A269837 -id:A269837 - cp's OEIS frontend

A271584 Irregular triangle read by rows: alternate (k-1)*k, k^2, for k = 0 to n.

0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 1, 4, 0, 0, 2, 6, 0, 1, 4, 9, 0, 0, 2, 6, 12, 0, 1, 4, 9, 16, 0, 0, 2, 6, 12, 20, 0, 1, 4, 9, 16, 25, 0, 0, 2, 6, 12, 20, 30, 0, 1, 4, 9, 16, 25, 36, 0, 0, 2, 6, 12, 20, 30, 42, 0, 1, 4, 9, 16, 25, 36, 49

Offset: 0

Paul Curtz, Apr 10 2016

This is the irregular triangle e(n) mentioned in A269837.
Last number: 0, 0, 0, 1, 2, 4, 6, 9, 12, 16, 20, 25, ... = A002620 with a third initial 0.
Row sums: 0, 0, 0, 1, 2, 5, 8, 14, 20, 30, 40, ... = A006918 with three initial 0.
Columns: the sum of the first two terms is A000384(n).

			Irregular triangle:
0,
0,
0, 0,
0, 1,
0, 0, 2,
0, 1, 4,
0, 0, 2, 6,
0, 1, 4, 9,
0, 0, 2, 6, 12,
0, 1, 4, 9, 16,
0, 0, 2, 6, 12, 20,
0, 1, 4, 9, 16, 25,
etc.

Cf. A000290, A000384, A002262, A006918, A161680, A269837.

Flatten[Transpose /@ Table[{k (k - 1), k^2}, {n, 0, 7}, {k, 0, n}]] (* Michael De Vlieger, Apr 10 2016 *)

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

Original entry on oeis.org

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

Paul Curtz, Apr 11 2016

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.

			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.

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

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

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

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

cp's OEIS Frontend

A271584 Irregular triangle read by rows: alternate (k-1)*k, k^2, for k = 0 to n.

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