A067956 Number of nodes in virtual, "optimal", chordal graphs of diameter 4 and degree n+1.

9, 16, 41, 66, 129, 192, 321, 450, 681, 912, 1289, 1666, 2241, 2816, 3649, 4482, 5641, 6800, 8361, 9922, 11969, 14016, 16641, 19266, 22569, 25872, 29961, 34050, 39041, 44032, 50049, 56066, 63241, 70416, 78889

Offset: 1

Author

S. Bujnowski & B. Dubalski (slawb(AT)atr.bydgoszcz.pl), Mar 08 2002

Keywords

• nonn
• easy

Examples

			For n=5, n is odd; t=3; a(5) = (2*3*(3+1)*(3^2+3+4)/3)+1 = ((6*4*16)/3)+1 = 129.
For n=6, n is even; t=3; a(6) = a(5) + ((2*3+1)*(2*t^2+2*t+3))/3 = 129 + (7*27)/3 = 192.
		

References

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

Links

• Michael De Vlieger, 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 4.
• Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Crossrefs

Cf. A006007.

Programs

Maple

for n from 1 to k do if ((n mod 2 ) = 1) then t := (n+1)/2; a[n] := ((2*(t*(t+1)*(t^2+t+4))/3)+1); else t := (n)/2; a[n] := ((2*(t*(t+1)*(t^2+t+4)/3)+1)+(2*t+1)*(2*t^2+2*t+3)/3); fi; print(a[n]); od;

Mathematica

Array[((2 #2 (#2 + 1) (#2^2 + #2 + 4))/3) + 1 + (Boole[EvenQ[#1]]*((2 #2 + 1) (2 #2^2 + 2 #2 + 3))/3) & @@ {#, (# + Boole[OddQ[#]])/2} &, 35] (* Michael De Vlieger, Jul 29 2022 *)

Formula

For n odd, t = (n+1)/2, a(n) = ((2*t*(t+1)*(t^2+t+4))/3)+1;

for n even, t = n/2, a(n) = (((2*t*(t+1)*(t^2+t+4))/3)+1)+((2*t+1)*(2*t^2+2*t+3))/3.

G.f.: -x*(9-2*x-9*x^2+6*x^3+11*x^4-6*x^5-3*x^6+2*x^7)/(1+x)^3/(x-1)^5 . - R. J. Mathar, Apr 07 2025