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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A291971 Triangle read by rows: T(n,k) = 3 * T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

Original entry on oeis.org

1, 0, 3, 0, 3, 0, 3, 9, 0, 3, 9, 0, 3, 18, 0, 3, 18, 27, 0, 3, 27, 27, 0, 3, 27, 54, 0, 3, 36, 81, 0, 3, 36, 108, 81, 0, 3, 45, 135, 81, 0, 3, 45, 189, 162, 0, 3, 54, 216, 243, 0, 3, 54, 270, 405, 0, 3, 63, 324, 486, 243, 0, 3, 63, 378, 729, 243, 0, 3, 72, 432, 891
Offset: 0

Views

Author

Seiichi Manyama, Sep 07 2017

Keywords

Examples

			First few rows are:
  1;
  0, 3;
  0, 3;
  0, 3,  9;
  0, 3,  9;
  0, 3, 18;
  0, 3, 18,  27;
  0, 3, 27,  27;
  0, 3, 27,  54;
  0, 3, 36,  81;
  0, 3, 36, 108, 81.

Links

Crossrefs

Row sums give A032308.
Columns 0-1 give A000007, A010701.
Cf. A008289 (m=1), A291970 (m=2), this sequence (m=3).

A362209 Irregular triangle read by rows: T(n, k) is the number of k X k matrices using all the integers from 1 to k^2 and having trace equal to n, with 1 <= k <= A003056(n).

Original entry on oeis.org

1, 0, 0, 4, 0, 4, 0, 8, 0, 4, 4320, 0, 4, 4320, 0, 0, 8640, 0, 0, 12960, 0, 0, 17280, 11496038400, 0, 0, 21600, 11496038400, 0, 0, 30240, 22992076800, 0, 0, 30240, 34488115200, 0, 0, 34560, 57480192000, 0, 0, 34560, 68976230400, 291948240981196800000
Offset: 1

Views

Author

Stefano Spezia, Apr 11 2023

Keywords

Examples

			Irregular triangle begins:
    1;
    0;
    0, 4;
    0, 4;
    0, 8;
    0, 4,  4320;
    0, 4,  4320;
    0, 0,  8640;
    0, 0, 12960;
    0, 0, 17280, 11496038400;
    0, 0, 21600, 11496038400;
    0, 0, 30240, 22992076800;
    0, 0, 30240, 34488115200;
    0, 0, 34560, 57480192000;
    0, 0, 34560, 68976230400, 291948240981196800000;
    ...
T(5,2) = 8 since we have:
    [1, 2]  [1, 3]  [4, 2]  [4, 3]
    [3, 4], [2, 4], [3, 1], [2, 1],
.
    [2, 1]  [2, 4]  [3, 1]  [3, 4]
    [4, 3], [1, 3], [4, 2], [1, 2].

Crossrefs

Cf. A000290, A003056 (row lengths), A345132, A362187, A362208.

Programs

  • Mathematica
    A362208[n_,k_] := Length[Select[Join@@Permutations/@Select[IntegerPartitions[n, All, Range[k^2]], UnsameQ@@#&], Length[#]==k&]]; Table[(k^2-k)!A362208[n,k],{n,15},{k,Floor[(Sqrt[8n+1]-1)/2]}]//Flatten

Formula

T(n, k) = A362187(k)*A362208(n, k).

A384999 Irregular triangle read by rows: T(n,k) is the total number of partitions of all numbers <= n with k designated summands, n >= 0, 0 <= k <= A003056(n).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 8, 1, 1, 15, 4, 1, 21, 13, 1, 33, 28, 1, 1, 41, 58, 4, 1, 56, 103, 13, 1, 69, 170, 35, 1, 87, 269, 77, 1, 1, 99, 404, 158, 4, 1, 127, 579, 298, 13, 1, 141, 810, 529, 35, 1, 165, 1116, 880, 86, 1, 189, 1470, 1431, 183, 1, 1, 220, 1935, 2214, 371, 4, 1, 238, 2475, 3348, 701, 13
Offset: 0

Views

Author

Omar E. Pol, Jul 22 2025

Keywords

Comments

When part i has multiplicity j > 0 exactly one part i is "designated".
The length of the row n is A002024(n+1) = 1 + A003056(n), hence the first positive element in column k is in the row A000217(k).
Column k gives the partial sums of the column k of A385001.
Columns converge to A210843 which is also the partial sums of A000716.

