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A067956
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Number of nodes in virtual, "optimal", chordal graphs of diameter 4 and degree n+1.
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1
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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
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OFFSET
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1,1
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REFERENCES
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Concrete Mathematics, R. L. Graham, D. E. Knuth, O. Patashnik, 1994, Addison-Wesley Company, Inc.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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.
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MAPLE
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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;
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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S. Bujnowski & B. Dubalski (slawb(AT)atr.bydgoszcz.pl), Mar 08 2002
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STATUS
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approved
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