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%I #5 Apr 14 2013 02:26:27
%S 1,4,6,4,8,12,10,16,9,12,20,24,14,24,30,16,16,28,36,40,18,32,42,48,25,
%T 20,36,48,56,60,22,40,54,64,70,36,24,44,60,72,80,84,26,48,66,80,90,96,
%U 49,28,52,72,88,100,108,112,30,56,78,96,110,120,126,64,32,60,84,104,120
%N A fold back triangular sequence for A003991: symmetrical folding and addition of.
%C Row sum is still A000292. These sequences are related to heights of Cartan A_n groups. I spent half the night trying to get this algorithm to work and just barely got this to do what I was doing with ease by hand. The other fold back sequences were analogs for this.
%F a0(n,m) = (n-m)*(1+m) Doubling of all elements a(n,m)=2*a0(n,m)-> m->Floor[n/2] except for the uneven middle element on odd sequences in n.
%e {1},
%e {4},
%e {6, 4},
%e {8, 12},
%e {10, 16, 9},
%e {12, 20, 24},
%e {14, 24, 30, 16},
%e {16, 28, 36, 40},
%e {18, 32, 42, 48, 25},
%e {20, 36, 48, 56, 60},
%e {22, 40, 54, 64, 70, 36},
%e {24, 44, 60, 72, 80, 84}
%t (* first A003991*) a = Table[Table[(n - i)*(1 + i), {i, 0, n - 1}], {n, 1, 20}]; (* then fold back from that*) Table[Table[If[ Mod[n, 2] == 1, a[[n]][[m]] + a[[n]][[Length[a[[n]]] - m]] - n, If[m - Floor[ n/2] == 0, (a[[n]][[m]] + a[[ n]][[Length[a[[n]]] - m]] - n)/ 2, a[[n]][[m]] + a[[n]][[Length[a[[n]]] - m]] - n]], {m, 1, Floor[n/ 2]}], {n, 1, Length[a]}]; Flatten[%]
%Y Cf. A000292, A003991.
%K nonn,tabf,uned
%O 1,2
%A _Roger L. Bagula_, Jul 13 2007
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