Examples

			Triangle begins:
---------------------------------------------
   n\k:   0    1     2      3     4    5   6
---------------------------------------------
   0 |    1;
   1 |    1,   1;
   2 |    1,   4;
   3 |    1,   8,    1;
   4 |    1,  15,    4;
   5 |    1,  21,   13;
   6 |    1,  33,   28,     1;
   7 |    1,  41,   58,     4;
   8 |    1,  56,  103,    13;
   9 |    1,  69,  170,    35;
  10 |    1,  87,  269,    77,    1;
  11 |    1,  99,  404,   158,    4;
  12 |    1, 127,  579,   298,   13;
  13 |    1, 141,  810,   529,   35;
  14 |    1, 165, 1116,   880,   86;
  15 |    1, 189, 1470,  1431,  183,   1;
  16 |    1, 220, 1935,  2214,  371,   4;
  17 |    1, 238, 2475,  3348,  701,  13;
  18 |    1, 277, 3156,  4894, 1269,  35;
  19 |    1, 297, 3921,  7036, 2187,  86;
  20 |    1, 339, 4866,  9871, 3639, 194;
  21 |    1, 371, 5906, 13629, 5872, 402,  1;
  ...

Links

Crossrefs

Columns: A000012 (k=0), A024916 (k=1).
Cf. A000217, A000716, A002024, A003056, A077285, A195825, A210843, A211970, A385001.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+add(expand(b(n-i*j, i-1)*j*x), j=1..n/i)))
    end:
g:= proc(n) option remember; `if`(n<0, 0, g(n-1)+b(n$2)) end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(g(n)):
seq(T(n), n=0..20);  # Alois P. Heinz, Jul 22 2025

A050601 Recursion counts for summation table A003056 with formula a(0,x) = x, a(y,0) = y, a(y,x) = a((y XOR x),2*(y AND x)).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 1, 2, 0, 0, 1, 2, 2, 1, 0, 0, 2, 1, 1, 1, 2, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 3, 2, 3, 1, 3, 2, 3, 0, 0, 1, 3, 3, 2, 2, 3, 3, 1, 0, 0, 2, 1, 3, 2, 1, 2, 3, 1, 2, 0, 0, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 0, 0, 3, 2, 3, 1, 3, 1, 3, 1, 3, 2, 3, 0, 0, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 0, 0, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 0
Offset: 0

Views

Author

Antti Karttunen, Jun 22 1999

Keywords

Crossrefs

Cf. A050600, A050602, A003056, A048720 (for the Maple implementation of trinv and XORnos, ANDnos)

Programs

  • Maple
    add2c := proc(a,b) option remember; if((0 = a) or (0 = b)) then RETURN(0); else RETURN(1+add_c(XORnos(a,b),2*ANDnos(a,b))); fi; end;
  • Mathematica
    trinv[n_] := Floor[(1/2)*(Sqrt[8*n + 1] + 1)];
Sum2c[a_, b_] := Sum2c[a, b] = If[0 == a || 0 == b, Return[0], Return[ Sum2c[BitXor[a, b], 2*BitAnd[a, b]] + 1]];
a[n_] := Sum2c[n - (1/2)*trinv[n]*(trinv[n] - 1), (trinv[n] - 1)*(trinv[ n]/2 + 1) - n];
Table[a[n], {n, 0, 120}](* Jean-François Alcover, Mar 07 2016, adapted from Maple *)

Formula

a(n) -> add2c( (n-((trinv(n)*(trinv(n)-1))/2)), (((trinv(n)-1)*(((1/2)*trinv(n))+1))-n) )

A166021 a(n) = 2*A000124(A003056(n-1)) if A002262(n-1)=0, otherwise a(n-1)+1.

Original entry on oeis.org

2, 4, 5, 8, 9, 10, 14, 15, 16, 17, 22, 23, 24, 25, 26, 32, 33, 34, 35, 36, 37, 44, 45, 46, 47, 48, 49, 50, 58, 59, 60, 61, 62, 63, 64, 65, 74, 75, 76, 77, 78, 79, 80, 81, 82, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121
Offset: 1

Views

Author

Antti Karttunen, Oct 05 2009

Keywords

Crossrefs

Complement of A136272.

A291955 Triangle read by rows: T(n,k) = T(n-k,k-1) - T(n-k,k+1) with T(0,0) = 1 for 0 <= k <= A003056(n).

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 1, 0, 0, 0, 0, -1, 0, 0, 1, 1, -1, 0, -1, 0, 0, 1, 0, 0, 2, -1, 0, 0, -2, -1, 1, 0, 0, 1, 1, -1, -1, 0, -1, -3, 2, 0, 0, 3, 2, -1, -1, 1, 0, -2, -3, 2, 1, 0, 0, 3, 4, -3, -1, 0, 0, -4, -4, 3, 2, -1, 0, 4
Offset: 0

Views

Author

Seiichi Manyama, Sep 06 2017

Keywords

Examples

			First few rows are:
  1;
  0,  1;
  0,  0;
  0,  0,  1;
  0, -1,  0;
  0,  0,  0;
  0,  0, -1,  1;
  0,  1,  0,  0;
  0,  0, -1,  0;
  0,  1,  1, -1;
  0, -1,  0,  0, 1.

Links

Crossrefs

Row sums give A291956.
Columns 0-1 give A000007, A049346.
Cf. A291904.

A291957 Triangle read by rows: T(n,k) = T(n-k,k-1) - T(n-k,k) + T(n-k,k+1) with T(0,0) = 1 for 0 <= k <= A003056(n).

