A185688 - OEIS

%I #26 Oct 03 2023 16:28:18

%S 2,-2,3,5,-2,18,-19,43,-30,62,-71,111,-82,130,-153,209,-158,222,-265,

%T 337,-258,338,-407,495,-382,478,-579,683,-530,642,-781,901,-702,830,

%U -1013,1149,-898,1042

%N First differences of A060819(n-4)*A060819(n).

%C The sequence b(n) = A060819(n-4)*A060819(n) is -3, -1, -3, 0, 5, 3, 21, 2, 45, 15, 77, 6, 117, 35, 165 for n>=1, an extension of A061037. Its first differences b(n+1)-b(n) = a(n) define the current sequence.

%C First differences of the quadrisection are a(4n+4)-a(4n) = 8+30*n.

%H G. C. Greubel, <a href="/A185688/b185688.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (-1,-1,-1,2,2,2,2,-1,-1,-1,-1).

%F a(2*n+1) + a(2*n+2) = 8*n.

%F G.f. ( x*(2+3*x^2+8*x^3+24*x^5+24*x^7+8*x^9-4*x^6+6*x^8+3*x^10) ) / ( (x-1)^2*(1+x)^3*(x^2+1)^3 ). - _R. J. Mathar_, Feb 16 2011

%F a(n) = -a(n-1) -a(n-2) -a(n-3) +2*a(n-4) +2*a(n-5) +2*a(n-6) +2*a(n-7) -a(n-8) -a(n-9) -a(n-10) -a(n-11).

%F a(n)=3*a(n-4) -3*a(n-8) +a(n-12). - _Paul Curtz_, Feb 17 2011

%F a(4n) + a(4n+1) +a(4n+2) +a(4n+3) = 2*n = A005843(n). - _Paul Curtz_, Feb 17 2011

%p A060819 := proc(n) n/igcd(n,4) ; end proc:

%p A185688b := proc(n) A060819(n-4)*A060819(n) ; end proc:

%p A185688 := proc(n) A185688b(n+1)-A185688b(n) ; end proc: # _R. J. Mathar_, Feb 16 2011

%t Rest[CoefficientList[Series[(x*(2 + 3*x^2 + 8*x^3 + 24*x^5 + 24*x^7 + 8*x^9 - 4*x^6 + 6*x^8 + 3*x^10))/((x - 1)^2*(1 + x)^3*(x^2 + 1)^3), {x, 0, 50}], x]] (* _G. C. Greubel_, Jul 10 2017 *)

%t LinearRecurrence[{-1,-1,-1,2,2,2,2,-1,-1,-1,-1},{2,-2,3,5,-2,18,-19,43,-30,62,-71},40] (* _Harvey P. Dale_, Oct 03 2023 *)

%o (PARI) Vec((2+3*x^2+8*x^3+24*x^5+24*x^7+8*x^9-4*x^6+6*x^8+3*x^10)/(x-1)^2/(1+x)^3/(x^2+1)^3+O(x^99)) \\ _Charles R Greathouse IV_, Feb 08 2012

%K sign,easy

%O 1,1

%A _Paul Curtz_, Feb 10 2011