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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A386823 Triangle read by rows: T(n,k) = numerator((n^2 - k^2)/(n^2 + k^2)), where 0 <= k < n.

Original entry on oeis.org

1, 1, 3, 1, 4, 5, 1, 15, 3, 7, 1, 12, 21, 8, 9, 1, 35, 4, 3, 5, 11, 1, 24, 45, 20, 33, 12, 13, 1, 63, 15, 55, 3, 39, 7, 15, 1, 40, 77, 4, 65, 28, 5, 16, 17, 1, 99, 12, 91, 21, 3, 8, 51, 9, 19, 1, 60, 117, 56, 105, 48, 85, 36, 57, 20, 21, 1, 143, 35, 15, 4, 119, 3, 95, 5, 7, 11, 23
Offset: 1

Views

Author

Stefano Spezia, Aug 04 2025

Keywords

Examples

			The triangle of the fractions begins as:
  1/1;
  1/1,   3/5;
  1/1,   4/5,  5/13;
  1/1, 15/17,   3/5,  7/25;
  1/1, 12/13, 21/29,  8/17,  9/41;
  1/1, 35/37,   4/5,   3/5,  5/13, 11/61;
  1/1, 24/25, 45/53, 20/29, 33/65, 12/37, 13/85;
  ...

Links

Crossrefs

Cf. A000012 (k=0), A000290, A005408, A066830 (k=1), A069011, A094728, A386824 (denominators).

Programs

  • Mathematica
    T[n_,k_]:=Numerator[(n^2-k^2)/(n^2+k^2)]; Table[T[n,k],{n,12},{k,0,n-1}]//Flatten

Formula

T(n,n-1) = A005804(n-1).

A386824 Triangle read by rows: T(n,k) = denominator((n^2 - k^2)/(n^2 + k^2)), where 0 <= k < n.

Original entry on oeis.org

1, 1, 5, 1, 5, 13, 1, 17, 5, 25, 1, 13, 29, 17, 41, 1, 37, 5, 5, 13, 61, 1, 25, 53, 29, 65, 37, 85, 1, 65, 17, 73, 5, 89, 25, 113, 1, 41, 85, 5, 97, 53, 13, 65, 145, 1, 101, 13, 109, 29, 5, 17, 149, 41, 181, 1, 61, 125, 65, 137, 73, 157, 85, 185, 101, 221
Offset: 1

Views

Author

Stefano Spezia, Aug 04 2025

Keywords

Examples

			The triangle of the fractions begins as:
  1/1;
  1/1,   3/5;
  1/1,   4/5,  5/13;
  1/1, 15/17,   3/5,  7/25;
  1/1, 12/13, 21/29,  8/17,  9/41;
  1/1, 35/37,   4/5,   3/5,  5/13, 11/61;
  1/1, 24/25, 45/53, 20/29, 33/65, 12/37, 13/85;
  ...

Links

Crossrefs

Cf. A000012 (k=0), A000290, A001844, A069011, A094728, A228564 (k=1), A243883 (k=2), A386823 (numerators).

Programs

  • Mathematica
    T[n_,k_]:=Denominator[(n^2-k^2)/(n^2+k^2)]; Table[T[n,k],{n,11},{k,0,n-1}]//Flatten

Formula

T(n,n-1) = A001844(n-1).

A095873 Triangle T(n,k) = (2*k-1)*(n+k-1)*(n-k+1) read by rows, 1<=k<=n.

Original entry on oeis.org

1, 4, 9, 9, 24, 25, 16, 45, 60, 49, 25, 72, 105, 112, 81, 36, 105, 160, 189, 180, 121, 49, 144, 225, 280, 297, 264, 169, 64, 189, 300, 385, 432, 429, 364, 225, 81, 240, 385, 504, 585, 616, 585, 480, 289, 100, 297, 480, 637, 756, 825
Offset: 1

Views

Author

Gary W. Adamson, Jun 10 2004

Keywords

Comments

Matrix square of A158405.

Examples

			[1 0 0 / 1 3 0 / 1 3 5]^2 = [1 0 0 / 4 9 0 / 9 24 25]. Delete the zeros and
read by rows:
1;
4, 9;
9, 24, 25;
16,45, 60, 49;
25,72,105,112, 81;

References

  • Albert H. Beiler, "Recreations in the Theory of Numbers", Dover, 1966.

Crossrefs

Cf. A094728, A095871, A095872.

Programs

  • Maple
    A095873 := proc(n,k)
        (2*k-1)*(n+k-1)*(n-k+1) ;
end proc:
seq(seq(A095873(n,k),k=1..n),n=1..13) ; # R. J. Mathar, Oct 30 2011
  • Mathematica
    Table[(2k-1)(n+k-1)(n-k+1),{n,10},{k,n}]//Flatten (* Harvey P. Dale, May 03 2018 *)

Formula

T(n,k) = (2*k-1)*A094728(n,k).
Sum_{k=1..n} T(n,k)= n*(n+1)*(3*n^2+n-1)/6 = A103220(n). - R. J. Mathar, Oct 30 2011

Extensions

Definition in closed form by R. J. Mathar, Oct 30 2011

A271668 Triangle read by rows. The first column is A000217(n+1). From the second row we apply - A002262(n) for the following terms of the row.

