cp's OEIS Frontend

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

%I A142705 #41 Mar 15 2024 02:19:35
%S A142705 0,3,2,15,6,35,12,63,20,99,30,143,42,195,56,255,72,323,90,399,110,483,
%T A142705 132,575,156,675,182,783,210,899,240,1023,272,1155,306,1295,342,1443,
%U A142705 380,1599,420,1763,462,1935,506,2115,552,2303,600,2499,650,2703,702
%N A142705 Numerator of 1/4 - 1/(2n)^2.
%C A142705 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.
%C A142705 Denominators are in A154615.
%C A142705 a(n) is the numerator of (n-1)*(n+1)/4. - _Altug Alkan_, Apr 19 2018
%H A142705 Vincenzo Librandi, <a href="/A142705/b142705.txt">Table of n, a(n) for n = 1..10000</a>
%H A142705 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-3,0,1).
%F A142705 a(n) = A061037(2*n).
%F A142705 a(n) = A070260(n-1), n>1.
%F A142705 a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6).
%F A142705 a(2^(n-1)) = a(1+A000225(n-1)) = 4^(n-1)-1 = A024036(n-1).
%F A142705 G.f.: x^2*(3+2x+6x^2-x^4)/(1-x^2)^3. - _R. J. Mathar_, Oct 24 2008
%F A142705 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
%F A142705 Sum_{n>=2} 1/a(n) = 3/2. - _Amiram Eldar_, Aug 11 2022
%t A142705 Numerator[Table[(1/4)*(1 - 1/n^2), {n,1,50}]] (* _G. C. Greubel_, Jul 20 2017 *)
%o A142705 (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
%o A142705 (PARI) for(n=1, 50, print1(numerator((1/4)*(1 - 1/n^2)), ", ")) \\ _G. C. Greubel_, Jul 20 2017
%o A142705 (PARI) a(n) = if(n%2,(n^2-1)/4,n^2-1); \\ _Altug Alkan_, Apr 19 2018
%Y A142705 Essentially the same as A070260. Cf. A078371 (second bisection of A061037), A142888 (first differences), A154615 (denominators), A225948.
%K A142705 nonn,easy,frac
%O A142705 1,2
%A A142705 _Paul Curtz_, Sep 24 2008
%E A142705 Edited by _R. J. Mathar_, Oct 24 2008