A091995 -id:A091995 - cp's OEIS frontend

A090894 Numbers in n-th downward diagonal of triangle T : 0; 1, 2; 3, 4, 5; 6, 7, 8, 9; ...

0, 1, 2, 3, 4, 6, 5, 7, 10, 8, 11, 15, 9, 12, 16, 21, 13, 17, 22, 28, 14, 18, 23, 29, 36, 19, 24, 30, 37, 45, 20, 25, 31, 38, 46, 55, 26, 32, 39, 47, 56, 66, 27, 33, 40, 48, 57, 67, 78, 34, 41, 49, 58, 68, 79, 91, 35, 42, 50, 59, 69, 80, 92, 105, 43, 51, 60, 70, 81, 93, 106, 120

Offset: 0

Philippe Deléham, Feb 25 2004

Also the skewed triangle read by nonzero columns:
1, 2
0, 3, 4, 5
0, 0, 6, 7, 8, 9
0, 0, 0, 10,11,12,13,14
... (offset 0,0), - Ralf Stephan, Mar 09 2014

A273751 Triangle of the natural numbers written by decreasing antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 6, 8, 10, 13, 9, 11, 14, 17, 21, 12, 15, 18, 22, 26, 31, 16, 19, 23, 27, 32, 37, 43, 20, 24, 28, 33, 38, 44, 50, 57, 25, 29, 34, 39, 45, 51, 58, 65, 73, 30, 35, 40, 46, 52, 59, 66, 74, 82, 91, 36, 41, 47, 53, 60, 67, 75, 83, 92, 101, 111

Offset: 1

Paul Curtz, May 30 2016

nonn

A permutation of the natural numbers.
a(n) and A091995(n) are different at the ninth term.
Antidiagonal sums: 1, 2, 7, 11, ... = A235355(n+1). Same idea.
Row sums: 1, 5, 16, 37, 72, 124, 197, 294, ... = 7*n^3/12 -n^2/8 +5*n/12 +1/16 -1/16*(-1)^n with g.f. x*(1+2*x+3*x^2+x^3) / ( (1+x)*(x-1)^4 ). The third difference is of period 2: repeat [3, 4].
Indicates the order in which electrons fill the different atomic orbitals (s,p,d,f,g,h). - Alexander Goebel, May 12 2020

			1,
2,   3,
4,   5,  7,
6,   8, 10, 13,
9,  11, 14, 17, 21,
12, 15, 18, 22, 26, 31,
16, 19, 23, 27, 32, 37, 43,
20, etc.

Cf. A002061 (right diagonal), A002620 (first column), A033638, A091995, A234305 (antidiagonals of the triangle).

A273751 := proc(n,k)
    option remember;
    if k = n then
        A002061(n) ;
    elif k > n or k < 0 then
        0;
    elif k = n-1 then
        procname(n-1,k)+k ;
    else
        procname(n-1,k+1)+1 ;
    end if;
end proc: # R. J. Mathar, Jun 13 2016

T[n_, k_] := T[n, k] = Which[k == n, n(n-1) + 1, k == n-1, (n-1)^2 + 1, k == 1, n + T[n-2, 1], 1 < k < n-1, T[n-1, k+1] + 1,True, 0];
Table[T[n, k], {n, 12}, {k, n}] // Flatten (* Jean-François Alcover, Jun 10 2016 *)

T(n, k) = (2 * (n+k)^2 + 7 + (-1)^(n-k)) / 8 - k. - Werner Schulte, Sep 27 2024

G.f.: x*y*(1 + x^4*y + x^2*(y - 1)*y + x^5*y^2 - x^3*y*(y + 2))/((1 - x)^3*(1 + x)*(1 - x*y)^3). - Stefano Spezia, Sep 28 2024

A092180 Permutation of primes generated by triangle shown below.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 13, 19, 31, 23, 37, 53, 29, 41, 59, 79, 43, 61, 83, 109, 47, 67, 89, 113, 157, 71, 97, 127, 163, 199, 73, 101, 131, 167, 211, 263, 103, 137, 173, 223, 269, 331, 107, 139, 179, 227, 271, 337, 401, 149, 181, 229, 277, 347, 409, 479, 151, 191, 233

Offset: 1

Giovanni Teofilatto, Apr 02 2004

2
5
11 13
19 23 29
37 41 43 47
59 61 67 71 73
a(n) = A000040(A091995(n)).

Cf. A000040, A091995.

Corrected and extended by Pab Ter (pabrlos(AT)yahoo.com), May 25 2004

cp's OEIS Frontend

A090894 Numbers in n-th downward diagonal of triangle T : 0; 1, 2; 3, 4, 5; 6, 7, 8, 9; ...

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A273751 Triangle of the natural numbers written by decreasing antidiagonals.

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A092180 Permutation of primes generated by triangle shown below.

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