A049778 - OEIS

%I #32 Sep 08 2022 08:44:58

%S 1,3,9,17,32,50,78,110,155,205,271,343,434,532,652,780,933,1095,1285,

%T 1485,1716,1958,2234,2522,2847,3185,3563,3955,4390,4840,5336,5848,

%U 6409,6987,7617,8265,8968,9690,10470,11270,12131

%N a(n) = Sum_{k=1..floor((n+1)/2)} T(n,2k-1), array T as in A049777.

%C Principal diagonal of the convolution array A213849. - _Clark Kimberling_, Jul 04 2012

%H G. C. Greubel, <a href="/A049778/b049778.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,1,2,-1).

%F G.f.: x*(1 + x + 2*x^2)/((1-x)^4*(1+x)^2). Pairwise sums of A023855. - _Ralf Stephan_, May 06 2004

%F a(n) = Sum_{k=1..n} k*ceiling(k/2). - _Vladeta Jovovic_, Apr 29 2006

%F Row sums of triangle A095800^2. - _Gary W. Adamson_, Dec 12 2007

%F a(n) = (3 + 10*n + 18*n^2 + 8*n^3 - 3*(-1)^n*(1 + 2*n))/48. - _R. J. Mathar_, Mar 03 2011

%F From _G. C. Greubel_, Dec 12 2019: (Start)

%F a(n) = m*(3*(n-1)*(n+2) - (m+1)*(4*m-7))/6, where m = floor((n+1)/2).

%F E.g.f.: ( (3+36*x+42*x^2+8*x^3)*exp(x) - 3*(1-2*x)*exp(-x) )/48. (End)

%p seq( (3 +10*n +18*n^2 +8*n^3 -3*(-1)^n*(1+2*n))/48, n=1..50); # _G. C. Greubel_, Dec 12 2019

%t Table[Floor[(n+1)/2]*(3*(n-1)*(n+2) -(1+Floor[(n+1)/2])*(4*Floor[(n+1)/2]-7))/6, {n,50}] (* _G. C. Greubel_, Dec 12 2019 *)

%o (PARI) vector(50, n, (3 +10*n +18*n^2 +8*n^3 -3*(-1)^n*(1+2*n))/48) \\ _G. C. Greubel_, Dec 12 2019

%o (Magma) [(3 +10*n +18*n^2 +8*n^3 -3*(-1)^n*(1+2*n))/48: n in [1..50]]; // _G. C. Greubel_, Dec 12 2019

%o (Sage) [(3 +10*n +18*n^2 +8*n^3 -3*(-1)^n*(1+2*n))/48 for n in (1..50)] # _G. C. Greubel_, Dec 12 2019

%o (GAP) List([1..50], n-> (3 +10*n +18*n^2 +8*n^3 -3*(-1)^n*(1+2*n))/48); # _G. C. Greubel_, Dec 12 2019

%Y Cf. A023855, A049777, A095800, A213849.

%K nonn

%O 1,2

%A _Clark Kimberling_