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

11, 20, 61, 102, 231, 360, 681, 1002, 1683, 2364, 3653, 4942, 7183, 9424, 13073, 16722, 22363, 28004, 36365, 44726, 56695, 68664, 85305, 101946, 124515, 147084, 177045, 207006, 246047, 285088, 335137, 385186, 448427, 511668, 590557, 669446

Offset: 1

Author

S. Bujnowski & B. Dubalski (slawb(AT)atr.bydgoszcz.pl), Mar 11 2002

Keywords

• nonn
• easy

Examples

			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
		

References

• Concrete Mathematics - R. L. Graham, D. E. Knuth, O. Patashnik, 1994,Addison-Wesley Company, Inc.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Ronald Cools, Ian H. Sloan, Minimial cubature formulae of trigonometric degree, Math. Comp. 65 (216) (1996) 1583-1600. Table 1 dimension 5.
• Index entries for linear recurrences with constant coefficients, signature (2,3,-8,-2,12,-2,-8,3,2,-1).

Crossrefs

Cf. A001847 (bisection), A035599 (bisection).

Programs

Maple

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;

Formula

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)

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]

(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

Extensions

G.f. proposed by Maksym Voznyy checked and corrected by R. J. Mathar, Sep 16 2009.