A226008 a(0) = 0; for n>0, a(n) = denominator(1/4 - 4/n^2).

0, 4, 4, 36, 1, 100, 36, 196, 16, 324, 100, 484, 9, 676, 196, 900, 64, 1156, 324, 1444, 25, 1764, 484, 2116, 144, 2500, 676, 2916, 49, 3364, 900, 3844, 256, 4356, 1156, 4900, 81, 5476, 1444, 6084, 400, 6724, 1764, 7396, 121, 8100

Offset: 0

Author

Paul Curtz, May 22 2013

Keywords

• nonn
• frac
• easy

Comments

Numerators are in A225948.

Repeated terms of A016826 are in the positions 1, 2, 3, 6, 5, 10, ... (A043547).

Examples

			a(0) = (-1+1)^2 = 0, a(1) = (-3+5)^2 = 4, a(2) = (-1+3)^2 = 4.
		

Crossrefs

Cf. A016754, A016802, A016826, A017090, A017138, A154615, A225948 (numerators).

Cf. A225975 (associated square roots).

Programs

Magma

[0] cat [Denominator(1/4-4/n^2): n in [1..50]]; // Bruno Berselli, May 23 2013

Mathematica

Join[{0},Table[Denominator[1/4 - 4/n^2], {n, 49}]] (* Alonso del Arte, May 22 2013 *)

Formula

a(n) = 3*a(n-8) -3*a(n-16) +a(n-24).

a(8n) = A016802(n), a(8n+4) = A016754(n).

a(4n) = A154615(n).

a(4n+1) = A017090(n).

a(4n+2) = a(2n+1) = A016826(n); a(2n) = A061038(n).

a(4n+3) = A017138(n).

From Bruno Berselli, May 23 2013: (Start)

G.f.: x*(4 +4*x +36*x^2 +x^3 +100*x^4 +36*x^5 +196*x^6 +16*x^7 +312*x^8 +88*x^9 +376*x^10 +6*x^11 +376*x^12 +88*x^13 +312*x^14 +16*x^15 +196*x^16 +36*x^17 +100*x^18 +x^19 +36*x^20 +4*x^21 +4*x^22)/(1-x^8)^3.

a(n) = n^2*(6*cos(3*Pi*n/4)+6*cos(Pi*n/4)-54*cos(Pi*n/2)-219*(-1)^n+293)/128.

a(n+9) = a(n+1)*((n+9)/(n+1))^2. (End)

Sum_{n>=1} 1/a(n) = 19*Pi^2/96. - Amiram Eldar, Aug 14 2022

Extensions

Edited by Bruno Berselli, May 23 2013