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.

Showing 1-10 of 17 results. Next

A128132 A natural number transform, companion to A127701.

Original entry on oeis.org

1, -1, 2, 0, -1, 3, 0, 0, -1, 4, 0, 0, 0, -1, 5, 0, 0, 0, 0, -1, 6, 0, 0, 0, 0, 0, -1, 7, 0, 0, 0, 0, 0, 0, -1, 8, 0, 0, 0, 0, 0, 0, 0, -1, 9, 0, 0, 0, 0, 0, 0, 0, 0, -1, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 12
Offset: 1

Views

Author

Gary W. Adamson, Feb 15 2007

Keywords

Comments

Binomial transform is A128133.

Examples

			Triangle T(n,k) (with rows n >= 1 and columns k = 1..n) begins:
   1;
  -1,  2;
   0, -1,  3;
   0,  0, -1,  4;
   0,  0,  0, -1,  5;
   0,  0,  0,  0, -1, 6;
   ...

Crossrefs

Cf. A127701, A128133, A128134.

Programs

  • Maple
    A128132 := proc(n,k)
    if n = k then
        n;
    elif k = n-1 then
        -1 ;
    else
        0 ;
    end if;
end proc: # R. J. Mathar, Apr 26 2016
  • Mathematica
    {1}~Join~Table[PadLeft[{-1, n}, n], {n, 2, 12}] // Flatten (* Michael De Vlieger, Apr 26 2016 *)

Formula

T(n,n) = n.
T(n,n-1) = -1.
T(n,k) = 0 for k <> n, n-1.

A128222 A127701 * A128174.

Original entry on oeis.org

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

Views

Author

Gary W. Adamson, Feb 19 2007

Keywords

Comments

Row sums = A128223: (1, 3, 7, 10, 17, 21, 31, 36, ...).

Examples

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

Crossrefs

Cf. A127701, A128174, A128223, A128222.

Programs

  • Mathematica
    a128222[n_, k_] := If[EvenQ[n-k], n, 1]/;1<=k<=n
a128222[r_] := Table[a128222[n, k], {n, 1, r}, {k, 1, n}]
TableForm[a128222[7]] (* triangle *)
Flatten[a128222[10]] (* data *) (* Hartmut F. W. Hoft, Mar 08 2017 *)

Formula

A127701 * A128174 as infinite lower triangular matrices. Odd rows: n terms of n, 1, n, ...; even rows: n terms of 1, n, 1, ...

Extensions

Inserted omitted values a(28) = 7 and a(29) = 1, Hartmut F. W. Hoft, Mar 08 2017

A127704 A054525 * A127701.

Original entry on oeis.org

1, 0, 2, -1, 1, 3, -1, -2, 1, 4, -1, 0, 0, 1, 5, 0, -3, -3, 0, 1, 6, -1, 0, 0, 0, 0, 1, 7, 0, 0, -1, -4, 0, 0, 1, 8, 0, -1, -3, 0, 0, 0, 0, 1, 9, 0, -2, 0, -1, -5, 0, 0, 0, 1, 10, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 11, 1, 2, -1, -4, -1, -6, 0, 0, 0, 0, 1, 12, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 13, 0, -2
Offset: 1

Views

Author

Gary W. Adamson, Jan 24 2007

Keywords

Comments

Moebius transform of A127701.
Row sums = A127705: (1, 2, 3, 2, 5, 1, 7, 4, 6, 3, ...)

Examples

			First few rows of the triangle are:
   1;
   0,  2;
  -1,  1,  3;
  -1, -2,  1,  4;
  -1,  0,  0,  1,  5;
   0, -3, -3,  0,  1,  6;
  -1,  0,  0,  0,  0,  1,  7;
  ...

Links

Crossrefs

Cf. A054525, A127701, A127705.

Programs

  • Maple
    A054525 := proc(n,k) if n>=1 and 1<=k and k <= n then if n mod k = 0 then numtheory[mobius](n/k) ; else 0 ; fi; else 0 ; fi; end: A127701 := proc(n,k) if n<1 or k<1 or k > n then 0 ; elif n = k then n; elif k+ 1 =n then 1; else 0 ; fi; end: A127704 := proc(n,k) add( A054525(n,i)*A127701(i,k),i=1..n) ; end: for n from 1 to 30 do for k from 1 to n do printf("%d,",A127704(n,k)) ; od: od: # R. J. Mathar, Jul 21 2009

Extensions

More terms from R. J. Mathar, Jul 21 2009

A127735 Triangle read by rows: A127701 * A002260 as infinite lower triangular matrices.

