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A067969 Number of nodes in virtual, "optimal", chordal graphs of diameter 5, degree =n+1.

%I A067969 #11 Apr 07 2025 05:13:03
%S A067969 11,20,61,102,231,360,681,1002,1683,2364,3653,4942,7183,9424,13073,
%T A067969 16722,22363,28004,36365,44726,56695,68664,85305,101946,124515,147084,
%U A067969 177045,207006,246047,285088,335137,385186,448427,511668,590557,669446
%N A067969 Number of nodes in virtual, "optimal", chordal graphs of diameter 5, degree =n+1.
%D A067969 Concrete Mathematics - R. L. Graham, D. E. Knuth, O. Patashnik, 1994,Addison-Wesley Company, Inc.
%H A067969 Robert Israel, <a href="/A067969/b067969.txt">Table of n, a(n) for n = 1..10000</a>
%H A067969 Ronald Cools, Ian H. Sloan, <a href="https://doi.org/10.1090/S0025-5718-96-00767-3">Minimial cubature formulae of trigonometric degree</a>, Math. Comp. 65 (216) (1996) 1583-1600. Table 1 dimension 5.
%H A067969 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (2,3,-8,-2,12,-2,-8,3,2,-1).
%F A067969 n - odd: t=(n+1)/2, a[n] := 4/15*t^5+2/3*t^4+8/3*t^3+10/3*t^2+46/15*t+1; n - even: t=n/2, a(n) := (4/15*t^5+2/3*t^4+8/3*t^3+10/3*t^2+46/15*t+1)+((2*(t*(t+1)*(t^2+t+4))/3)+1)
%F A067969 G.f.: x*(11-2*x-12*x^2+8*x^3+26*x^4-12*x^5-12*x^6+8*x^7+3*x^8-2*x^9)/ ((1+x)^4 * (x-1)^6) [From Maksym Voznyy (voznyy(AT)mail.ru), Jul 28 2009]
%F A067969 (n+1)*a(n) -2*a(n-1) -18*a(n-2) -2*a(n-3) +(-n+1)*a(n-4)=0. - _R. J. Mathar_, Apr 07 2025
%e A067969 a(5)=231 n=odd, t=3, a(5)=324/5+54+72+30+46/5+1=231 a(6)=360 n=even, t=3, a(6)=231+(24*16)/3+1=231+128+1=360
%p A067969 for n from 1 to k do if ((n mod 2 ) = 1) then t := (n+1)/2; a[n] := 4/15*t^5+2/3*t^4+8/3*t^3+10/3*t^2+46/15*t+1; else t := n/2; a[n] := ((4/15*t^5+2/3*t^4+8/3*t^3+10/3*t^2+46/15*t+1)+((2*(t*(t+1)*(t^2+t+4))/3)+1)); fi; print(a[n]); od;
%Y A067969 Cf. A001847 (bisection), A035599 (bisection).
%K A067969 nonn,easy
%O A067969 1,1
%A A067969 S. Bujnowski & B. Dubalski (slawb(AT)atr.bydgoszcz.pl), Mar 11 2002
%E A067969 G.f. proposed by Maksym Voznyy checked and corrected by R. J. Mathar, Sep 16 2009.