

%e Fibonacci(10+1) = 89 = 25+64, so a(10)=2.

%t Array[If[First[#] > 0, 1, Length@ First@ Split@ # + 1] &@ SquaresR[Range@ 4, Fibonacci@ #] &, 50, 2] (* _Michael De Vlieger_, Nov 13 2018, after _Harvey P. Dale_ at A002828 *)

%o (PARI)

%o istwo(n:int) = { my(f); if(n<3, return(n>=0); ); f=factor(n>>valuation(n, 2)); for(i=1, #f[, 1], if(bitand(f[i, 2], 1)==1&&bitand(f[i, 1], 3)==3, return(0))); 1 };

%o isthree(n:int) = { my(tmp=valuation(n, 2)); bitand(tmp, 1)||bitand(n>>tmp, 7)!=7 };

%o A002828(n) = if(issquare(n), !!n, if(istwo(n), 2, 4-isthree(n))); \\ From A002828

%o A103266(n) = A002828(fibonacci(1+n)); \\ _Antti Karttunen_, Nov 10 2018

%Y Cf. A000045, A002828.

%K nonn

%O 1,2

%A _Giovanni Teofilatto_, Mar 20 2005

%E Corrected and extended by _John W. Layman_, Mar 30 2005

%E Extended by _Ray Chandler_, May 16 2005