Original entry on oeis.org

1, 3, 4, 4, 8, 9, 5, 10, 15, 16, 6, 12, 18, 24, 25, 7, 14, 21, 28, 35, 36, 8, 16, 24, 32, 40, 48, 49, 9, 18, 27, 36, 45, 54, 63, 64, 10, 20, 30, 40, 50, 60, 70, 80, 81, 11, 22, 33, 44, 55, 66, 77, 88, 99, 100, 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 121, 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 144
Offset: 1

Views

Author

Gary W. Adamson, Jan 26 2007

Keywords

Examples

			First few rows of the triangle are:
1;
3, 4;
4, 8, 9;
5, 10, 15, 16;
6, 12, 18, 24, 25;
...

Crossrefs

Cf. A002260, A127701, A127736.

Formula

Row sums = A127736: (1, 7, 21, 46, 85, 141, ...).
T(n,n) = n^2. T(n,k) = k*(n+1), 1<=kR. J. Mathar, Jul 21 2009

Extensions

A-number of left factor in the definition corrected by R. J. Mathar, Jul 21 2009
a(19) corrected and more terms from Georg Fischer, Jun 05 2023

A128221 A128174 * A127701.

Original entry on oeis.org

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

Views

Author

Gary W. Adamson, Feb 19 2007

Keywords

Comments

Row sums = A024206: (1, 3, 5, 8, 11, 15, 19, ...). A128222 = A127701 * A128174.
Table T(n,k) = n, if k is odd, 1 if k is even; n, k > 0, read by antidiagonals. -Boris Putievskiy, Jan 30 2013

Examples

			From _Boris Putievskiy_, Jan 30 2013: (Start)
The start of the sequence as a table:
  1, 1, 1, 1, 1, 1, 1, ...
  2, 1, 2, 1, 2, 1, 2, ...
  3, 1, 3, 1, 3, 1, 3, ...
  4, 1, 4, 1, 4, 1, 4, ...
  5, 1, 5, 1, 5, 1, 5, ...
  6, 1, 6, 1, 6, 1, 6, ...
  7, 1, 7, 1, 7, 1, 7, ...
  ...
(End)
First few rows of the triangle are:
  1;
  1, 2;
  1, 1, 3;
  1, 2, 1, 4;
  1, 1, 3, 1, 5;
  1, 2, 1, 4, 1, 6;
  1, 1, 3, 1, 5, 1, 7;
  ...

Links

Crossrefs

Cf. A128174, A127701, A024206, A128222.

Programs

  • Mathematica
    a128221[n_, k_] := If[EvenQ[n-k], k, 1]/;1<=k<=n
a128221[r_] := Table[a128221[n, k], {n, 1, r}, {k, 1, n}]
TableForm[a128221[7]] (* triangle *)
Flatten[a128221[10]] (* data *) (* Hartmut F. W. Hoft, Mar 08 2017 *)
t[r_, c_] := If[ OddQ@ c, r, 1]; Table[t[k, n - k + 1], {n, 13}, {k, n}] // Flatten (* Robert G. Wilson v, Mar 09 2017 *)

Formula

A128174 * A127701 as infinite lower triangular matrices. By columns, k-th column = k, 1, k, ...; k=1,2,3,...
From Boris Putievskiy, Jan 30 2013: (Start)
As table T(n,k) = (1+(-1)^k)/2 - (-1+(-1)^k)*n/2.
As linear sequence a(n) = (1+(-1)^A004736(n))/2 - (-1+(-1)^A004736(n))*A002260(n)/2. a(n) = (1+(-1)^j)/2 - (-1+(-1)^j)*i/2,
where i = n-t*(t+1)/2, j = (t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). (End)

Extensions

More terms from Robert G. Wilson v, Mar 09 2017

A128219 A000012 * A127701. a(1) = 1, a(2) = 2, a(3) = 2; by rows, n-1 terms of 2, 3, 4, ... followed by "n".

Original entry on oeis.org

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

Views

Author

Gary W. Adamson, Feb 19 2007

Keywords

Examples

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

Crossrefs

Cf. A000012, A127701, A034856 (row sums), A128220.

Programs

  • Mathematica
    trm[i_]:=Join[Range[2,i],{i}]; Flatten[Table[trm[n],{n,13}]] (* Harvey P. Dale, Nov 14 2012 *)

Formula

A000012 * A127701 as infinite lower triangular matrices.

