A279385 -id:A279385 - cp's OEIS frontend

A280223 Precipice of n: descending by the main diagonal of the pyramid described in A245092, a(n) is the height difference between the n-th level (starting from the top) and the level of the next terrace.

1, 2, 1, 2, 1, 2, 1, 1, 3, 2, 1, 3, 2, 1, 1, 2, 1, 2, 1, 4, 3, 2, 1, 1, 3, 2, 1, 2, 1, 2, 1, 3, 2, 1, 1, 4, 3, 2, 1, 2, 1, 3, 2, 1, 3, 2, 1, 1, 1, 4, 3, 2, 1, 2, 1, 4, 3, 2, 1, 3, 2, 1, 1, 2, 1, 4, 3, 2, 1, 2, 1, 5, 4, 3, 2, 1, 3, 2, 1, 1, 3, 2, 1, 4, 3, 2, 1, 2, 1, 1, 5, 4, 3, 2, 1, 2, 1, 1, 1, 4, 3, 2, 1, 4, 3, 2, 1

Offset: 1

Omar E. Pol, Dec 29 2016

nonn

The structure of the stepped pyramid arises after the 90-degree-zig-zag folding of the diagram of the isosceles triangle A237593.
The terraces at the n-th level of the pyramid are also the parts of the symmetric representation of sigma(n).
The stepped pyramid is also one of the 3D-quadrants of the stepped pyramid described in A244050.
Note that if a(n) > 1 then the next k terms are the first k positive integers in decreasing order, where k = a(n) - 1.
For more information about the precipices see A277437 and A280295.
a(n) is also the number of numbers >= n whose largest Dyck paths of the symmetric representation of sigma share the same point at the main diagonal of the diagram. For more information see A237593.

			Descending by the main diagonal of the stepped pyramid, for the levels 9, 10 and 11 we have that the next terrace is in the 12th level, so a(9) = 12 - 9 = 3, a(10) = 12 - 10 = 2, and a(11) = 12 - 11 = 1.

Omar E. Pol, Perspective view of the pyramid (first 16 levels)

Cf. A000203, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A240542, A244050, A245092, A259179, A262626, A277437, A279286, A279385, A280295.

More terms from Omar E. Pol, Jan 02 2017

A277437 Square array read by antidiagonals upwards in which T(n,k) is the n-th number j such that, descending by the main diagonal of the pyramid described in A245092, the height difference between the level j (starting from the top) and the level of the next terrace is equal to k.

Original entry on oeis.org

1, 3, 2, 5, 4, 9, 7, 6, 12, 20, 8, 10, 21, 36, 72, 11, 13, 25, 50, 91, 144, 14, 16, 32, 56, 112

Offset: 1

Omar E. Pol, Dec 29 2016

This is a permutation of the natural numbers.
Column k lists the numbers with precipice k. For more information about the precipices see A280223 and A280295.
The structure of the stepped pyramid arises after the 90-degree-zig-zag folding of the diagram of the isosceles triangle A237593.
The terraces at the m-th level of the pyramid are also the parts of the symmetric representation of sigma(m), m >= 1.
The stepped pyramid is also one of the 3D-quadrants of the stepped pyramid described in A244050.
If a number m is in the column k and k > 1 then m + 1 is the column k - 1.
The largest Dyck path of the symmetric representations of next k - 1 positive integers greater than T(n,k) shares the middle point of the largest Dyck path of the symmetric representation of sigma(T(n,k)). For more information see A237593.

			The corner of the square array begins:
   1,  2,  9, 20, 72, 144,
   3,  4, 12, 36, 91,
   5,  6, 21, 50,
   7, 10, 25,
   8, 13,
  11,
  ...
T(1,6) = 144 because it is the smallest number with precipice 6.

Row 1 gives A280295.
Column 1 gives A276112.
Cf. A000203, A071562, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A240542, A244050, A245092, A259179, A262626, A279286, A279385, A280223.

T(n,1) = A071562(n+1) - 1.

a(20)-a(26) from Omar E. Pol, Jan 02 2017

A280295 Smallest number with precipice n. Descending by the main diagonal of the pyramid described in A245092, the height difference between the level a(n) (starting from the top) and the level of the next terrace is equal to n.

Original entry on oeis.org

1, 2, 9, 20, 72, 144

Offset: 1

Omar E. Pol, Dec 31 2016

The structure of the stepped pyramid arises after the 90-degree-zig-zag folding of the diagram of the isosceles triangle A237593.
The terraces at the k-th level of the pyramid are also the parts of the symmetric representation of sigma(k), k >= 1.
The stepped pyramid is also one of the 3D-quadrants of the stepped pyramid described in A244050.
For more information about the precipices see A277437 and A280223.
Is this sequence infinite?

			a(3) = 9 because descending by the main diagonal of the pyramid, the height difference between the level 9 and the level of the next terrace is equal to 3, and 9 is the smallest number with this property.

