A142705 Numerator of 1/4 - 1/(2n)^2.

0, 3, 2, 15, 6, 35, 12, 63, 20, 99, 30, 143, 42, 195, 56, 255, 72, 323, 90, 399, 110, 483, 132, 575, 156, 675, 182, 783, 210, 899, 240, 1023, 272, 1155, 306, 1295, 342, 1443, 380, 1599, 420, 1763, 462, 1935, 506, 2115, 552, 2303, 600, 2499, 650, 2703, 702

Offset: 1

Author

Paul Curtz, Sep 24 2008

Keywords

• nonn
• easy
• frac

Comments

Read modulo 10 (the last digits), a sequence with period length 10 results: 0, 3, 2, 5, 6, 5, 2, 3, 0, 9. Read modulo 9, a sequence with period length 18 results.

Denominators are in A154615.

a(n) is the numerator of (n-1)*(n+1)/4. - Altug Alkan, Apr 19 2018

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Crossrefs

Essentially the same as A070260. Cf. A078371 (second bisection of A061037), A142888 (first differences), A154615 (denominators), A225948.

Programs

Magma

[-(3/4)*(-1)^n*n-(3/8)*(-1)^n*n^2+(5/8)*n^2+(5/4)*n: n in [0..60]]; // Vincenzo Librandi, Jul 02 2011

Mathematica

Numerator[Table[(1/4)*(1 - 1/n^2), {n,1,50}]] (* G. C. Greubel, Jul 20 2017 *)

PARI

for(n=1, 50, print1(numerator((1/4)*(1 - 1/n^2)), ", ")) \\ G. C. Greubel, Jul 20 2017

PARI

a(n) = if(n%2,(n^2-1)/4,n^2-1); \\ Altug Alkan, Apr 19 2018

Formula

a(n) = A061037(2*n).

a(n) = A070260(n-1), n>1.

a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6).

a(2^(n-1)) = a(1+A000225(n-1)) = 4^(n-1)-1 = A024036(n-1).

G.f.: x^2*(3+2x+6x^2-x^4)/(1-x^2)^3. - R. J. Mathar, Oct 24 2008

E.g.f.: 1 + (1/4)*((4*x^2 + x - 4)*cosh(x) + (x^2 + 4*x -1)*sinh(x)). - G. C. Greubel, Jul 20 2017

Sum_{n>=2} 1/a(n) = 3/2. - Amiram Eldar, Aug 11 2022

Extensions

Edited by R. J. Mathar, Oct 24 2008