A171712 Triangle T(n,k) read by rows. Coloring of sectors in a circle.

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, 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, 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

Offset: 1

Author

Mats Granvik, Dec 16 2009

Keywords

• nonn
• tabl

Comments

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.

Examples

			Table begins:
  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, 1, 2;
		

Links

• G. C. Greubel, Rows n = 1..100 of triangle, flattened

Crossrefs

Cf. A158478.

Programs

GAP

T:= function(n,k)
    if k=1 then return 1;
    elif k=n then return (5-(-1)^n)/2;
    else return (3+(-1)^k)/2; fi; end;
Flat(List([1..15], n-> List([1..n], k-> T(n,k) ))); # G. C. Greubel, Nov 29 2019

Magma

function T(n,k)
  if k eq 1 then return 1;
  elif k eq n then return (5-(-1)^n)/2;
  else return (3+(-1)^k)/2;  end if; return T; end function;
[T(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Nov 29 2019

Maple

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

Mathematica

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 *)

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

Sage

def T(n, k):
    if (k==1): return 1
    elif (k==n): return (5-(-1)^n)/2
    else: return (3+(-1)^k)/2
[[T(n, k) for k in (1..n)] for n in (1..15)] # G. C. Greubel, Nov 29 2019

Formula

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