A059403 - OEIS

%I #34 Dec 29 2024 21:01:11

%S 0,0,0,1,4,9,20,36,64,100,156,225,324,441,600,784,1024,1296,1640,2025,

%T 2500,3025,3660,4356,5184,6084,7140,8281,9604,11025,12656,14400,16384,

%U 18496,20880,23409,26244,29241,32580,36100,40000,44100,48620,53361,58564,64009

%N Quarter-squared applied twice.

%H Harvey P. Dale, <a href="/A059403/b059403.txt">Table of n, a(n) for n = 0..1000</a> (first 501 terms from Harry J. Smith)

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (2, 1, -4, 2, 0, -2, 4, -1, -2, 1).

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

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

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

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

%F 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

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

%t 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 *)

%o (PARI) a(n) = { (n^2\4)^2\4 } \\ _Harry J. Smith_, Jun 26 2009

%Y Cf. A008233 for an alternative approach.

%K easy,nonn

%O 0,5

%A _Henry Bottomley_, Mar 21 2001