Original entry on oeis.org

1, 0, 1, 0, -1, 0, 1, 1, 0, 0, -1, 0, -1, 0, 0, 1, 1, 1, 0, 0, -1, -1, 0, -1, 1, 0, 0, 2, 0, 0, 0, -2, -2, 0, 1, 0, 0, 2, 1, -1, 0, 2, 0, 0, 0, 0, -2, -1, -1, 0, 0, 1, 2, 0, -1, 0, 1, -2, 0, 2, 1, 0, -3, -1, 0, 0, -1, 0, 2, 3, 1, -1, 0, 0, 1, -2, 0, 1, 0, 0, -3, 0
Offset: 0

Views

Author

Seiichi Manyama, Sep 06 2017

Keywords

Examples

			First few rows are:
  1;
  0,  1;
  0, -1;
  0,  1,  1;
  0,  0, -1;
  0, -1,  0;
  0,  1,  1,  1;
  0,  0, -1, -1;
  0, -1,  1,  0;
  0,  2,  0,  0;
  0, -2, -2,  0, 1.

Links

Crossrefs

Row sums give A291958.
Columns 0-1 give A000007, (-1)*A111165 (for n>0).

A291968 Triangle read by rows: T(n,k) = (k+1) * T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

Original entry on oeis.org

1, 0, 2, 0, 2, 0, 2, 6, 0, 2, 6, 0, 2, 12, 0, 2, 12, 24, 0, 2, 18, 24, 0, 2, 18, 48, 0, 2, 24, 72, 0, 2, 24, 96, 120, 0, 2, 30, 120, 120, 0, 2, 30, 168, 240, 0, 2, 36, 192, 360, 0, 2, 36, 240, 600, 0, 2, 42, 288, 720, 720, 0, 2, 42, 336, 1080, 720, 0, 2, 48, 384
Offset: 0

Views

Author

Seiichi Manyama, Sep 07 2017

Keywords

Examples

			First few rows are:
  1;
  0, 2;
  0, 2;
  0, 2,  6;
  0, 2,  6;
  0, 2, 12;
  0, 2, 12, 24;
  0, 2, 18, 24;
  0, 2, 18, 48;
  0, 2, 24, 72;
  0, 2, 24, 96, 120.

Links

Crossrefs

Row sums give A072576.
Columns 0-1 give A000007, A007395.
Cf. A216652.

Formula

G.f. of column k: (k+1)! * x^(k*(k+1)/2) / Product_{j=1..k} (1-x^j).

A291969 Triangle read by rows: T(n,k) = k * (T(n-k,k-1) + T(n-k,k)) with T(0,0) = 1 for 0 <= k <= A003056(n).

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 6, 0, 1, 6, 6, 0, 1, 14, 6, 0, 1, 14, 18, 0, 1, 30, 36, 0, 1, 30, 60, 24, 0, 1, 62, 96, 24, 0, 1, 62, 198, 72, 0, 1, 126, 270, 144, 0, 1, 126, 474, 336, 0, 1, 254, 780, 480, 120, 0, 1, 254, 1188, 1080, 120, 0, 1, 510, 1800
Offset: 0

Views

Author

Seiichi Manyama, Sep 07 2017

Keywords

Examples

			First few rows are:
  1;
  0, 1;
  0, 1;
  0, 1,  2;
  0, 1,  2;
  0, 1,  6;
  0, 1,  6,  6;
  0, 1, 14,  6;
  0, 1, 14, 18;
  0, 1, 30, 36;
  0, 1, 30, 60, 24.

Links

Crossrefs

Row sums give A204858.
Columns 0-1 give A000007, A000012.
Cf. A291960.

Formula

G.f. of column k: k! * x^(k*(k+1)/2) / Product_{j=1..k} (1-j*x^j).

A292130 Triangle read by rows: T(n,k) = (-3) * T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

Original entry on oeis.org

1, 0, -3, 0, -3, 0, -3, 9, 0, -3, 9, 0, -3, 18, 0, -3, 18, -27, 0, -3, 27, -27, 0, -3, 27, -54, 0, -3, 36, -81, 0, -3, 36, -108, 81, 0, -3, 45, -135, 81, 0, -3, 45, -189, 162, 0, -3, 54, -216, 243, 0, -3, 54, -270, 405, 0, -3, 63, -324, 486, -243, 0, -3, 63, -378
Offset: 0

Views

Author

Seiichi Manyama, Sep 09 2017

Keywords

Examples

			First few rows are:
  1;
  0, -3;
  0, -3;
  0, -3,  9;
  0, -3,  9;
  0, -3, 18;
  0, -3, 18,  -27;
  0, -3, 27,  -27;
  0, -3, 27,  -54;
  0, -3, 36,  -81;
  0, -3, 36, -108, 81.

Links

Crossrefs

Row sums give A292128.
Columns 0-1 give A000007, (-1)*A010701.
Cf. A291971.
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