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

0,5

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).

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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A008233 for an alternative approach.

Sequence in context: A164931 A066186 A346558 * A009909 A009910 A060494

Adjacent sequences: A059400 A059401 A059402 * A059404 A059405 A059406

Keyword

easy,nonn

Author

Henry Bottomley, Mar 21 2001

Status

approved