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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A253902 Write numbers 1, then 2^2 down to 1, then 3^2 down to 1, then 4^2 down to 1 and so on.

Original entry on oeis.org

1, 4, 3, 2, 1, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1
Offset: 1

Views

Author

Mikael Aaltonen, Jan 18 2015

Keywords

Comments

Triangle read by rows in which row n lists the first n^2 positive integers in decreasing order, n >= 1. - Omar E. Pol, Jan 20 2015

Examples

			From _Omar E. Pol_, Jan 20 2015: (Start)
Written as an irregular triangle in which row lengths are successive squares, the sequence begins:
   1;
   4,  3,  2,  1;
   9,  8,  7,  6,  5,  4,  3, 2, 1;
  16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1;
  ...
(End)

Crossrefs

Cf. A000290, A064866, A074279.

Programs

  • Maple
    T:= n-> (t-> seq(t-i, i=0..t-1))(n^2):
seq(T(n), n=1..6);  # Alois P. Heinz, Nov 05 2024
  • Mathematica
    a253902[n_] := Flatten@ Table[Reverse[Range[i^2]], {i, n}]; a253902[6] (* Michael De Vlieger, Jan 19 2015 *)
  • PARI
    lista(nn=10) = {for (n=1, nn, forstep(k=n^2, 1, -1, print1(k, ", ");););} \\ Michel Marcus, Jan 20 2015
  • Python
    from sympy import integer_nthroot
def A253902(n): return (k:=(m:=integer_nthroot(3*n,3)[0])+(6*n>m*(m+1)*((m<<1)+1))+2)*(k*((k<<1)-9)+13)//6-n # Chai Wah Wu, Nov 05 2024

Formula

For 1 <= n <= 650, a(n) = -n + (t + 2)*(2*t^2 - t + 3)/6, where t = floor((3*n)^(1/3)+1/2).
a(n) = k*(2*k^2-9*k+13)/6-n where k = floor((3*n)^(1/3))+3 if n>A000330(floor((3*n)^(1/3))) and k = floor((3*n)^(1/3))+2 otherwise. - Chai Wah Wu, Nov 05 2024

A345018 For each n, append to the sequence n^2 consecutive integers, starting from n.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 4, 5, 6, 7, 8, 9, 10, 11, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41
Offset: 1

Views

Author

Paolo Xausa, Jun 05 2021

Keywords

Comments

Irregular triangle read by rows T(n,k) in which row n lists the integers from n to n + n^2 - 1, with n >= 1.

Examples

			Written as an irregular triangle T(n,k) the sequence begins:
  n\k|  1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16 ...
  ---+---------------------------------------------------------------
   1 |  1;
   2 |  2,  3,  4,  5;
   3 |  3,  4,  5,  6,  7,  8,  9, 10, 11;
   4 |  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19;
  ...

Links

Crossrefs

Column 1: A000027.
Right border: A028387.
Row lengths: A000290.
Row sums: A255499.
Cf. A064866, A074279.

Programs

  • Maple
    T:= n-> (t-> seq(n+i, i=0..t-1))(n^2):
seq(T(n), n=1..6);  # Alois P. Heinz, Nov 05 2024
  • Mathematica
    Table[Range[n,n^2+n-1],{n,6}] (* Paolo Xausa, Sep 05 2023 *)
  • PARI
    row(n) = vector(n^2, k, n+k-1); \\ Michel Marcus, Jun 08 2021
  • Python
    from sympy import integer_nthroot
def A345018(n): return n-1+(k:=(m:=integer_nthroot(3*n,3)[0])+(6*n>m*(m+1)*((m<<1)+1)))*(k*(3-(k<<1))+5)//6 # Chai Wah Wu, Nov 05 2024

Formula

T(n,k) = n + k - 1, with n >= 1 and 1 <= k <= n^2.

A065222 Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the hexagonal numbers. The final elements of the rows form a(n).

Original entry on oeis.org

1, 2, 5, 3, 8, 14, 6, 14, 23, 5, 16, 28, 41, 10, 25, 41, 58, 10, 29, 49, 70, 1, 24, 48, 73, 99, 6, 34, 63, 93, 124, 3, 36, 70, 105, 141, 178, 26, 65, 105, 146, 188, 231, 44, 89, 135, 182, 230, 3, 53, 104, 156, 209, 263, 318, 49, 106, 164, 223, 283, 344, 28, 91, 155, 220
Offset: 0

Views

Author

Floor van Lamoen, Oct 22 2001

Keywords

Examples

			The array begins
....1
...1.2
..3.4.5
.6.1.2.3
4.5.6.7.8

Crossrefs

Cf. A000384.
Mini-index to these sequences: A064766, A064865, A064866, A065221-A655234 are all of the same type. See A064766 for a detailed explanation.

A065223 Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the heptagonal numbers (A000566). The final elements of the rows form a(n).

Original entry on oeis.org

1, 2, 5, 2, 7, 13, 2, 10, 19, 29, 6, 18, 31, 45, 5, 21, 38, 56, 75, 14, 35, 57, 80, 104, 17, 43, 70, 98, 127, 9, 40, 72, 105, 139, 174, 21, 58, 96, 135, 175, 216, 23, 66, 110, 155, 201, 248, 10, 59, 109, 160, 212, 265, 319, 32, 88, 145, 203, 262, 322, 383, 42, 105
Offset: 0

Views

Author

Floor van Lamoen, Oct 22 2001

Keywords

Crossrefs

Mini-index to these sequences: A064766, A064865, A064866, A065221-A655234 are all of the same type. See A064766 for a detailed explanation.

