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 10 results.

A128134 A128132 * A007318.

Original entry on oeis.org

1, 1, 2, 2, 5, 3, 3, 10, 11, 4, 4, 17, 27, 19, 5, 5, 26, 54, 56, 29, 6, 6, 37, 95, 130, 100, 41, 7, 7, 50, 153, 260, 265, 162, 55, 8, 8, 65, 231, 469, 595, 483, 245, 71, 9, 9, 82, 332, 784, 1190, 1204, 812, 352, 89, 10
Offset: 1

Views

Author

Gary W. Adamson, Feb 15 2007

Keywords

Comments

A007318 * A128132 = A128133. Row sums = A128135: (1, 3, 10, 28, 72, 176, ...).

Examples

			Triangle T(n,k) (with rows n >= 1 and columns k = 1..n) begins:
  1;
  1,  2;
  2,  5,  3;
  3, 10, 11,   4;
  4, 17, 27,  19,   5;
  5, 26, 54,  56,  29,  6;
  6, 37, 95, 130, 100, 41, 7;
  ...

Crossrefs

Cf. A002522, A007318, A028387, A128132, A128133, A128135, A164845, A332697.

Formula

A128132 * A007318 as infinite lower triangular matrices (assuming the top of the Pascal triangle A007318 is shifted from (0,0) to (1,1)).
From Petros Hadjicostas, Jul 26 2020: (Start)
T(n,k) = n*binomial(n-1, k-1) - binomial(n-2, k-1)*[n <> k] for 1 <= k <= n, where [ ] is the Iverson bracket.
Bivariate o.g.f.: x*y*(1 - x + x^2*(1 + y))/(1 - x*(1 + y))^2.
T(n,k) = T(n-1,k) + T(n-1,k-1) + binomial(n-1,k-1) for 2 <= k <= n with (n,k) <> (2,2).
T(n,k) = 2*T(n-1,k) - T(n-2,k) - T(n-2,k-2) + 2*T(n-1,k-1) - 2*T(n-2,k-1) for 3 <= k <= n with (n,k) <> (3,3).
T(n,1) = n - 1 for n >= 2.
T(n,2) = A002522(n-1) for n >= 2.
T(n,3) = A164845(n-3) for n >= 3.
T(n,4) = A332697(n-3) for n >= 4.
T(n,n) = n for n >= 1.
T(n,n-1) = A028387(n-2) for n >= 2. (End)

A128133 Binomial transform of A128132.

Original entry on oeis.org

1, 0, 2, -1, 3, 3, -2, 3, 8, 4, -3, 2, 14, 15, 5, -4, 0, 20, 35, 24, 6, -5, -3, 25, 65, 69, 35, 7, -6, -7, 28, 105, 154, 119, 48, 8, -7, -12, 28, 154, 294, 308, 188, 63, 9, -8, -18, 24, 210, 504, 672, 552, 279, 80, 10
Offset: 1

Views

Author

Gary W. Adamson, Feb 15 2007

Keywords

Comments

Row sums = A005183: (1, 2, 5, 13, 33, 81, 193, ...).
A128132 * A007318 = A128134.

Examples

			First few rows of the triangle:
   1;
   0,  2;
  -1,  3,  3;
  -2,  3,  8,  4;
  -3,  2, 14, 15,  5;
  -4,  0, 20, 35, 24,  6;
  -5, -3, 25, 65, 69, 35,  7;
  ...

Crossrefs

Cf. A128132, A007318, A128134, A005183.

Formula

A007318 * A128132 as infinite lower triangular matrices.

