|
%I #6 Dec 04 2019 15:08:35
%S 0,0,1,9,42,140,383,925,2056,4316,8705,17069,32810,62192,116743,
%T 217673,404000,747496,1380177,2544865,4688186,8631620,15886111,
%U 29230725,53776968,98926372,181971057,334716197,615660634,1132400520
%N a(n) = a(n-1)+a(n-2)+a(n-3)+(8*n^3-48*n^2+112*n-96)/3 with a(0)=0, a(1)=0 and a(2)=1.
%C The a(n+2) represent the Kn15 and Kn25 sums of the square array of Delannoy numbers A008288. See A180662 for the definition of these knight and other chess sums.
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (5,-9,7,-3,3,-3,1).
%F a(n) = a(n-1)+a(n-2)+a(n-3)+(8*n^3-48*n^2+112*n-96)/3 with a(0)=0, a(1)=0 and a(2)=1.
%F a(n) = a(n-1)+A001590(n+7)-(12+4*n+4*n^2) with a(0)=0.
%F a(n) = sum(A008412(m)*A000073(n-m),m=0..n).
%F a(n+2) = add(A008288(n-k+4,k+4),k=0..floor(n/2)).
%F GF(x) = (x^2*(1+x)^4)/((1-x)^4*(1-x-x^2-x^3)).
%p nmax:=29: a(0):=0: a(1):=0: a(2):=1: for n from 3 to nmax do a(n):= a(n-1)+a(n-2)+a(n-3)+(8*n^3-48*n^2+112*n-96)/3 od: seq(a(n),n=0..nmax);
%t RecurrenceTable[{a[0]==a[1]==0,a[2]==1,a[n]==a[n-1]+a[n-2]+a[n-3]+(8n^3-48n^2+112n-96)/3},a,{n,30}] (* or *) LinearRecurrence[{5,-9,7,-3,3,-3,1},{0,0,1,9,42,140,383},30] (* _Harvey P. Dale_, Dec 04 2019 *)
%Y Cf. A000073 (Kn11 & Kn21), A089068 (Kn12 & Kn22), A180668 (Kn13 & Kn23), A180669 (Kn14 & Kn24), A180670 (Kn15 & Kn25).
%K easy,nonn
%O 0,4
%A _Johannes W. Meijer_, Sep 21 2010
|
