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.

User: Binay Krishna Maity

Binay Krishna Maity's wiki page.

Binay Krishna Maity has authored 4 sequences.

A385439 Row sums of triangle A385865.

Original entry on oeis.org

1, 2, 6, 10, 13, 14, 20, 42, 45, 44, 50, 78, 77, 70, 104, 136, 165, 152, 150, 182, 209, 184, 172, 350, 325, 342, 322, 406, 365, 372, 400, 484, 561, 490, 582, 666, 665, 572, 580, 820, 805, 860, 770, 930, 897, 846, 824, 1274, 1325, 1156
Offset: 2

Views

Author

Binay Krishna Maity, Jul 30 2025

Keywords

Crossrefs

Cf. A385865.

Programs

  • PARI
    a(n) = sum(k=1, n-1, (k^2-1) % n + 1); \\ Michel Marcus, Aug 01 2025

A386319 Triangle read by rows where row n is the start, corner and end vertex numbers of a triangular spiral with n sides on a triangular grid, starting from 1 and working inwards (0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 5, 6, 1, 4, 7, 9, 10, 1, 5, 9, 12, 14, 15, 1, 6, 11, 15, 18, 20, 21, 1, 7, 13, 18, 22, 25, 27, 28, 1, 8, 15, 21, 26, 30, 33, 35, 36, 1, 9, 17, 24, 30, 35, 39, 42, 44, 45, 1, 10, 19, 27, 34, 40, 45, 49, 52, 54, 55, 1, 11, 21, 30, 38, 45, 51, 56, 60, 63, 65, 66, 1, 12, 23, 33, 42, 50, 57, 63, 68, 72, 75, 77, 78
Offset: 0

Views

Author

Binay Krishna Maity, Jul 18 2025

Keywords

Comments

The first 2 sides are length n-1 so that T(n,1) = 1 + (n-1) and T(n,2) = 1 + 2*(n-1) and then the side lengths decrease by 1 each time as it spirals in (ending at triangular number A000217(n) when n>=1).
These sides mesh to fill the triangle as they go inwards, and can also be thought of going outwards tracing out the sides of the triangle.
The resulting vertex numbers are 1 together with row n of A141419.
Row n=1 is taken as a side of length 0 so the start and end numbers are both 1 (which is not really a spiral but is consistent with the formula and two points 1,2 would be even less like a triangle filled by a spiral).

Examples

			Triangle begins:
--------------------------------------
   n\k  0   1   2   3   4   5   6   7
--------------------------------------
   0|   1;
   1|   1,  1;
   2|   1,  2,  3;
   3|   1,  3,  5,  6;
   4|   1,  4,  7,  9, 10;
   5|   1,  5,  9, 12, 14, 15;
   6|   1,  6, 11, 15, 18, 20, 21;
   7|   1,  7, 13, 18, 22, 25, 27, 28;
  ...
For n = 2 the spiral is 2 sides of length 1 so row [1, 2, 3],
   1 --- 2
       /
     3
For n = 4 the spiral is:
   1  2  3  4
    9  10  5
      8  6
        7
The start, corner and end vertices are [1, 4, 7, 9, 10].

Crossrefs

Columns: A000012 (k=0), A000027 (k=1), A144396 (k=3).
Cf. A179865(n+1) (main diagonal), A056520 (row sums).
Cf. A000217, A141419.

Programs

  • Mathematica
    T[n_,k_]:=If[k==0,1,k(2n-k+1)/2];Table[T[n,k],{n,0,12},{k,0,n}]//Flatten (* James C. McMahon, Jul 31 2025 *)

Formula

T(n,0) = 1.
T(n,k) = k*(2*n - k + 1)/2 for k >= 1.

A385866 Triangle read by rows where T(n,k), for 1 <= k < n, is the row number where (n-k)^2 occurs in an n X n grid filled rowwise with the numbers 1 to n^2.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 4, 2, 1, 1, 5, 3, 2, 1, 1, 6, 4, 3, 2, 1, 1, 7, 5, 4, 2, 2, 1, 1, 8, 6, 4, 3, 2, 1, 1, 1, 9, 7, 5, 4, 3, 2, 1, 1, 1, 10, 8, 6, 5, 4, 3, 2, 1, 1, 1, 11, 9, 7, 6, 5, 3, 3, 2, 1, 1, 1, 12, 10, 8, 7, 5, 4, 3, 2, 2, 1, 1, 1, 13, 11, 9, 8, 6, 5, 4, 3, 2, 2, 1, 1, 1
Offset: 2

Views

Author

Binay Krishna Maity, Jul 10 2025

Keywords

Comments

Rows are numbered starting from 1.

Examples

			      k=1  2  3  4  5  6
  n=2:  1
  n=3:  2, 1
  n=4:  3, 1, 1
  n=5:  4, 2, 1, 1
  n=6:  5, 3, 2, 1, 1
  n=7:  6, 4, 2, 2, 1, 1
For n = 5, the grid is:
  1  2  3  4  5
  6  7  8  9 10
 11 12 13 14 15
 16 17 18 19 20
 21 22 23 24 25
The squares (n-k)^2 = {16, 9, 4, 1} are in row numbers {4, 2, 1, 1} respectively.

Crossrefs

Cf. A385865 (column position).

Programs

  • Mathematica
    T[n_,k_]:=Floor[((n-k)^2-1)/n]+1;Table[T[n,k],{n,14},{k,n-1}]//Flatten (* James C. McMahon, Jul 17 2025 *)

Formula

T(n, k) = floor(((n-k)^2 - 1) / n) + 1.

A385865 Triangle read by rows where T(n,k), for 1 <= k < n, is the column number where (n-k)^2 occurs in an n X n grid filled rowwise with the numbers 1 to n^2.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 4, 4, 1, 1, 4, 3, 4, 1, 1, 4, 2, 2, 4, 1, 1, 4, 1, 8, 1, 4, 1, 1, 4, 9, 7, 7, 9, 4, 1, 1, 4, 9, 6, 5, 6, 9, 4, 1, 1, 4, 9, 5, 3, 3, 5, 9, 4, 1, 1, 4, 9, 4, 1, 12, 1, 4, 9, 4, 1, 1, 4, 9, 3, 12, 10, 10, 12, 3, 9, 4, 1, 1, 4, 9, 2, 11, 8, 7, 8, 11, 2, 9, 4, 1, 1
Offset: 2

Views

Author

Binay Krishna Maity, Jul 10 2025

Keywords

Comments

Columns are numbered starting from 1.

Examples

			      k=1  2  3  4  5  6
  n=2:  1
  n=3:  1, 1
  n=4:  1, 4, 1
  n=5:  1, 4, 4, 1
  n=6:  1, 4, 3, 4, 1
  n=7:  1, 4, 2, 2, 4, 1
For n = 5, the grid is
    1  2  3  4  5
    6  7  8  9 10
   11 12 13 14 15
   16 17 18 19 20
   21 22 23 24 25
The squares (n-k)^2 = {16, 9, 4, 1} are in column numbers {1, 4, 4, 1} respectively.

Crossrefs

Cf. A385866.

Programs

  • Mathematica
    T[n_,k_]:=Mod[k^2-1,n]+1;Table[T[n,k],{n,0,14},{k,n-1}]//Flatten (* James C. McMahon, Jul 16 2025 *)
  • PARI
    row(n) = vector(n-1, k, (k^2-1) % n + 1); \\ Michel Marcus, Jul 11 2025

Formula

T(n, k) = ((k^2 - 1) mod n) + 1.

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