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%I A103266 #20 Nov 21 2018 03:18:07
%S A103266 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 A103266 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 A103266 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 A103266 Minimal number of squares needed to sum to Fibonacci(n+1).
%C A103266 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 A103266 Hardy and Wright, An Introduction to the Theory of Numbers, Fourth Ed., Oxford, Section 20.10.
%H A103266 Hans Havermann, <a href="/A103266/b103266.txt">Table of n, a(n) for n = 1..1400</a> (terms 1..465 from Antti Karttunen)
%F A103266 a(n) = A002828(A000045(n+1)).
%e A103266 Fibonacci(10+1) = 89 = 25+64, so a(10)=2.
%t A103266 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 A103266 (PARI)
%o A103266 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 A103266 isthree(n:int) = { my(tmp=valuation(n, 2)); bitand(tmp, 1)||bitand(n>>tmp, 7)!=7 };
%o A103266 A002828(n) = if(issquare(n), !!n, if(istwo(n), 2, 4-isthree(n))); \\ From A002828
%o A103266 A103266(n) = A002828(fibonacci(1+n)); \\ _Antti Karttunen_, Nov 10 2018
%Y A103266 Cf. A000045, A002828.
%K A103266 nonn
%O A103266 1,2
%A A103266 _Giovanni Teofilatto_, Mar 20 2005
%E A103266 Corrected and extended by _John W. Layman_, Mar 30 2005
%E A103266 Extended by _Ray Chandler_, May 16 2005

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