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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A350634 Products of the parts s,t in each partition of k (= 2,3,...) into two parts, ordered by increasing k and then by increasing values of s*t (see example).

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 5, 8, 9, 6, 10, 12, 7, 12, 15, 16, 8, 14, 18, 20, 9, 16, 21, 24, 25, 10, 18, 24, 28, 30, 11, 20, 27, 32, 35, 36, 12, 22, 30, 36, 40, 42, 13, 24, 33, 40, 45, 48, 49, 14, 26, 36, 44, 50, 54, 56, 15, 28, 39, 48, 55, 60, 63, 64, 16, 30, 42, 52, 60, 66, 70, 72
Offset: 1

Views

Author

Wesley Ivan Hurt, Jan 09 2022

Keywords

Comments

If b > 0 and c > 0 are the integer coefficients of a monic quadratic x^2 + b*x + c, it has integer roots if its discriminant d^2 = b^2 - 4c is a perfect square. This sequence is the values of c for increasing b sorted by b then c. The first pair of (b, c) = (2, 1) and has d = 0. The n-th pair of (b, c) = (A027434(n),a(n)) and has d = A082375(n-1). - Frank M Jackson, Jan 22 2024

Examples

			---------------------------------------------------------------------------
The products of the parts start: 1*1, 1*2, 1*3, 2*2, 1*4, 2*3, etc., which are precisely the values of a(n): 1, 2, 3, 4, 4, 6, ...
                                                                     [1,9]
                                                     [1,7]   [1,8]   [2,8]
                                     [1,5]   [1,6]   [2,6]   [2,7]   [3,7]
                     [1,3]   [1,4]   [2,4]   [2,5]   [3,5]   [3,6]   [4,6]
     [1,1]   [1,2]   [2,2]   [2,3]   [3,3]   [3,4]   [4,4]   [4,5]   [5,5]
  k    2       3       4       5       6       7       8       9      10
---------------------------------------------------------------------------

Crossrefs

Cf. A027434, A082375, A122197, A199474, A339399.

Programs

  • Mathematica
    Times@@@Flatten[Table[IntegerPartitions[k, {2}], {k, 2, 100}], 1] (* Frank M Jackson, Jan 22 2024 *)
lst={}; Do[If[IntegerQ[d=Sqrt[b^2-4c]], AppendTo[lst, c]], {b, 1, 100}, {c, 1, b^2/4}]; lst (* Frank M Jackson, Jan 22 2024 *)

Formula

a(n) = A122197(n) * A199474(n).
a(n) = A339399(2n-1) * A339399(2n).
a(n) = ((A027434(n))^2 - (A082375(n))^2)/4. - Frank M Jackson, Jan 22 2024

A270861 Irregular triangle read by rows: numerators of the coefficients of polynomials J(2n-1,z) = Sum_(k=1,2, .. n) ((n+1)^2 - k + (n+1-k)*z^n)*z^(k-1)/k.

Original entry on oeis.org

3, 1, 8, 7, 2, 1, 15, 7, 13, 3, 1, 1, 24, 23, 22, 21, 4, 3, 2, 1, 35, 17, 11, 8, 31, 5, 2, 1, 1, 1, 48, 47, 46, 45, 44, 43, 6, 5, 4, 3, 2, 1, 63, 31, 61, 15, 59, 29, 57, 7, 3, 5, 1, 3, 1, 1, 80, 79, 26, 77, 76, 25, 74, 73, 8, 7, 2, 5, 4, 1, 2, 1
Offset: 1

Views

Author

Paul Curtz, Mar 24 2016

Keywords

Comments

Irregular triangle of fractions:
3, 1,
8, 7/2, 2, 1/2,
15, 7, 13/3, 3, 1, 1/3,
24, 23/2, 22/3, 21/4, 4, 3/2, 2/3, 1/4,
35, 17, 11, 8, 31/5, 5, 2, 1, 1/2, 1/5,
48, 47/2, 46/3, 45/4, 44/5, 43/6, 6, 5/2, 4/3, 3/4, 2/5, 1/6.
etc.
First column: A005563; T(n, 1) = A005563(n).
Main diagonal: T(n, n) - n = n^2+1 = A002522(n).
The first upper diagonal is T(n, n+1) = n.
Consider TT(n, k) = k*T(n, k) for k = 1 to n:
3,
8, 7,
15, 14, 13,
24, 23, 22, 21,
etc.
Row sums: 3, 8+7, ... , are the positive terms of A059270; that is A059270(n).

Examples

			Irregular triangle:
3,   1,
8,   7,  2,  1,
15,  7, 13,  3,  1,  1,
24, 23, 22, 21,  4,  3, 2, 1,
35, 17, 11,  8, 31,  5, 2, 1, 1, 1
48, 47, 46, 45, 44, 43, 6, 5, 4, 3, 2, 1
etc.
Second half part by row: A112543.

Links

Crossrefs

Cf. A002260, A002378, A002522, A005563, A059270, A112543, A122197, A004736.

Programs

  • Mathematica
    row[n_] := CoefficientList[Sum[(((n + 1)^2 - k + (n + 1 - k)*z^n))*z^(k - 1)/k, {k, n}], z]; Table[row[n] // Numerator, {n, 1, 9}] // Flatten (* Jean-François Alcover, Apr 07 2016 *)

A158440 Triangle T(n,k) read by rows: row n contains n times n+1 followed by n 1's.

Original entry on oeis.org

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

Views

Author

Paul Curtz, Mar 19 2009

Keywords

Comments

These are essentially the unreduced numerators of the coefficients of P(2n-1,x) of A130679, with denominators represented by A122197.

Examples

			The triangle has 2n columns in row n. It starts:
  2, 1;
  3, 3, 1, 1;
  4, 4, 4, 1, 1, 1;
  5, 5, 5, 5, 1, 1, 1, 1;

Crossrefs

Cf. A003057.

Programs

  • Mathematica
    Flatten[Table[Join[PadRight[{},n,n+1],PadRight[{},n,1]],{n,12}]] (* Harvey P. Dale, Feb 18 2013 *)

Extensions

Edited by R. J. Mathar, Apr 09 2009

A347912 a(n) = Sum_{k=1..n} k - floor(sqrt(k)+1/2) * floor(sqrt(k-1)).

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 9, 11, 14, 15, 17, 20, 21, 23, 26, 30, 31, 33, 36, 40, 41, 43, 46, 50, 55, 56, 58, 61, 65, 70, 71, 73, 76, 80, 85, 91, 92, 94, 97, 101, 106, 112, 113, 115, 118, 122, 127, 133, 140, 141, 143, 146, 150, 155, 161, 168, 169, 171, 174, 178, 183, 189, 196, 204
Offset: 1

Views

Author

Wesley Ivan Hurt, Sep 18 2021

Keywords

Comments

Partial sums of A122197.

Crossrefs

Cf. A122197.

Programs

  • Mathematica
    Table[Sum[k - Floor[Sqrt[k] + 1/2]*Floor[Sqrt[k - 1]], {k, n}], {n, 100}]
Previous Showing 11-14 of 14 results.

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