A059403 Quarter-squared applied twice.

0, 0, 0, 1, 4, 9, 20, 36, 64, 100, 156, 225, 324, 441, 600, 784, 1024, 1296, 1640, 2025, 2500, 3025, 3660, 4356, 5184, 6084, 7140, 8281, 9604, 11025, 12656, 14400, 16384, 18496, 20880, 23409, 26244, 29241, 32580, 36100, 40000, 44100, 48620, 53361, 58564, 64009

Offset: 0

Author

Henry Bottomley, Mar 21 2001

Keywords

• easy
• nonn

Examples

			a(9)=100 since the ninth quarter-square is 20 and the twentieth quarter-square is 100.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000 (first 501 terms from Harry J. Smith)
• Index entries for linear recurrences with constant coefficients, signature (2, 1, -4, 2, 0, -2, 4, -1, -2, 1).

Crossrefs

Cf. A008233 for an alternative approach.

Programs

Mathematica

Floor[Floor[Range[0,50]^2/4]^2/4] (* or *) LinearRecurrence[{2,1,-4,2,0,-2,4,-1,-2,1},{0,0,0,1,4,9,20,36,64,100},50] (* Harvey P. Dale, Dec 13 2014 *)

PARI

a(n) = { (n^2\4)^2\4 } \\ Harry J. Smith, Jun 26 2009

Formula

a(n) = floor(floor(n^2/4)^2/4) = A002620(A002620(n)).

a(4*n) = 4n^4; a(4*n+1) = n^2*(2*n+1)^2;

a(4*n+2) = 2*n*(n+1)*(2*n*(n+1)+1); a(4*n+3) = (n+1)^2*(2*n+1)^2.

a(2n) = A060494(2n); a(2n-1) = A060494(2n-1)-A011861(n).

G.f.: x^3*(1 + 2*x + 2*x^3 + x^4)/((1 - x)^5*(1 + x)^3*(1 + x^2)). - R. J. Mathar, Sep 09 2008