Original entry on oeis.org

1, 3, 3, 6, 6, 5, 10, 10, 9, 7, 15, 15, 14, 12, 9, 21, 21, 20, 18, 15, 11, 28, 28, 27, 25, 22, 18, 13, 36, 36, 35, 33, 30, 26, 21, 15, 45, 45, 44, 42, 39, 35, 30, 24, 17, 55, 55, 54, 52, 49, 45, 40, 34, 27, 19, 66, 66, 65, 63, 60, 56, 51, 45, 38, 30, 21
Offset: 0

Views

Author

Paul Curtz, Apr 12 2016

Keywords

Comments

Row sums: A084990(n+1).
A158405(n) = A002262(n) + A002260(n). See the formula.
(Without its first column, A094728 is A120070, which could be built from positive A005563 and -A158894.)

Examples

			a(0) = 1, a(1) = 3, a(2) =3-0 = 3,  a(3) = 6, a(4) =6-0= 6, a(5) =6-1= 5, ... .
Triangle:
1,
3,   3,
6,   6,  5,
10, 10,  9,  7,
15, 15, 14, 12,  9,
21, 21, 20, 18, 15, 11,
28, 28, 27, 25, 22, 18, 13,
36, 36, 35, 33, 30, 26, 21, 15,
etc.

Crossrefs

Cf. A000096, A000217, A002260, A002262, A005408, A005563, A049780, A084990, A094728, A120070, A158405, A158894.

Programs

  • Mathematica
    Table[(n^2 - n)/2 - Prepend[Accumulate@ Range[0, n - 3], 0], {n, 12}] // Flatten (* Michael De Vlieger, Apr 12 2016 *)

Formula

a(n) = A094728(n+1) - A049780(n).

A212649 a(n) = floor(Sum_{k=0..n-1} sqrt(n^2 - k^2)).

Original entry on oeis.org

1, 3, 8, 13, 21, 30, 41, 53, 67, 82, 99, 118, 138, 159, 183, 207, 234, 262, 291, 322, 355, 389, 425, 462, 501, 542, 584, 628, 673, 720, 768, 818, 870, 923, 977, 1034, 1091, 1151, 1212, 1274, 1338, 1404, 1471, 1540, 1610, 1682, 1756, 1831, 1908, 1986, 2066
Offset: 1

Views

Author

Philippe Deléham, Mar 07 2013

Keywords

Comments

Limit_{n->oo} a(n)/n^2 = Pi/4 = 0.78539816...

Examples

			A094728(4) is (16, 15, 12, 7). Hence, a(4) = floor(sqrt(16) + sqrt(15) + sqrt(12) + sqrt(7)) = floor(13.9828...) = 13.

Crossrefs

Cf. A003881, A094728.

Programs

  • Mathematica
    Table[Floor[Sum[Sqrt[n^2 - k^2], {k, 0, n - 1}]], {n, 60}] (* T. D. Noe, Mar 19 2013 *)

Formula

a(n) = floor(Sum_{k=0..n-1} sqrt(A094728(n,k))).

Extensions

a(16)-a(50) from Giovanni Resta, Mar 19 2013

A348675 a(n) = Sum_{k=0..n-1} Omega(n^2-k^2).

Original entry on oeis.org

0, 3, 6, 10, 14, 18, 21, 27, 31, 35, 39, 44, 48, 54, 58, 64, 68, 74, 77, 83, 87, 91, 96, 102, 107, 112, 117, 123, 127, 132, 135, 144, 149, 153, 158, 164, 167, 173, 178, 184, 190, 195, 199, 205, 210, 215, 219, 227, 231, 238, 242, 247, 252, 258, 262, 269, 273, 278
Offset: 1

Views

Author

Oscar Granfeldt, Oct 29 2021

Keywords

Examples

			For n = 3, row 3 of A094728 is 9, 8, 5, and a(3) = Omega(9)+Omega(8)+Omega(5) = 2+3+1 = 6.

Crossrefs

Cf. A001222, A094728.

Programs

  • Maple
    a := n -> add(NumberTheory:-NumberOfPrimeFactors(n*n - k*k), k = 0..n-1):
seq(a(n), n = 1..58);
  • Mathematica
    a[n_] := Sum[PrimeOmega[n^2 - k^2], {k, 0, n - 1}]; Array[a, 60] (* Amiram Eldar, Oct 30 2021 *)
  • PARI
    a(n) = sum(k=0, n-1, bigomega(n^2-k^2)); \\ Michel Marcus, Oct 30 2021

Formula

a(n) = Sum_{k=0..n-1} A001222(A094728(n,k)).
Previous Showing 11-16 of 16 results.

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