Alcuin's Sequence

Author

Eric W. Weisstein
March 27, 2006

This notebook downloaded from http://mathworld.wolfram.com/notebooks/IntegerSequences/AlcuinsSequence.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/AlcuinsSequence.html.

©2006 Wolfram Research, Inc. except for portions noted otherwise

Series

Series[(1/((1-(x^2))(1-(x^3))(1-(x^4)))),{x,0,5}]

<<DiscreteMath`RSolve`

SeriesTerm[(1/((1-(x^2))(1-(x^3))(1-(x^4)))),{x,0,n}]

FullSimplify[ComplexExpand[%],n∈Integers&&n>0]

Subtract@@@Table[{Cos[((n π)/2)]-Sin[((n π)/2)],(-1)^Floor[(n+1)/2]},{n,20}]//Union

Table[(1/288) (107+54 n+6 (n^2)+(((-1))^n) (81+18 n+64 Cos[((n π)/3)])+36 (((-1))^(Floor[(n+1)/2]))),{n,0,20}]

Pein

v5.2

FullSimplify[Subtract@@{
(1/288) (107+81 (((-1))^k)+(2^(5-k)) ((((-1-I (Sqrt[3])))^k)+(((-1+I (Sqrt[3])))^k))+6 k (9+3 (((-1))^k)+k)+36( Cos[((k π)/2)]-Sin[((k π)/2)]))/.k->n,
(1/288) (107+54 n+6 (n^2)+(((-1))^n) (81+18 n+64 Cos[((n π)/3)])+36 Cos[((n π)/2)]-36 Sin[((n π)/2)])
},n∈Integers]

v6

FullSimplify[Subtract@@{
(1/288) (107+81 (((-1))^k)+(2^(5-k)) ((((-1-I (Sqrt[3])))^k)+(((-1+I (Sqrt[3])))^k))+6 k (9+3 (((-1))^k)+k)+36( Cos[((k π)/2)]-Sin[((k π)/2)]))/.k->n,
(1/288) (107+54 n+6 (n^2)+(((-1))^n) (81+18 n+64 Cos[((n π)/3)])+36 Cos[((n π)/2)]-36 Sin[((n π)/2)])
},n∈Integers]

Integer Triangles

Table[Round[((n^2)/12)]-Floor[(n/4)]Floor[((n+2)/4)],{n,20}]

Table[Round[(1/48)If[EvenQ[n],(n^2),(((n+3))^2)]],{n,20}]

Plot

<<Utilities`Plot`

NaryPlot[Table[(1/288) (107+54 n+6 (n^2)+(((-1))^n) (81+18 n+64 Cos[((n π)/3)])+36 (((-1))^(Floor[(n+1)/2]))),{n,0,128}]]