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Triangle T(n,k) read by rows. Coloring of sectors in a circle.
1

%I #13 Sep 08 2022 08:45:50

%S 1,1,2,1,2,3,1,2,1,2,1,2,1,2,3,1,2,1,2,1,2,1,2,1,2,1,2,3,1,2,1,2,1,2,

%T 1,2,1,2,1,2,1,2,1,2,3,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,3,1,2,

%U 1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,3,1,2,1,2,1,2,1,2,1,2,1,2,1,2

%N Triangle T(n,k) read by rows. Coloring of sectors in a circle.

%C One row equals a coloring of n sectors in a circle and each number in the k-th column represents a color in the k-th sector of the circle. No pair of adjacent sectors can have the same color. The smallest numbers are chosen as colors and they are ordered from smallest to largest.

%H G. C. Greubel, <a href="/A171712/b171712.txt">Rows n = 1..100 of triangle, flattened</a>

%F T(n, k) = (3 + (-1)^k)/2 with T(n, 1) = 1 and T(n, n) = (5 - (-1)^n)/2 for n >= 2. - _G. C. Greubel_, Nov 29 2019

%e Table begins:

%e 1;

%e 1, 2;

%e 1, 2, 3;

%e 1, 2, 1, 2;

%e 1, 2, 1, 2, 3;

%e 1, 2, 1, 2, 1, 2;

%e 1, 2, 1, 2, 1, 2, 3;

%e 1, 2, 1, 2, 1, 2, 1, 2;

%p seq(seq( `if`(k=1, 1, `if`(k=n, (5-(-1)^n)/2, (3+(-1)^k)/2 )), k=1..n), n=1..15); # _G. C. Greubel_, Nov 29 2019

%t T[n_, k_]:= If[k==1, 1, If[k==n, (5-(-1)^n)/2, (3+(-1)^k)/2]]; Table[T[n, k], {n, 15}, {k, n}]//Flatten (* _G. C. Greubel_, Nov 29 2019 *)

%o (PARI) T(n,k) = if(k==1, 1, if(k==n, (5-(-1)^n)/2, (3+(-1)^k)/2 )); \\ _G. C. Greubel_, Nov 29 2019

%o (Magma)

%o function T(n,k)

%o if k eq 1 then return 1;

%o elif k eq n then return (5-(-1)^n)/2;

%o else return (3+(-1)^k)/2; end if; return T; end function;

%o [T(n,k): k in [1..n], n in [1..15]]; // _G. C. Greubel_, Nov 29 2019

%o (Sage)

%o def T(n, k):

%o if (k==1): return 1

%o elif (k==n): return (5-(-1)^n)/2

%o else: return (3+(-1)^k)/2

%o [[T(n, k) for k in (1..n)] for n in (1..15)] # _G. C. Greubel_, Nov 29 2019

%o (GAP)

%o T:= function(n,k)

%o if k=1 then return 1;

%o elif k=n then return (5-(-1)^n)/2;

%o else return (3+(-1)^k)/2; fi; end;

%o Flat(List([1..15], n-> List([1..n], k-> T(n,k) ))); # _G. C. Greubel_, Nov 29 2019

%Y Cf. A158478.

%K nonn,tabl

%O 1,3

%A _Mats Granvik_, Dec 16 2009