cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

Original entry on oeis.org

1, 2, 9, 20, 72, 144
Offset: 1

Views

Author

Omar E. Pol, Dec 31 2016

Keywords

Comments

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?

Examples

			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.

Links

Crossrefs

Row 1 of A277437.
Cf. A000203, A196020, A235791, A236104, A237591, A237593, A240542, A244050, A245092, A259179, A262626, A276112, A277437, A279286, A279385, A280223.

Extensions

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

Views

Author

Omar E. Pol, Jan 02 2017

Keywords

Comments

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)

Examples

			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)

Links

Crossrefs

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.

Programs

  • Mathematica
    (* 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 *)

Formula

a(n) = A071562(n+1) - 1.
a(n) = Sum_{i=1..n} A280919(i), n >= 1. - Hartmut F. W. Hoft, Feb 02 2022

A244367 Main diagonal of A244580.

Original entry on oeis.org

1, 2, 4, 6, 6, 8, 9, 12, 12, 15, 15, 16, 18, 20, 20, 24, 24, 25, 28, 28, 30, 30, 32, 35, 35, 36, 40, 40, 42, 42, 45, 45, 48, 48, 49, 50, 54, 54, 56, 56, 60, 60, 63, 63, 64, 66, 66, 70, 70, 72, 72, 72, 77, 77, 80, 80, 81, 84, 84, 88, 88, 90, 90, 91, 91
Offset: 1

Views

Author

Omar E. Pol, Jul 05 2014

Keywords

Comments

a(n) is the number associated to the n-th cell of the main diagonal in the diagram of the symmetry of sigma. For more information see A237593 and A237270.
Records give A071562.
First differences are A259179. - Omar E. Pol, Apr 21 2020

Links

Crossrefs

Cf. A000203, A071562, A196020, A236104, A237270, A237271, A237593, A240542, A259179.

Programs

  • PARI
    See Links section.

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

Views

Author

Omar E. Pol, Feb 19 2018

Keywords

Crossrefs

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

Views

Author

Omar E. Pol, Feb 19 2018

Keywords

Comments

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

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

Omar E. Pol, Feb 19 2018

Keywords

Examples

			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;
...

Crossrefs

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.

Formula

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