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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A133113 A128174 * A007318 * A133080.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 6, 4, 4, 1, 9, 6, 11, 4, 1, 12, 9, 24, 11, 6, 1, 16, 12, 46, 24, 22, 6, 1, 20, 16, 80, 46, 62, 22, 8, 1, 25, 20, 130, 80, 148, 62, 37, 8, 1, 30, 25, 200, 130, 314, 148, 128, 37, 10, 1
Offset: 1

Views

Author

Gary W. Adamson, Sep 14 2007

Keywords

Comments

Row sums = 2^n - 1, A000225: (1, 3, 7, 15, 31, ...).

Examples

			First few rows of the triangle:
   1;
   2,  1;
   4,  2,  1;
   6,  4,  4,  1;
   9,  6, 11,  4,  1;
  12,  9, 24, 11,  6,  1;
  16, 12, 46, 24, 22,  6,  1;
  20, 16, 80, 46, 62, 22,  8,  1;
  ...

Crossrefs

Cf. A133080, A007318, A128174, A000225.

Formula

A128174 * A007318 * A133080 as infinite lower triangular matrices.

A134444 (A000012 * A128174 + A128174 * A000012) - A000012.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 5, 3, 3, 1, 1, 5, 5, 3, 3, 1, 1, 7, 5, 5, 3, 3, 1, 1, 7, 7, 5, 5, 3, 3, 1, 1, 9, 7, 7, 5, 5, 3, 3, 1, 1, 9, 9, 7, 7, 5, 5, 3, 3, 1, 1
Offset: 0

Views

Author

Gary W. Adamson, Oct 25 2007

Keywords

Comments

Row sums = A000982: (1, 2, 5, 8, 13, 18, 25, ...).

Examples

			First few rows of the triangle:
  1;
  1, 1;
  3, 1, 1;
  3, 3, 1, 1;
  5, 3, 3, 1, 1;
  5, 5, 3, 3, 1, 1;
  ...

Crossrefs

Cf. A128174, A000012, A000982.

Formula

(A000012 * A128174 + A128174 * A000012) - A000012, as infinite lower triangular matrices. Triangle read by rows, n-th row = n terms of odd numbers repeated, starting from the right.

A144152 Triangle read by rows: A128174 * X; X = an infinite lower triangular matrix with a shifted Fibonacci sequence: (1, 1, 1, 2, 3, 5, 8, ...) in the main diagonal and the rest zeros.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 1, 0, 2, 1, 0, 1, 0, 3, 0, 1, 0, 2, 0, 5, 1, 0, 1, 0, 3, 0, 8, 0, 1, 0, 2, 0, 5, 0, 13, 1, 0, 1, 0, 3, 0, 8, 0, 21, 0, 1, 0, 2, 0, 0, 5, 0, 13, 0, 34, 1, 0, 1, 0, 3, 0, 8, 0, 21, 0, 55
Offset: 1

Views

Author

Gary W. Adamson, Sep 12 2008

Keywords

Comments

The original definition was: Eigentriangle, row sums = Fibonacci numbers.
Even n rows are composed of odd-indexed Fibonacci numbers interpolated with zeros.
Odd n rows are composed of even-indexed Fibonacci numbers with alternate zeros.
Sum of n-th row terms = rightmost term of next row, = F(n-1). Row sums = F(n).

Examples

			First few rows of the triangle =
  1;
  0,  1;
  1,  0,  1;
  0,  1,  0,  2;
  1,  0,  1,  0,  3
  0,  1,  0,  2,  0,  5;
  1,  0,  1,  0,  3,  0,  8;
  0,  1,  0,  2,  0,  5,  0, 13;
  1,  0,  1,  0,  3,  0,  8,  0, 21;
  ...
Row 5 = (1, 0, 1, 0, 3) = termwise products of (1, 0, 1, 0, 1) and (1, 1, 1, 2, 3).

Crossrefs

Cf. A000045, A128174.

Programs

  • PARI
    MT(n,k) = (1+(-1)^(n-k))/2;
MF(n,k) = n--; k--; if (n==k, if (n==0, 1, fibonacci(n)), 0);
tabl(nn) = {my(T=matrix(nn,nn, n, k, MT(n,k))); my(F=matrix(nn,nn, n, k, MF(n,k))); my(P=T*F); matrix(nn, nn, n, k, if (n>=k, P[n,k], 0));} \\ Michel Marcus, Mar 08 2021

Formula

A128174 = the matrix: (1; 0,1; 1,0,1; 0,1,0,1; ...). These operations are equivalent to termwise products of n terms of A128174 matrix row terms and an equal number of terms in (1, 1, 1, 2, 3, 5, 8, ...).