A128139 Triangle read by rows: matrix product A004736 * A128132.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 4, 5, 4, 1, 5, 7, 7, 5, 1, 6, 9, 10, 9, 6, 1, 7, 11, 13, 13, 11, 7, 1, 8, 13, 16, 17, 16, 13, 8, 1, 9, 15, 19, 21, 21, 19, 15, 9, 1, 10, 17, 22, 25, 26, 25, 22, 17, 10
Offset: 0

Views

Author

Gary W. Adamson, Feb 16 2007

Keywords

Comments

A077028 with the final term in each row omitted.
Interchanging the factors in the matrix product leads to A128140 = A128132 * A004736.
From Gary W. Adamson, Jul 01 2012: (Start)
Alternatively, antidiagonals of an array A(n,k) of sequences with arithmetic progressions as follows:
1, 2, 3, 4, 5, 6, ...
1, 3, 5, 7, 9, 11, ...
1, 4, 7, 10, 13, 16, ...
1, 5, 9, 13, 17, 21, ...
... (End)
From Gary W. Adamson, Jul 02 2012: (Start)
A summation generalization for Sum_{k>=1} 1/(A(n,k)*A(n,k+1)) (formulas copied from A002378, A000466, A085001, A003185):
1 = 1/(1)*(2) + 1/(2)*(3) + 1/(3)*(4) + ...
1 = 2/(1)*(3) + 2/(3)*(5) + 2/(5)*(7) + ...
1 = 3/(1)*(4) + 3/(4)*(7) + 3/(7)*(10) + ...
1 = 4/(1)*(5) + 4/(5)*(9) + 4/(9)*(13) + ...
...
As a summation of terms equating to a definite integral:
Integral_{0..1} dx/(1+x) = ... 1 - 1/2 + 1/3 - 1/4 + ... = log(2).
Integral_{0..1} dx/(1+x^2) = 1 - 1/3 + 1/5 - 1/7 + ... = Pi/4 (see A157142)
Integral_{0..1} dx/(1+x^3) = 1 - 1/4 + 1/7 - 1/10 + ... (see A016777)
Integral_{0..1} dx/(1+x^4) = 1 - 1/5 + 1/9 - 1/13 + ... (see A016813). (End)

Examples

			First few rows of the triangle:
  1;
  1,  2;
  1,  3,  3;
  1,  4,  5,  4;
  1,  5,  7,  7,  5;
  1,  6,  9, 10,  9,  6;
  1,  7, 11, 13, 13, 11,  7;
  1,  8, 13, 16, 17, 16, 13,  8;
  1,  9, 15, 19, 21, 21, 19, 15,  9;
  1, 10, 17, 22, 25, 26, 25, 22, 17, 10;
  ...

Crossrefs

Cf. A004736, A128132, A128140, A004006 (row sums).

Formula

A004736 * A128132 as infinite lower triangular matrices.
T(n,k) = k*(1+n-k)+1 = 1 + A094053(n+1,1+n-k). - R. J. Mathar, Jul 09 2012

A128136 A128132 * A002260.

Original entry on oeis.org

1, 1, 4, 2, 4, 9, 3, 6, 9, 16, 4, 8, 12, 16, 25, 5, 10, 15, 20, 25, 36, 6, 12, 18, 24, 30, 36, 49, 7, 14, 21, 28, 35, 42, 49, 64, 8, 16, 24, 32, 40, 48, 56, 64, 81, 9, 18, 27, 36, 45, 54, 63, 72, 81, 100
Offset: 1

Views

Author

Gary W. Adamson, Feb 16 2007

Keywords

Comments

Row sums = A006003: (1, 5, 15, 34, 65, ...).
A002260 * A128132 = A128137.

Examples

			First few rows of the triangle:
  1;
  1,  4;
  2,  4,  9;
  3,  6,  9, 16;
  4,  8, 12, 16, 25;
  5, 10, 15, 20, 25, 36;
  6, 12, 18, 24, 30, 36, 49;
  ...

Crossrefs

Cf. A128132, A002260, A006003, A128137.

Formula

A128132 * A002260 as infinite lower triangular matrices.

A128137 A002260 * A128132.

Original entry on oeis.org

1, -1, 4, -1, 1, 9, -1, 1, 5, 16, -1, 1, 5, 11, 25, -1, 1, 5, 11, 19, 36, -1, 1, 5, 11, 19, 29, 49, -1, 1, 5, 11, 19, 29, 41, 64, -1, 1, 5, 11, 19, 29, 41, 55, 81, -1, 1, 5, 11, 19, 29, 41, 55, 71, 100
Offset: 1

Views

Author

Gary W. Adamson, Feb 16 2007

Keywords

Comments

Deleting the right and left borders gives terms of A028387: (1, 5, 11, 29, 29, 41, 55, 71, ...). Row sums = A064999: (1, 3, 9, 21, 41, ...).