A128220 Triangle, A127701 * A000012.

Original entry on oeis.org

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

Views

Author

Gary W. Adamson, Feb 19 2007

Keywords

Comments

Row sums = A028387: (1, 5, 11, 19, 29, 41, 55, ...) A000012 * A127701 = A128219.

Examples

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

Crossrefs

Cf. A128219, A000012, A127701, A028387.

Formula

A127701 * A000012 as infinite lower triangular matrices. Triangle read by rows: a(1) = 1; n-th row = (n-1) terms of (n+1) followed by "n".

A133981 Triangle read by rows: A000012 * A127701 + A127701 * A000012 - A000012 as infinite lower triangular matrices.

Original entry on oeis.org

1, 4, 3, 5, 6, 5, 6, 7, 8, 7, 7, 8, 9, 10, 9, 8, 9, 10, 11, 12, 11, 9, 10, 11, 12, 13, 14, 13, 10, 11, 12, 13, 14, 15, 16, 15, 11, 12, 13, 14, 15, 16, 17, 18, 17, 12, 13, 14, 15, 16, 17, 18, 19, 20, 19, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 21, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 23
Offset: 1

Views

Author

Gary W. Adamson, Sep 30 2007

Keywords

Comments

Row sums = A133694: (1, 7, 16, 28, 43, 61, ...).

Examples

			First few rows of the triangle:
   1;
   4,  3;
   5,  6,  5;
   6,  7,  8,  7;
   7,  8,  9, 10,  9;
   8,  9, 10, 11, 12, 11;
   9, 10, 11, 12, 13, 14, 13;
  10, 11, 12, 13, 14, 15, 16, 15;
  11, 12, 13, 14, 15, 16, 17, 18, 17;
  ...

Crossrefs

Cf. A127701, A133694.

Extensions

a(26) = 13 corrected and more terms from Georg Fischer, Jun 07 2023

A127737 A002260 * A127701.

Original entry on oeis.org

1, 3, 4, 3, 7, 9, 3, 7, 13, 16, 3, 7, 13, 21, 25, 3, 7, 13, 21, 31, 36, 3, 7, 13, 21, 31, 43, 49, 3, 7, 13, 21, 31, 43, 57, 64, 3, 7, 13, 21, 31, 43, 57, 73, 81, 3, 7, 13, 21, 31, 43, 57, 73, 91, 100
Offset: 1

Views

Author

Gary W. Adamson, Jan 27 2007

Keywords

Comments

Deleting the right border (1, 4, 9, 16, ...), rows tend to A002061 starting (3, 7, 13, 21, 31, ...). Row sums = A108766: (1, 7, 19, 39, 69, 111, ...).

Examples

			First few rows of the triangle:
  1;
  3, 4;
  3, 7,  9;
  3, 7, 13, 16;
  3, 7, 13, 21, 25;
  ...

Crossrefs

Cf. A002260, A127701, A108766, A002061.

Formula

A002260 * A127701 as infinite lower triangular matrices.

A127738 Triangle read by rows: the matrix product A004736 * A127701 of two triangular matrices.

Original entry on oeis.org

1, 3, 2, 5, 5, 3, 7, 8, 7, 4, 9, 11, 11, 9, 5, 11, 14, 15, 14, 11, 6, 13, 17, 19, 19, 17, 13, 7, 15, 20, 23, 24, 23, 20, 15, 8, 17, 23, 27, 29, 29, 27, 23, 17, 9, 19, 26, 31, 34, 35, 34, 31, 26, 19, 10
Offset: 1

Views

Author

Gary W. Adamson, Jan 27 2007

Keywords

Comments

Left column = A028387: (1, 5, 11, 19, 29, 41, 55, ...).

Examples

			First few rows of the triangle:
   1;
   3,  2;
   5,  5,  3;
   7,  8,  7,  4;
   9, 11, 11,  9,  5;
  11, 14, 15, 14, 11,  6;
  13, 17, 19, 19, 17, 13,  7;
  ...

Crossrefs

Cf. A004736, A127701, A008778 (row sums), A028387.

Formula

T(n,k) = Sum_{j=k..n} A004736(n,j)*A127701(j,k). - R. J. Mathar, Aug 31 2022
T(n,k) = k+(k+1)*(n-k) = n+k*(n-k) = n +A094053(n,k) = A059036(n,k). - R. J. Mathar, Aug 31 2022
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