Extensions

Moved a comment to the Name section. - Omar E. Pol, Mar 08 2021

A128177 A128174 * A004736 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 6, 4, 2, 1, 9, 6, 4, 2, 1, 12, 9, 6, 4, 2, 1, 16, 12, 9, 6, 4, 2, 1, 20, 16, 12, 9, 6, 4, 2, 1, 25, 20, 16, 12, 9, 6, 4, 2, 1, 30, 25, 20, 16, 12, 9, 6, 4, 2, 1, 36, 30, 25, 20, 16, 12, 9, 6, 4, 2, 1, 42, 36, 30, 25, 20, 16, 12, 9, 6, 4, 2, 1
Offset: 1

Views

Author

Gary W. Adamson, Feb 17 2007

Keywords

Comments

n-th row has n nonzero terms of A002620: (1, 2, 4, 6, 9, 12, 16, ...) in reverse.
Row sums = A002623: (1, 3, 7, 13, 22, 34, 50, ...).

Examples

			First few rows of the triangle:
   1;
   2, 1;
   4, 2, 1;
   6, 4, 2, 1;
   9, 6, 4, 2, 1;
  12, 9, 6, 4, 2, 1;
  ...

Crossrefs

Cf. A128174, A004736, A002623, A002620.

Programs

  • Maple
    seq(seq(floor((n-k+2)^2/4), k=1..n), n=1..20); # Ridouane Oudra, Mar 23 2024
  • Mathematica
    T[n_,k_]:=Floor[(n-k+2)^2/4];Table[T[n,k],{n,12},{k,n}]//Flatten (* James C. McMahon, Jan 05 2025 *)
  • PARI
    lista(nn) = {t128174 = matrix(nn, nn, n, k, (k<=n)*(1+(-1)^(n-k))/2); t004736 = matrix(nn, nn, n, k, (k<=n)*(n - k + 1)); t128177 = t128174*t004736; for (n = 1, nn, for (k = 1, n, print1(t128177[n, k], ", ");););} \\ Michel Marcus, Feb 11 2014

Formula

From Ridouane Oudra, Mar 23 2024: (Start)
T(n, k) = A002620(n-k+2), with 1 <= k <= n;
T(n, k) = floor((n-k+2)^2/4);
T(n, k) = (1/2)*floor((n-k+2)^2/2);
T(n, k) = (1/8)*(2*(n-k+2)^2 + (-1)^(n-k) - 1). (End)

Extensions

Partially edited and more terms from Michel Marcus, Feb 11 2014

A128186 A051731 * A128174.

Original entry on oeis.org

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

Views

Author

Gary W. Adamson, Feb 17 2007

Keywords

Comments

Left column = A001227.
Row sums = A079247: (1, 2, 3, 4, 4, 7, 5, 8, 8, 10, ...).
A128187 = A128174 * A051731.

Examples

			First few rows of the triangle:
  1;
  1, 1;
  2, 0, 1;
  1, 2, 0, 1;
  2, 0, 1, 0, 1;
  2, 2, 1, 1, 0, 1;
  2, 0, 1, 0, 1, 0, 1;
  ...

Crossrefs

Cf. A128174, A051731, A128187, A079247, A001227.

Formula

A051731 * A128174 as infinite lower triangular matrices.

A129240 A128174 * A129234.

Original entry on oeis.org

1, 2, 2, 4, 0, 3, 6, 5, 0, 4, 9, 0, 3, 0, 5, 12, 9, 4, 4, 0, 6, 16, 0, 3, 0, 5, 0, 7, 20, 14, 4, 9, 0, 6, 0, 8, 25, 0, 8, 0, 5, 0, 7, 0, 9, 30, 20, 4, 9, 6, 6, 0, 8, 0, 10
Offset: 1

Views

Author

Gary W. Adamson, Apr 05 2007

Keywords

Comments

Left border = A002620 starting (1, 2, 4, 6, 9, 12, 16, ...).

Examples

			First few rows of the triangle:
   1;
   2, 2;
   4, 0, 3;
   6, 5, 0, 4;
   9, 0, 3, 0, 5;
  12, 9, 4, 4, 0, 6;
  16, 0, 3, 0, 5, 0, 7;
  ...

Crossrefs

Cf. A128174, A129234, A002620.

Formula

A128174 * A129234 as infinite lower triangular matrices.

A129560 A054523 * A128174.