Examples

			First few rows of the triangle:
   1;
  -1, 4;
  -1, 1, 9;
  -1, 1, 5, 16;
  -1, 1, 5, 11, 25;
  -1, 1, 5, 11, 19, 36;
  -1, 1, 5, 11, 19, 29, 49;
  ...

Crossrefs

Cf. A002260, A128132, A028387, A128136, A064999.

Formula

A002260 * A128132 as infinite lower triangular matrices.

A128138 A000012 * A128132.

Original entry on oeis.org

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

Views

Author

Gary W. Adamson, Feb 16 2007

Keywords

Examples

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

Crossrefs

Cf. A000012, A128132, A000124 (row sums).

Programs

  • Mathematica
    Table[Delete[Range[0, n], -2], {n, 14}] // Flatten (* or *)
Table[If[k == n - 1, k + 1, k], {n, 14}, {k, 0, n - 1}] (* Michael De Vlieger, Apr 26 2016 *)

Formula

A000012 * A128132 as infinite lower triangular matrices.
T(n,n) = n.
T(n,k) = k-1, 0

A128140 A128132 * A004736.

Original entry on oeis.org

1, 3, 2, 7, 5, 3, 13, 10, 7, 4, 21, 17, 13, 9, 5, 31, 26, 21, 16, 11, 6, 43, 37, 31, 25, 19, 13, 7, 57, 50, 43, 36, 29, 22, 15, 8, 73, 65, 57, 49, 41, 33, 25, 17, 9, 91, 82, 73, 64, 55, 46, 37, 28, 19, 10
Offset: 1

Views

Author

Gary W. Adamson, Feb 16 2007

Keywords

Comments

Row sums = A006003, starting (1, 5, 15, 34, 65, 111, ...). Left border = A002061: (1, 3, 7, 13, 21, 31, 43, ...) A128139 = A004736 * A128132

Examples

			First few rows of the triangle:
   1;
   3,  2;
   7,  5,  3;
  13, 10,  7,  4;
  21, 17, 13,  9,  5;
  31, 26, 21, 16, 11,  6;
  43, 37, 31, 25, 19, 13,  7;
  ...

Crossrefs

Cf. A128132, A004736, A006003, A004006, A128139.

Formula

A128132 * A004736 as infinite lower triangular matrices.

A128141 A122432 (unsigned) * A128132.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 4, 9, 8, 4, 5, 14, 15, 11, 5, 6, 20, 24, 21, 14, 6, 7, 27, 35, 34, 27, 17, 7, 8, 35, 48, 50, 44, 33, 20, 8, 9, 44, 63, 69, 65, 54, 39, 23, 9, 10, 54, 80, 91, 90, 80, 64, 45, 26, 10
Offset: 1

Views

Author

Gary W. Adamson, Feb 16 2007

Keywords

Comments

Row sums = A006522 starting (1, 4, 11, 25, 50, 91, ...). A128142 = A128132 * A122432.

Examples

			First few rows of the triangle:
  1;
  2,  2;
  3,  5,  3;
  4,  9,  8,  4;
  5, 14, 15, 11,  5;
  6, 20, 24, 21, 14,  6;
  7, 27, 35, 34, 27, 17,  7;
  ...

Crossrefs

Cf. A122432, A128132, A006522, A128142, A006325.

Formula

Unsigned A122432 * A128132 as infinite lower triangular matrices; where unsigned A122432 = (1; 3, 1; 6, 3, 1; ...).

A128142 A128132 * A122432 (unsigned).