Omar E. Pol, Perspective view of the pyramid (first 16 steps)

Row 1 of A277437.

Cf. A000203, A196020, A235791, A236104, A237591, A237593, A240542, A244050, A245092, A259179, A262626, A276112, A277437, A279286, A279385, A280223.

a(6) from Omar E. Pol, Jan 02 2017

A276112 Numbers with precipice 1: descending by the main diagonal of the pyramid described in A245092, the height difference between the level a(n) (starting from the top) and the level of the next terrace is equal to 1.

Original entry on oeis.org

1, 3, 5, 7, 8, 11, 14, 15, 17, 19, 23, 24, 27, 29, 31, 34, 35, 39, 41, 44, 47, 48, 49, 53, 55, 59, 62, 63, 65, 69, 71, 76, 79, 80, 83, 87, 89, 90, 95, 97, 98, 99, 103, 107, 109, 111, 116, 119, 120, 125, 127, 129, 131, 134, 139, 142, 143, 149, 152, 153, 155, 159

Offset: 1

Omar E. Pol, Jan 02 2017

nonn

The structure of the stepped pyramid arises after the 90-degree-zig-zag folding of the diagram of the isosceles triangle A237593.
The terraces at the k-th level of the pyramid are also the parts of the symmetric representation of sigma(k).
The stepped pyramid is also one of the 3D-quadrants of the stepped pyramid described in A244050.
For more information about the precipices see A277437, A280223 and A280295.
From Hartmut F. W. Hoft, Feb 02 2022: (Start)
Also partial sums of A280919.
a(n) is also the largest number of a Dyck path that crosses the diagonal at point A282131(n) which also is the rightmost number in each nonzero row of the irregular triangle in A279385. (End)

			From _Hartmut F. W. Hoft_, Feb 02 2022: (Start)
      n: 1  2  3  4  5  6  7  8  9 10 11 12 13 14 index.
A282131: 1  2  3  5  6  7  9 11 12 13 15 17 18 20 position on diagonal.
A276112: 1  3  5  7  8 11 14 15 17 19 23 24 27 29 max index of Dyck path.
A280919: 1  2  2  2  1  3  3  1  2  2  4  1  3  2 paths at diag position.
(End)

Omar E. Pol, Perspective view of the stepped pyramid (first 16 levels)

Column 1 of A277437.
Cf. A000203, A071562, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A240542, A244050, A245092, A259179, A262626, A279286, A279385, A280223, A280295.
Cf. A280919, A282131.

(* last computed value of a280919[ ] is dropped to avoid a potential undercount of crossings *)
a240542[n_] := Sum[(-1)^(k+1)Ceiling[(n+1)/k-(k+1)/2], {k, 1, Floor[-1/2+1/2 Sqrt[8n+1]]}]
a280919[n_] := Most[Map[Length, Split[Map[a240542, Range[n]]]]]
A276112[160] (* Hartmut F. W. Hoft, Feb 02 2022 *)

a(n) = A071562(n+1) - 1.

a(n) = Sum_{i=1..n} A280919(i), n >= 1. - Hartmut F. W. Hoft, Feb 02 2022

A299472 a(n) is the sum of all divisors of all numbers k whose associated largest Dyck path contains the point (n,n) in the diagram of the symmetric representation of sigma(k) described in A237593, or 0 if no such k exists.

Original entry on oeis.org

1, 7, 13, 0, 20, 15, 43, 0, 66, 0, 24, 49, 59, 0, 134, 0, 60, 113, 0, 86, 0, 104, 165, 0, 48, 245, 0, 132, 0, 224, 0, 198, 0, 124, 57, 317, 0, 192, 0, 350, 0, 326, 0, 104, 211, 0, 434, 0, 216, 0, 0, 647, 0, 344, 0, 186, 331, 0, 584, 0, 270, 0, 234, 0, 672, 0, 350, 171, 0, 156, 639, 0, 672, 0, 390, 0, 368, 0, 956

Offset: 1

Omar E. Pol, Feb 19 2018

nonn

Row sums of A299693.

Cf. A196020, A235791, A236104, A237048, A237591, A237593, A240542, A245092, A259179, A276112, A277437, A279286, A279385, A280919, A280223, A282131, A282197, A280295, A281012.

