A222740 Denominators of 1/16 - 1/(4 + 8*n)^2.

1, 18, 50, 49, 81, 242, 338, 225, 289, 722, 882, 529, 625, 1458, 1682, 961, 1089, 2450, 2738, 1521, 1681, 3698, 4050, 2209, 2401, 5202, 5618, 3025, 3249, 6962, 7442, 3969, 4225, 8978, 9522, 5041, 5329, 11250, 11858, 6241

Offset

0,2

Comments

Denominators of the reduced fractions A064038(n)/a(n) = 0/1, 1/18, 3/50, 3/49, 5/81, 15/242, 21/338, 14/225, 18/289, ... .

Also, A064038 and a(n) are related to the sequence of period 4: repeat 1, 2, 2, 1.

Links

Table of n, a(n) for n=0..39.

Index entries for linear recurrences with constant coefficients, signature (3,-6,10,-12,12,-10,6,-3,1).

Formula

a(n) = A014695(n) * A016754(n).

a(n) = 16*A064038(n+1) + A014695(n).

a(n) = A061042(4+8*n).

a(2n+2) - a(2n+1) = 32*A026741(n+1).

G.f.: ( -1 - 15*x - 2*x^2 + 3*x^3 - 66*x^4 + 3*x^5 - 2*x^6 - 15*x^7 - x^8 ) / ( (x-1)^3*(x^2+1)^3 ). - R. J. Mathar, Jun 04 2013

a(n) = (3-sqrt(2)*cos((2*n+1)*Pi/4))*(2*n+1)^2/2. - Wesley Ivan Hurt, Oct 04 2018

Example

a(0) = 1*1, a(1) = 2*9 = 18, a(2) = 2*25 = 50, a(3) = 1*49 = 49.
a(0) = 16*0 + 1 = 1, a(1) = 16*1 + 2 = 18, a(2) = 16*3 + 2 = 50, a(3) = 16*3 + 1 = 49.

Mathematica

Table[1/16-1/(4+8n)^2, {n, 0, 40}]//Denominator (* or *) LinearRecurrence[ {3, -6, 10, -12, 12, -10, 6, -3, 1}, {1, 18, 50, 49, 81, 242, 338, 225, 289}, 40] (* Harvey P. Dale, Aug 30 2021 *)

Crossrefs

Sequence in context: A234956 A135189 A178398 * A335377 A317258 A071365

Adjacent sequences: A222737 A222738 A222739 * A222741 A222742 A222743

Keyword

nonn,frac,easy

Author

Paul Curtz, May 29 2013

Status

approved