A103266 Minimal number of squares needed to sum to Fibonacci(n+1).

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, 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, 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

Offset: 1

Author

Giovanni Teofilatto, Mar 20 2005

Keywords

• nonn

Comments

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

Examples

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

References

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

Links

• Hans Havermann, Table of n, a(n) for n = 1..1400 (terms 1..465 from Antti Karttunen)

Crossrefs

Cf. A000045, A002828.

Programs

Mathematica

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 *)

PARI

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 };
isthree(n:int) = { my(tmp=valuation(n, 2)); bitand(tmp, 1)||bitand(n>>tmp, 7)!=7 };
A002828(n) = if(issquare(n), !!n, if(istwo(n), 2, 4-isthree(n))); \\ From A002828
A103266(n) = A002828(fibonacci(1+n)); \\ Antti Karttunen, Nov 10 2018

Formula

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

Extensions

Corrected and extended by John W. Layman, Mar 30 2005

Extended by Ray Chandler, May 16 2005