A299482 Numbers m such that in the diagram of the symmetric representation of sigma(k) described in A237593 there is no Dyck path that contains the point (m,m), where both k and m are positive integers.

Original entry on oeis.org

4, 8, 10, 14, 16, 19, 21, 24, 27, 29, 31, 33, 37, 39, 41, 43, 46, 48, 50, 51, 53, 55, 58, 60, 62, 64, 66, 69, 72, 74, 76, 78, 80, 82, 83, 84, 87, 90, 92, 94, 96, 98, 100, 101, 103, 105, 107, 109, 111, 114, 116, 119, 121, 123, 124, 125, 127, 129, 131, 133, 135, 138, 141, 143, 145, 147, 149, 151, 153

Offset: 1

Omar E. Pol, Feb 19 2018

nonn

Indices of the rows that contain a zero in the triangle A279385.

a(n) is the index of the n-th zero in A259179; i.e. A259179(a(n)) = 0. - Hartmut F. W. Hoft, Aug 07 2020

Complement of A282131.
Cf. A196020, A235791, A236104, A237048, A237591, A237593, A244050, A245092, A262626, A279385.
Cf. A240542, A259179.

a240542[n_] := Sum[(-1)^(k+1)*Ceiling[(n+1)/k - (k+1)/2], {k, 1, Floor[(Sqrt[8n+1]-1)/2]}]
a299482[n_] := Module[{t=Table[0, n], k=1, d=1}, While[d<=n, t[[d]]+=1; d=a240542[++k]]; Flatten[Position[t, 0]]]
a299482[153] (* Hartmut F. W. Hoft, Aug 07 2020 *)

A299693 Irregular triangle read by rows in which row n lists the total sum of the divisors of all numbers k such that the largest Dyck path of the symmetric representation of sigma(k) contains the point (n,n); or row n is 0 if no such k exists.

Original entry on oeis.org

1, 3, 4, 7, 6, 0, 12, 8, 15, 13, 18, 12, 0, 28, 14, 24, 0, 24, 31, 18, 39, 20, 0, 42, 32, 36, 24, 0, 60, 31, 42, 40, 0, 56, 30, 0, 72, 32, 63, 48, 54, 0, 48, 91, 38, 60, 56, 0, 90, 42, 0, 96, 44, 84, 0, 78, 72, 48, 0, 124, 57, 93, 72, 98, 54, 0, 120, 72, 0, 120, 80, 90, 60, 0, 168, 62, 96, 0, 104, 127, 84, 0

Offset: 1

Omar E. Pol, Feb 19 2018

			Triangle begins:
   1;
   3,  4;
   7,  6;
   0;
  12,  8;
  15;
  13, 18, 12;
   0;
  28, 14, 24;
   0;
  24;
  31, 18;
  39, 20;
   0;
  42, 32, 36, 24;
   0;
...

Nonzero terms give A000203.
Row sums give A299472.
Cf. A259179(n) is the number of positive terms in row n.
Cf. A071562, A196020, A235791, A236104, A237048, A237591, A237593, A240542, A244050, A245092, A276112, A277437, A279286, A279385, A280919, A280223, A282131, A282197, A280295, A281012.

T(n,m) = A000203(A279385(n,m)) if A279385(n,m) > 0, otherwise T(n,m) = 0.

cp's OEIS Frontend

A280223 Precipice of n: descending by the main diagonal of the pyramid described in A245092, a(n) is the height difference between the n-th level (starting from the top) and the level of the next terrace.

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A277437 Square array read by antidiagonals upwards in which T(n,k) is the n-th number j such that, descending by the main diagonal of the pyramid described in A245092, the height difference between the level j (starting from the top) and the level of the next terrace is equal to k.

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A280295 Smallest number with precipice n. Descending by the main diagonal of the pyramid described in A245092, the height difference between the level a(n) (starting from the top) and the level of the next terrace is equal to n.

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A276112 Numbers with precipice 1: descending by the main diagonal of the pyramid described in A245092, the height difference between the level a(n) (starting from the top) and the level of the next terrace is equal to 1.

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A299472 a(n) is the sum of all divisors of all numbers k whose associated largest Dyck path contains the point (n,n) in the diagram of the symmetric representation of sigma(k) described in A237593, or 0 if no such k exists.

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A299482 Numbers m such that in the diagram of the symmetric representation of sigma(k) described in A237593 there is no Dyck path that contains the point (m,m), where both k and m are positive integers.

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Mathematica

A299693 Irregular triangle read by rows in which row n lists the total sum of the divisors of all numbers k such that the largest Dyck path of the symmetric representation of sigma(k) contains the point (n,n); or row n is 0 if no such k exists.

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