A118487 Least number of squares that add up to Lucas numbers L(n).

2, 1, 3, 1, 4, 3, 2, 2, 4, 3, 3, 4, 3, 2, 3, 3, 4, 3, 3, 2, 4, 4, 3, 4, 3, 3, 3, 3, 4, 3, 2, 2, 4, 3, 3, 4, 3, 2, 3, 3, 4, 3, 3, 2, 4, 4, 3, 4, 3, 2, 3, 3, 4, 3, 3, 3, 4, 3, 3, 4, 3, 2, 3, 3, 4, 3, 3, 2, 4, 4, 3, 4, 3, 2, 3, 3, 4, 3, 2, 2, 4, 3, 3, 4, 3, 3, 3, 3, 4, 3, 3, 2, 4, 4, 3, 4, 3, 3, 3, 3, 4

Offset: 1

Author

Jonathan Vos Post, May 16 2006

Keywords

• easy
• nonn

Comments

By the "Four Squares theorem", a(n) <= 4. Any positive integer not of the form 4^k(8m+7) is the sum 3 or fewer squares. See also: A000032 Lucas numbers. See also: A103266 Minimal number of squares needed to sum to Fibonacci(n+1). See also: A000045 Fibonacci numbers: F(n) = F(n-1) + F(n-2), F(0) = 0, F(1) = 1, F(2) = 1, ... See also: A002828 Least number of squares that add up to n.

Examples

			a(4) = 4 because L(4) = 7 = 2^2 + 2^2 + 1^1 + 1^1 is the minimum representation as sum of squares, in this case of 4 squares.
a(20) = 4 because L(20) = 15127 = 74^2 + 73^2 + 59^2 + 29^2.
a(30) = 2 because L(30) = 1860498 = 1077^2 + 837^2.
a(100) = 4 because L(100) = 16930663951^2 + 16706810102^2 + 13499760391^2 + 6637953271^2.
		

References

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

Crossrefs

Cf. A000032, A000045, A002828, A103266.

Formula

a(n) = A002828(A000032(n)).