A065224 Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the octagonal numbers. The final elements of the rows form a(n).

Original entry on oeis.org

1, 2, 5, 1, 6, 12, 19, 6, 15, 25, 36, 8, 21, 35, 50, 1, 18, 36, 55, 75, 96, 22, 45, 69, 94, 120, 14, 42, 71, 101, 132, 164, 21, 55, 90, 126, 163, 201, 15, 55, 96, 138, 181, 225, 270, 36, 83, 131, 180, 230, 281, 333, 45, 99, 154, 210, 267, 325, 384, 36, 97, 159, 222
Offset: 0

Views

Author

Floor van Lamoen, Oct 22 2001

Keywords

Crossrefs

Cf. A000567.
Mini-index to these sequences: A064766, A064865, A064866, A065221-A655234 are all of the same type. See A064766 for a detailed explanation.

A065225 Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the 9-gonal (nonagonal) numbers. The final elements of the rows form a(n).

Original entry on oeis.org

1, 2, 5, 9, 5, 11, 18, 2, 11, 21, 32, 44, 11, 25, 40, 56, 73, 16, 35, 55, 76, 98, 10, 34, 59, 85, 112, 140, 15, 45, 76, 108, 141, 175, 6, 42, 79, 117, 156, 196, 237, 18, 61, 105, 150, 196, 243, 291, 15, 65, 116, 168, 221, 275, 330, 386, 47, 105, 164, 224, 285, 347, 410
Offset: 0

Views

Author

Floor van Lamoen, Oct 22 2001

Keywords

Crossrefs

Cf. A001106.
Mini-index to these sequences: A064766, A064865, A064866, A065221-A655234 are all of the same type. See A064766 for a detailed explanation.

A065226 Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the decagonal numbers. The final elements of the rows form a(n).

Original entry on oeis.org

1, 2, 5, 9, 4, 10, 17, 25, 7, 17, 28, 40, 1, 15, 30, 46, 63, 81, 15, 35, 56, 78, 101, 125, 24, 50, 77, 105, 134, 164, 20, 52, 85, 119, 154, 190, 227, 33, 72, 112, 153, 195, 238, 282, 30, 76, 123, 171, 220, 270, 321, 3, 56, 110, 165, 221, 278, 336, 395, 4, 65, 127, 190
Offset: 0

Views

Author

Floor van Lamoen, Oct 22 2001

Keywords

Crossrefs

Cf. A001107.
Mini-index to these sequences: A064766, A064865, A064866, A065221-A655234 are all of the same type. See A064766 for a detailed explanation.

A065227 Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the triangular numbers. The first elements of the rows form a(n).

Original entry on oeis.org

1, 1, 3, 3, 1, 6, 2, 9, 2, 11, 21, 11, 23, 8, 22, 1, 17, 34, 7, 26, 46, 12, 34, 57, 15, 40, 66, 15, 43, 72, 11, 42, 74, 2, 36, 71, 107, 24, 62, 101, 5, 46, 88, 131, 22, 67, 113, 160, 37, 86, 136, 187, 49, 102, 156, 1, 57, 114, 172, 231, 60, 121, 183, 246, 57, 122, 188, 255
Offset: 0

Views

Author

Floor van Lamoen, Oct 22 2001

Keywords

Crossrefs

Mini-index to these sequences: A064766, A064865, A064866, A065221-A655234 are all of the same type. See A064766 for a detailed explanation.

A065229 Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the pentagonal numbers. The first elements of the rows form a(n).

Original entry on oeis.org

1, 1, 3, 1, 5, 10, 4, 11, 19, 6, 16, 27, 4, 17, 31, 46, 11, 28, 46, 65, 15, 36, 58, 81, 13, 38, 64, 91, 2, 31, 61, 92, 124, 12, 46, 81, 117, 154, 16, 55, 95, 136, 178, 11, 55, 100, 146, 193, 241, 43, 93, 144, 196, 249, 16, 71, 127, 184, 242, 301, 31, 92, 154, 217, 281, 346
Offset: 0

Views

Author

Floor van Lamoen, Oct 22 2001

Keywords

Crossrefs

Mini-index to these sequences: A064766, A064865, A064866, A065221-A655234 are all of the same type. See A064766 for a detailed explanation.

A065230 Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the hexagonal numbers. The first elements of the rows form a(n).

Original entry on oeis.org

1, 1, 3, 6, 4, 9, 15, 7, 15, 24, 6, 17, 29, 42, 11, 26, 42, 59, 11, 30, 50, 71, 2, 25, 49, 74, 100, 7, 35, 64, 94, 125, 4, 37, 71, 106, 142, 179, 27, 66, 106, 147, 189, 1, 45, 90, 136, 183, 231, 4, 54, 105, 157, 210, 264, 319, 50, 107, 165, 224, 284, 345, 29, 92, 156, 221
Offset: 0

Views

Author

Floor van Lamoen, Oct 22 2001

Keywords

Crossrefs

Mini-index to these sequences: A064766, A064865, A064866, A065221-A655234 are all of the same type. See A064766 for a detailed explanation.
Previous Showing 11-20 of 24 results. Next

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