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A142705 Numerator of 1/4 - 1/(2n)^2. 8

%I #41 Mar 15 2024 02:19:35

%S 0,3,2,15,6,35,12,63,20,99,30,143,42,195,56,255,72,323,90,399,110,483,

%T 132,575,156,675,182,783,210,899,240,1023,272,1155,306,1295,342,1443,

%U 380,1599,420,1763,462,1935,506,2115,552,2303,600,2499,650,2703,702

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

%C 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 Denominators are in A154615.

%C a(n) is the numerator of (n-1)*(n+1)/4. - _Altug Alkan_, Apr 19 2018

%H Vincenzo Librandi, <a href="/A142705/b142705.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-3,0,1).

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

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

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

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

%F G.f.: x^2*(3+2x+6x^2-x^4)/(1-x^2)^3. - _R. J. Mathar_, Oct 24 2008

%F 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 Sum_{n>=2} 1/a(n) = 3/2. - _Amiram Eldar_, Aug 11 2022

%t Numerator[Table[(1/4)*(1 - 1/n^2), {n,1,50}]] (* _G. C. Greubel_, Jul 20 2017 *)

%o (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 (PARI) for(n=1, 50, print1(numerator((1/4)*(1 - 1/n^2)), ", ")) \\ _G. C. Greubel_, Jul 20 2017

%o (PARI) a(n) = if(n%2,(n^2-1)/4,n^2-1); \\ _Altug Alkan_, Apr 19 2018

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

%K nonn,easy,frac,changed

%O 1,2

%A _Paul Curtz_, Sep 24 2008

%E Edited by _R. J. Mathar_, Oct 24 2008

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