Original entry on oeis.org

1, 1, 1, 3, 0, 1, 3, 2, 0, 1, 7, 0, 1, 0, 1, 5, 4, 1, 1, 0, 1, 13, 0, 1, 0, 1, 0, 1, 9, 6, 1, 2, 0, 1, 0, 1, 19, 0, 3, 0, 1, 0, 1, 0, 1, 13, 10, 1, 2, 1, 1, 0, 1, 0, 1
Offset: 1

Views

Author

Gary W. Adamson, Apr 20 2007

Keywords

Comments

Row sums = A002620: (1, 2, 4, 6, 9, 12, 16, 20, ...). Left column = A106477: (1, 1, 3, 3, 7, 5, 13, 9, 19, ...).

Examples

			First few rows of the triangle:
   1;
   1, 1;
   3, 0, 1;
   3, 2, 0, 1;
   7, 0, 1, 0, 1;
   5, 4, 1, 1, 0, 1;
  13, 0, 1, 0, 1, 0, 1;
  ...

Crossrefs

Cf. A054523, A128174, A106477, A002620.

Formula

A054523 * A128174 as infinite lower triangular matrices.

A129569 A129360 * A128174.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1
Offset: 1

Views

Author

Gary W. Adamson, Apr 22 2007

Keywords

Comments

Row sums = A055034: (1, 1, 1, 2, 2, 2, 3, 4, 3, 4, ...).

Examples

			First few rows of the triangle:
  1;
  0, 1;
  0, 0, 1;
  0, 1, 0, 1;
  0, 0, 1, 0, 1;
  0, 0, 0, 1, 0, 1;
  0, 0, 1, 0, 1, 0, 1;
  ...

Crossrefs

Cf. A129360, A128174, A055034.

Formula

A129360 * A128174 as infinite lower triangular matrices.

A131226 Triangle read by rows: 3*A002260 - 2*A128174 as infinite lower triangular matrices.

Original entry on oeis.org

1, 3, 4, 1, 6, 7, 3, 4, 9, 10, 1, 6, 7, 12, 13, 3, 4, 9, 10, 15, 16, 1, 6, 7, 12, 13, 18, 19, 3, 4, 9, 10, 15, 16, 21, 22, 1, 6, 7, 12, 13, 18, 19, 24, 25, 3, 4, 9, 10, 15, 16, 21, 22, 27, 28, 1, 6, 7, 12, 13, 18, 19, 24, 25, 30, 31, 3, 4, 9, 10, 15, 16, 21, 22, 27, 28, 33, 34
Offset: 1

Views

Author

Gary W. Adamson, Jun 20 2007

Keywords

Comments

Row sums = A095894: (1, 7, 14, 26, 39, 57, ...).
A131225 = 2*A002260 - A128174.

Examples

			First few rows of the triangle:
  1;
  3, 4;
  1, 6, 7;
  3, 4, 9, 10;
  1, 6, 7, 12, 13;
  3, 4, 9, 10, 15, 16;
  1, 6, 7, 12, 13, 18, 19;
  ...

Crossrefs

Cf. A002260, A095894, A128174, A131225.

Extensions

a(47), a(49) corrected and more terms from Georg Fischer, Jun 07 2023

A131230 Triangle read by rows: 2*A130296 - A128174.

Original entry on oeis.org

1, 4, 1, 5, 2, 1, 8, 1, 2, 1, 9, 2, 1, 2, 1, 12, 1, 2, 1, 2, 1, 13, 2, 1, 2, 1, 2, 1, 16, 1, 2, 1, 2, 1, 2, 1, 17, 2, 1, 2, 1, 2, 1, 2, 1, 20, 1, 2, 1, 2, 1, 2, 1, 2, 1, 21, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 24, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 25, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1
Offset: 1

Views

Author

Gary W. Adamson, Jun 20 2007

Keywords

Comments

Left column = A042948, numbers congruent to {1,0} mod 4: (1, 4, 5, 8, 9, 12, ...).
Row sums = A047383, numbers congruent to {1,5} mod 7: (1, 5, 8, 12, 15, ...).

Examples

			First few rows of the triangle:
   1;
   4, 1;
   5, 2, 1;
   8, 1, 2, 1;
   9, 2, 1, 2, 1;
  12, 1, 2, 1, 2, 1;
  ...

Crossrefs

Cf. A047383, A042948, A128174, A131231, A130296.

Extensions

Incorrect formula removed and more terms from Georg Fischer, Jun 08 2023
Previous Showing 31-40 of 59 results. Next

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