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a(n) = T(n,n-2), array T as in A055801.
12

%I #28 Sep 08 2022 08:45:01

%S 1,1,1,2,3,4,6,7,10,11,15,16,21,22,28,29,36,37,45,46,55,56,66,67,78,

%T 79,91,92,105,106,120,121,136,137,153,154,171,172,190,191,210,211,231,

%U 232,253,254,276,277,300,301,325,326,351,352,378,379,406,407,435

%N a(n) = T(n,n-2), array T as in A055801.

%C For n>2, a(n)+a(n+1) seems to be A002620(n+1)+1.

%H Colin Barker, <a href="/A055802/b055802.txt">Table of n, a(n) for n = 2..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).

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

%F a(n) = A114220(n-1), n>=3. - _R. J. Mathar_, Feb 03 2013

%F From _Colin Barker_, Jan 27 2016: (Start)

%F a(n) = (2*n^2 +2*(-1)^n*n -6*n -11*(-1)^n +11)/16 for n>2.

%F a(n) = (n^2 - 2*n)/8 for n>2 and even.

%F a(n) = (n^2 - 4*n + 11)/8 for n odd. (End)

%F E.g.f.: (4*x*(x-2) + x*(x-3)*cosh(x) + (x^2 -x +11)*sinh(x))/8. - _G. C. Greubel_, Jan 23 2020

%p seq( `if`(n==2, 1, (2*n^2 -6*n +11 +(-1)^n*(2*n -11))/16), n=2..65); # _G. C. Greubel_, Jan 23 2020

%t CoefficientList[Series[(1 -2*x^2 +x^3 +2*x^4 -x^5)/((1-x)^3*(1+x)^2), {x,0,65}], x] (* _Wesley Ivan Hurt_, Jan 20 2017 *)

%t Table[If[n==2,1, (2*n^2 -6*n +11 +(-1)^n*(2*n -11))/16], {n,2,65}] (* _G. C. Greubel_, Jan 23 2020 *)

%o (PARI) Vec(x^2*(1-2*x^2+x^3+2*x^4-x^5)/((1-x)^3*(1+x)^2) + O(x^65)) \\ _Charles R Greathouse IV_, Feb 03 2013

%o (PARI) vector(65, n, my(m=n+1); if(m==2, 1, (2*m^2 -6*m +11 +(-1)^m*(2*m -11))/16)) \\ _G. C. Greubel_, Jan 23 2020

%o (Magma) [1] cat [(2*n^2 -6*n +11 +(-1)^n*(2*n -11))/16: n in [3..65]]; // _G. C. Greubel_, Jan 23 2020

%o (Sage) [1]+[(2*n^2 -6*n +11 +(-1)^n*(2*n -11))/16 for n in (3..65)] # _G. C. Greubel_, Jan 23 2020

%o (GAP) Concatenation([1], List([3..65], n-> (2*n^2 -6*n +11 +(-1)^n*(2*n -11))/16 )); # _G. C. Greubel_, Jan 23 2020

%Y Cf. A002620, A134519.

%Y Cf. A055801, A055803, A055804, A055805, A055806.

%K nonn,easy

%O 2,4

%A _Clark Kimberling_, May 28 2000