Original entry on oeis.org

1, 5, 2, 15, 8, 3, 34, 21, 11, 4, 65, 44, 27, 14, 5, 111, 80, 54, 33, 17, 6, 175, 132, 95, 64, 39, 20, 7, 260, 203, 153, 110, 74, 45, 23, 8, 369, 296, 231, 174, 125, 84, 51, 26, 9, 505, 414, 332, 259, 195, 140, 94, 57, 29, 10
Offset: 1

Views

Author

Gary W. Adamson, Feb 16 2007

Keywords

Comments

Row sums = A006325 starting (1, 7, 26, 70, 155, ...).
Left border = A006003: (1, 5, 15, 34, 65, 111, ...).
A128141 = A122432 * A128132.

Examples

			First few rows of the triangle:
    1;
    5,   2;
   15,   8,  3;
   34,  21, 11,  4;
   65,  44, 27, 14,  5;
  111,  80, 54, 33, 17,  6;
  175, 132, 95, 64, 39, 20,  7;
  ...

Crossrefs

Cf. A128132, A122432, A006325, A128141, A006003.

Formula

A128132 * A122432 (unsigned); as infinite lower triangular matrices.

A128135 Row sums of A128134.

Original entry on oeis.org

1, 3, 10, 28, 72, 176, 416, 960, 2176, 4864, 10752, 23552, 51200, 110592, 237568, 507904, 1081344, 2293760, 4849664, 10223616, 21495808, 45088768, 94371840, 197132288, 411041792, 855638016, 1778384896, 3690987520, 7650410496, 15837691904, 32749125632, 67645734912, 139586437120, 287762808832
Offset: 1

Views

Author

Gary W. Adamson, Feb 16 2007

Keywords

Comments

Conjecture: a(n)/a(n-1) tends to sqrt(5). (E.g., a(10)/a(9) = 2.235294....)
The conjecture is false. The fraction a(n)/a(n-1) tends to 2 as n grows. - Philipp Zumstein (zuphilip(AT)inf.ethz.ch), Oct 05 2009
This sequence is a subsequence of a greedily and recursively defined sequence (see links). - Sela Fried, Aug 30 2024
For n>=2, a(n) is the total number of ones in runs of ones of length >=3 over all binary strings of length n+1. - Félix Balado, Aug 06 2025

Examples

			a(4) = 28 = sum of row 4 of A128134 = 3 + 10 + 11 + 4.

Links

Crossrefs

Cf. A128132, A128133, A128134, A134315.

Programs

  • Magma
    I:=[1, 3, 10]; [n le 3 select I[n] else 4*Self(n-1)-4*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Jun 28 2012
  • Mathematica
    CoefficientList[Series[(1-x+2*x^2)/(1-2*x)^2,{x,0,40}],x] (* Vincenzo Librandi, Jun 28 2012 *)
LinearRecurrence[{4,-4},{1,3,10},40] (* Harvey P. Dale, May 26 2023 *)
  • PARI
    a(n)=if(n<=2,[1,3][n],2*a(n-1)+2^(n-1)); /* Joerg Arndt, Sep 29 2012 */

Formula

Row sums of A128134.
Equals A134315 * [1, 2, 3, ...]. - Gary W. Adamson, Oct 19 2007
a(n) = 2*a(n-1) + 2^(n-1) for n >= 2. - Philipp Zumstein (zuphilip(AT)inf.ethz.ch), Oct 05 2009
From Colin Barker, May 29 2012: (Start)
a(n) = 2^(n - 2)*(2*n - 1) for n > 1.
a(n) = 4*a(n-1) - 4*a(n-2) for n > 3.
G.f.: x*(1 - x + 2*x^2)/(1 - 2*x)^2. (End)
G.f.: (1 - G(0))/2 where G(k) = 1 - (2*k + 2)/(1 - x/(x - (k + 1)/G(k+1))) (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 06 2012
From Amiram Eldar, Aug 05 2020: (Start)
Sum_{n>=1} 1/a(n) = 2*sqrt(2)*arcsinh(1) - 1.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*sqrt(2)*arccot(sqrt(2)) - 1. (End)

Extensions

More terms from Philipp Zumstein (zuphilip(AT)inf.ethz.ch), Oct 05 2009
Incorrect formula deleted by Colin Barker, May 29 2012
Showing 1-10 of 10 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.