A377226 - OEIS

%I #18 Oct 21 2024 04:40:06

%S 1,5,-3,9,13,-7,17,21,-11,25,29,-15,33,37,-19,41,45,-23,49,53,-27,57,

%T 61,-31,65,69,-35,73,77,-39,81,85,-43,89,93,-47,97,101,-51,105,109,

%U -55,113,117,-59,121,125,-63,129,133,-67,137,141,-71,145,149,-75,153,157,-79,161

%N Take the sequence of the signed denominators of Leibniz series for Pi/4 (cf. A157142) and permute the terms so that a negative term follows every two positive terms and the absolute difference between two consecutive terms of the same sign is 4.

%D Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.4.1, p. 20.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,2,0,0,-1).

%F a(n) = 2*a(n-3) - a(n-6) for n > 5.

%F a(n)= (9 + 12*n - 6*n*A061347(n) + 6*(2 + 3*n)*A049347(n+2))/9.

%F G.f.: (1 + 5*x - 3*x^2 + 7*x^3 + 3*x^4 - x^5)/(1 - x^3)^2.

%F E.g.f.: exp(-x/2)*(3*exp(3*x/2)*(3 + 4*x) + 12*x*cos(sqrt(3)*x/2) + 4*sqrt(3)*(2 - 3*x)*sin(sqrt(3)*x/2))/9.

%F Sum_{n>=0} 1/a(n) = (log(2) + Pi)/4 = A377227.

%t LinearRecurrence[{0,0,2,0,0,-1},{1,5,-3,9,13,-7},61]

%Y Cf. A003881, A005408, A049347, A061347, A131561, A157142, A377227.

%K sign,easy

%O 0,2

%A _Stefano Spezia_, Oct 20 2024