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 1-4 of 4 results.

A124934 Numbers of the form 4mn - m - n, where m, n are positive integers.

Original entry on oeis.org

2, 5, 8, 11, 12, 14, 17, 19, 20, 23, 26, 29, 30, 32, 33, 35, 38, 40, 41, 44, 47, 50, 52, 53, 54, 56, 59, 61, 62, 63, 65, 68, 71, 74, 75, 77, 80, 82, 83, 85, 86, 89, 90, 92, 95, 96, 98, 101, 103, 104, 107, 109, 110, 113, 116, 117, 118, 119, 122, 124, 125, 128, 129, 131
Offset: 1

Views

Author

Nick Hobson, Nov 13 2006

Keywords

Comments

a(n) misses the squares since (2x)^2 + 1 = (4m - 1)(4n - 1) is impossible.
a(n) misses the triangular numbers since (2x + 1)^2 + 1 = 2(4m - 1)(4n - 1) is impossible.
Taking m = k(k - 1)/2, n = k(k + 1)/2 gives 4mn - m - n = (k^2 - 1)^2 - 1, so a(n) is one less than a square infinitely often.
Complement of A094178; A125203(a(n)) > 0; union of A125217 and A125218; range of A125199. - Reinhard Zumkeller, Nov 24 2006

Examples

			a(1) = 2 because 2 = 4*1*1 - 1 - 1 is the smallest value in the sequence.

References

  • L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. Dover Publications, Inc., Mineola, NY, 2005, p. 401.

Links

Crossrefs

Cf. A000290, A000217.

Programs

  • Haskell
    import Data.List (findIndices)
a124934 n = a124934_list !! (n-1)
a124934_list = map (+ 1) $ findIndices (> 0) a125203_list
-- Reinhard Zumkeller, Jan 02 2013

Extensions

More terms from Reinhard Zumkeller, Nov 24 2006

A125203 Number of ways to write n as 4*x*y - x - y with 1<=x<=y.

Original entry on oeis.org

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

Views

Author

Reinhard Zumkeller, Nov 24 2006

Keywords

Comments

a(A094178(n))=0; a(A124934(n))>0; a(A125217(n))=1; a(A125218(n))>1.

Links

Crossrefs

Cf. A094178, A124934, A125217, A125218.
Cf. A193773.

Programs

  • Haskell
    a125203 n = length [() | x <- [1 .. (n + 1) `div` 3],
                         let (y,m) = divMod (x + n) (4 * x - 1),
                         x <= y, m == 0]
-- Reinhard Zumkeller, Jan 02 2013
  • Mathematica
    a[n_] := Solve[1<=x<=y && n == 4 x y - x - y, {x, y}, Integers] // Length;
Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Oct 12 2021 *)

A125217 Numbers that can be written uniquely as 4*x*y-x-y with 1<=x<=y.

Original entry on oeis.org

2, 5, 8, 11, 12, 14, 17, 19, 20, 23, 29, 30, 32, 33, 35, 38, 40, 44, 50, 52, 53, 54, 59, 61, 62, 63, 65, 75, 77, 80, 82, 83, 85, 90, 92, 95, 98, 103, 104, 109, 113, 117, 118, 119, 122, 124, 125, 129, 132, 134, 137, 138, 143, 145, 147, 149, 151, 158, 159, 162, 164, 167
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 24 2006

Keywords

Comments

A125203(a(n)) = 1.
A124934 is the union of this sequence and A125218.

Links

Programs

  • Haskell
    import Data.List (elemIndices)
a125217 n = a125217_list !! (n-1)
a125217_list = map (+ 1) $ elemIndices 1 a125203_list
-- Reinhard Zumkeller, Jan 02 2013

A380644 Numbers that can be expressed as 4*j*k+j+k, j,k >= 1, as well as 4*j*k-j-k, j,k >= 2.

Original entry on oeis.org

26, 41, 47, 56, 61, 68, 71, 74, 86, 89, 96, 101, 107, 110, 116, 128, 131, 140, 146, 151, 152, 155, 159, 161, 166, 173, 176, 182, 185, 191, 194, 201, 206, 208, 209, 215, 221, 224, 236, 239, 242, 250, 251, 257, 261, 263, 266, 271, 272, 278, 281, 290, 293, 296, 299
Offset: 1

Views

Author

Hugo Pfoertner, Jan 29 2025

Keywords

Examples

			a(1) = 26, because 26 = 4*1*5 + 1 + 5 = 4*2*4 - 2 - 4.
a(2) = 41: 41 = 4*1*8 + 1 + 8 = 4*3*4 - 3 - 4.

Crossrefs

Cf. A054520, A124934, A125217, A125218, A380140, A380509, A380572.
Showing 1-4 of 4 results.

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