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

%I A067956 #12 Apr 07 2025 05:01:22
%S A067956 9,16,41,66,129,192,321,450,681,912,1289,1666,2241,2816,3649,4482,
%T A067956 5641,6800,8361,9922,11969,14016,16641,19266,22569,25872,29961,34050,
%U A067956 39041,44032,50049,56066,63241,70416,78889
%N A067956 Number of nodes in virtual, "optimal", chordal graphs of diameter 4 and degree n+1.
%D A067956 Concrete Mathematics, R. L. Graham, D. E. Knuth, O. Patashnik, 1994, Addison-Wesley Company, Inc.
%H A067956 Michael De Vlieger, <a href="/A067956/b067956.txt">Table of n, a(n) for n = 1..10000</a>
%H A067956 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 4.
%H A067956 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-6,0,6,-2,-2,1).
%F A067956 For n odd, t = (n+1)/2, a(n) = ((2*t*(t+1)*(t^2+t+4))/3)+1;
%F A067956 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.
%F A067956 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
%e A067956 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.
%e A067956 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.
%p A067956 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;
%t A067956 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 *)
%Y A067956 Cf. A006007.
%K A067956 nonn,easy
%O A067956 1,1
%A A067956 S. Bujnowski & B. Dubalski (slawb(AT)atr.bydgoszcz.pl), Mar 08 2002