A136107 -id:A136107 - cp's OEIS frontend

A351285 Irregular triangle read by rows in which row n gives the y-values of the solutions of the equation 2(x - 1)y - (x - 3)x = 2n for 0 < x <= y.

2, 3, 4, 5, 3, 6, 7, 4, 8, 9, 5, 4, 10, 11, 6, 12, 5, 13, 7, 14, 5, 15, 8, 6, 16, 17, 9, 18, 7, 6, 19, 10, 20, 6, 21, 11, 8, 22, 7, 23, 12, 24, 9, 25, 13, 7, 26, 8, 27, 14, 10, 7, 28, 29, 15, 30, 11, 9, 8, 31, 16, 32, 33, 17, 12, 8, 34, 10, 35, 18, 9, 8, 36, 13

Offset: 3

Stefano Spezia, Feb 06 2022

Equivalently, the n-th row gives the row indices corresponding to n in the triangle A351153.

			Triangle begins:
  2;
  3;
  4;
  5, 3;
  6;
  7, 4;
  8;
  9, 5, 4;
  ...

Cf. A341830, A351153, A136107(row length or solutions number), A351284 (x-values).

Mathematica
```
Table[r={};For[d=1,d
				
```

A368072 Number of representations of n as the difference of two positive tetrahedral numbers.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2

Offset: 1

Ilya Gutkovskiy, Dec 10 2023

nonn

			a(36) = 2: 36 = 56 - 20 = 120 - 84.

Eric Weisstein's World of Mathematics, Tetrahedral Number

Cf. A000292, A136107, A307666, A368073, A368076.

A334077 a(n) is the smallest positive integer that can be expressed as the difference of two positive triangular numbers in at least n ways.

Original entry on oeis.org

2, 5, 9, 27, 45, 63, 105, 135, 225, 315, 315, 315, 945, 945, 945, 945, 1575, 1575, 2835, 2835, 3465, 3465, 3465, 3465, 10395, 10395, 10395, 10395, 10395, 10395, 10395, 10395, 17325, 17325, 17325, 17325, 31185, 31185, 31185, 31185, 45045, 45045, 45045, 45045

Offset: 1

Ilya Gutkovskiy, Apr 13 2020

nonn

Eric Weisstein's World of Mathematics, Triangular Number

Cf. A000217, A038547, A052346, A053587, A136107, A136108, A214771, A334078.

A368952 Irregular triangle T(n,k) read by rows: row n lists the larger number in each pair of triangular numbers (a, b) satisfying a - b = n.

Original entry on oeis.org

1, 3, 6, 3, 10, 15, 6, 21, 6, 28, 10, 36, 45, 15, 10, 55, 10, 66, 21, 78, 15, 91, 28, 105, 15, 120, 36, 21, 15, 136, 153, 45, 171, 28, 21, 190, 55, 210, 21, 231, 66, 36, 21, 253, 28, 276, 78, 300, 45, 325, 91, 28, 351, 36, 378, 105, 55, 28, 406, 28, 435, 120, 465, 66, 45, 36

Offset: 1

Hartmut F. W. Hoft, Jan 10 2024

The length of row n in the triangle is A001227(n) and its first column T(n, 1) is ordered. Also, A001227(n) = number of 1s in row n of the triangle of A237048 = length of row n in the triangle of A280851. The records of row lengths in the triangle form sequence A038547.

			For n=3 with 0 <= k <= 6, sqrt((2*k + 1)^2 - 8*3) has integer values for k=2, 3, so that the pairs of triangular numbers are (3, 0) and (6, 3), and row 3 of the triangle consists of 6 and 3.
The first 20 rows of the irregular triangle:
   n| k:   1     2     3     4
  -----------------------------
   1|      1
   2|      3
   3|      6     3
   4|     10
   5|     15     6
   6|     21     6
   7|     28    10
   8|     36
   9|     45    15    10
  10|     55    10
  11|     66    21
  12|     78    15
  13|     91    28
  14|    105    15
  15|    120    36    21    15
  16|    136
  17|    153    45
  18|    171    28    21
  19|    190    55
  20|    210    21
  ...

Cf. A000217, A001227, A038547, A136107, A237048, A280851.

a000217[k_] := k (k+1)/2
triangle[n_] := Map[a000217, Select[Range[a000217[n], 0, -1], IntegerQ[Sqrt[(2#+1)^2 -8n]]&]]
a368952[n_] := Flatten[Map[triangle, Range[n]]]
a368952[30]

n = A000217(x) - A000217(y), x > y >= 0, precisely when sqrt( (2*x + 1)^2 - 8*n ) is an integer.

A368364 a(n) = number of s with n^k-n^2 <= s <= n^k-1, k >= 3, such that a comma sequence in base n with initial term s will not reach n^k.

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 8, 11, 12, 14, 16, 18, 20, 23, 24, 26, 29, 31, 33, 36, 38, 40, 42, 45, 47, 51, 52, 54, 58, 60, 61, 65, 67, 71, 73, 75, 77, 81, 83, 85, 89, 91, 93, 98, 100, 102, 104, 107, 110, 114, 116, 118, 122, 125, 127, 131, 133, 135, 139, 141, 143, 149, 150, 154

Offset: 2

N. J. A. Sloane, Jan 19 2024

nonn

Conjectured to have g.f. (Sum_{n>=1} x^((n^2+3*n)/2)/(1-x^n) - x^2)/(1-x). [Corrected by N. J. A. Sloane, May 14 2024]
a(n) is independent of k provided k >= 3.
This is conjectured to equal A368363(n) - 1. Normally that would be enough to rule out this sequence. However, it is included because it is at present the only one of the nearly 100 OEIS entries based on comma sequences which has a connection with a sequence not connected with comma sequences.
(In the (virtual) graph that shows connections between OEIS entries, this sequence is the sole node at present that connects the component containing A121805 to the rest of the graph.)

			In base 10, a(10) = 8 values of s hit a landmine before reaching safety.

Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, arXiv:2401.14346, Fibonacci Quarterly 62:3 (2024), 215-232.
Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, Local copy.
N. J. A. Sloane, Eric Angelini's Comma Sequence, Experimental Math Seminar, Rutgers Univ., January 18, 2024, Youtube video; Slides

Cf. A121805, A136107, A368363.

cp's OEIS Frontend

A351285 Irregular triangle read by rows in which row n gives the y-values of the solutions of the equation 2*(x - 1)*y - (x - 3)*x = 2*n for 0 < x <= y.

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A368072 Number of representations of n as the difference of two positive tetrahedral numbers.

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A334077 a(n) is the smallest positive integer that can be expressed as the difference of two positive triangular numbers in at least n ways.

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A368952 Irregular triangle T(n,k) read by rows: row n lists the larger number in each pair of triangular numbers (a, b) satisfying a - b = n.

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A368364 a(n) = number of s with n^k-n^2 <= s <= n^k-1, k >= 3, such that a comma sequence in base n with initial term s will not reach n^k.

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A351285 Irregular triangle read by rows in which row n gives the y-values of the solutions of the equation 2(x - 1)y - (x - 3)x = 2n for 0 < x <= y.