A271647 Irregular triangle read by rows: the natural numbers from right to left.

1, 2, 4, 3, 6, 5, 9, 8, 7, 12, 11, 10, 16, 15, 14, 13, 20, 19, 18, 17, 25, 24, 23, 22, 21, 30, 29, 28, 27, 26, 36, 35, 34, 33, 32, 31, 42, 41, 40, 39, 38, 37, 49, 48, 47, 46, 45, 44, 43, 56, 55, 54, 53, 52, 51, 50, 64, 63, 62, 61, 60, 59, 58, 57

Offset: 1

Author

Paul Curtz, Apr 11 2016

Keywords

• nonn
• tabf

Comments

A permutation of the natural numbers. Mentioned as d(n) in A269837.

Difference table:

1, 2, 4, 3, 6, 5, 9, 8, 7, 12, 11, 10, 16, 15, 14, 13, 20, 19, 18, ...

1, 2, -1, 3, -1, 4, -1, -1, 5, -1, -1, 6, -1, -1, -1, 7, -1, -1, -1, ...

1, -3, 4, -4, 5, -5, 0, 6, -6, 0, 7, -7, 0, 0, 8, -8, 0, 0, 9, ...

etc.

Examples

			Irregular triangle:
1,
2,
4,   3,
6,   5,
9,   8,  7,
12, 11, 10,
16, 15, 14, 13,
20, 19, 18, 17,
25, 24, 23, 22, 21,
30, 29, 28, 27, 26,
etc.
		

Crossrefs

Cf. A000027, A002620, A024206, A033638, A075356, A235355, A269837, A271584

Programs

Maple

count:= 0:
for r from 1 to 20 do
  d:= ceil(r/2);
  for i from 0 to d-1 do A[r,i]:= count+ d-i od;
  count:= count+d;
od:
seq(seq(A[r,i],i=0..ceil(r/2)-1),r=1..20); # Robert Israel, Apr 11 2016

Mathematica

Table[Reverse@ Range[Floor[n/2]] + Floor[(n - 1)^2/4], {n, 16}] // Flatten (* Michael De Vlieger, Apr 11 2016 *)

Formula

With offset=0, a(n) = A271584(n) + A269837(n)

Empirical g.f. as triangle: (1-y*x^3+y^2*x^4-2*y*x^4-y^2*x^5+y*x^5+y^2*x^7)*x/((1+x)*(1-x)^3*(1-y*x^2)^3). - Robert Israel, Apr 11 2016