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.

A134575 A051731 * A127733.

Original entry on oeis.org

1, 1, 4, 1, 0, 9, 1, 4, 0, 16, 1, 0, 0, 0, 25, 1, 4, 9, 0, 0, 36, 1, 0, 0, 0, 0, 0, 49, 1, 4, 0, 16, 0, 0, 0, 64, 1, 0, 9, 0, 0, 0, 0, 0, 81, 1, 4, 0, 0, 25, 0, 0, 0, 0, 100
Offset: 1

Views

Author

Gary W. Adamson, Nov 02 2007

Keywords

Comments

Row sums = A001157: (1, 5, 10, 21, 26, 25, 50, 50, ...).

Examples

			First few rows of the triangle:
  1;
  1,  4;
  1,  0,  9;
  1,  4,  0, 16;
  1,  0,  0,  0, 25;
  1,  4,  9,  0,  0, 36;
  1,  0,  0,  0,  0,  0, 49;
  ...

Crossrefs

Cf. A051731, A127733, A001157.

Formula

A051731 * A127733 as infinite lower triangular matrices.

A134576 A127733 * A051731.

Original entry on oeis.org

1, 4, 4, 9, 0, 9, 16, 16, 0, 16, 25, 0, 0, 0, 25, 36, 36, 36, 0, 0, 36, 49, 0, 0, 0, 0, 0, 49, 64, 64, 0, 64, 0, 0, 0, 64, 81, 0, 81, 0, 0, 0, 0, 0, 81, 100, 100, 0, 0, 100, 0, 0, 0, 0, 100
Offset: 1

Views

Author

Gary W. Adamson, Nov 02 2007

Keywords

Comments

Row sums = A034714: (1, 8, 18, 48, 50, 144, ...).

Examples

			First few rows of the triangle:
   1;
   4,  4;
   9,  0,  9;
  16, 16,  0, 16;
  25,  0,  0,  0, 25;
  36, 36, 36,  0,  0, 36;
  49,  0,  0,  0,  0,  0, 49;
  ...

Crossrefs

Cf. A051731, A127733, A034714.

Formula

A127733 * A051731 as infinite lower triangular matrices.
Triangle read by rows: replace 1's in n-th row of A051731 with n^2.

A135065 A127733 * A007318 as infinite lower triangular matrices.

Original entry on oeis.org

1, 4, 4, 9, 18, 9, 16, 48, 48, 16, 25, 100, 150, 100, 25, 36, 180, 360, 360, 180, 36, 49, 294, 735, 980, 735, 294, 49, 64, 448, 1344, 2240, 2240, 1344, 448, 64, 81, 648, 2268, 4536, 5670, 4536, 2268, 648, 81, 100, 900, 3600, 8400, 12600, 12600, 8400, 3600
Offset: 0

Views

Author

Gary W. Adamson, Nov 16 2007

Keywords

Comments

A135065 * [1/1, 1/2, 1/3, ...] = A066524: (1, 6, 21, 60, 155, ...).
Triangle T(n,k), 0 <= k <= n, read by rows, given by (4, -7/4, 17/28, -32/119, 7/17, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (4, -7/4, 17/28, -32/119, 7/17, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 27 2011

Examples

			First few rows of the triangle:
   1;
   4,   4;
   9,  18,   9;
  16,  48,  48,  16;
  25, 100, 150, 100,  25;
  36, 180, 360, 360, 180,  36;
  49, 294, 735, 980, 735, 294,  49;

Links

Crossrefs

Cf. A000290, A127733, A066524, A014477 (row sums), A084938.

Programs

  • Maple
    with(combstruct):for n from 0 to 11 do seq(n*m*count(Combination(n), size=m), m = 1 .. n) od; # Zerinvary Lajos, Apr 09 2008
  • Mathematica
    Flatten[Table[Binomial[n,k](n+1)^2,{n,0,10},{k,0,n}]] (* Harvey P. Dale, Jul 12 2013 *)

Formula

T(n,k) = binomial(n,k)*(n+1)^2 = A007318(n,k)*A000290(n+1). - Philippe Deléham, Oct 27 2011
T(n-1,k-1) = Sum_{i=-k..k} (-1)^i*(k^2-i^2)*binomial(n,k+i)*binomial(n,k-i). - Mircea Merca, Apr 05 2012
G.f.: (-1 - x - x*y)/(x + x*y - 1)^3. - R. J. Mathar, Aug 12 2015

Extensions

Corrected by Zerinvary Lajos, Apr 09 2008

A093995 n^2 appears n times, triangle read by rows.

Original entry on oeis.org

1, 4, 4, 9, 9, 9, 16, 16, 16, 16, 25, 25, 25, 25, 25, 36, 36, 36, 36, 36, 36, 49, 49, 49, 49, 49, 49, 49, 64, 64, 64, 64, 64, 64, 64, 64, 81, 81, 81, 81, 81, 81, 81, 81, 81, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121
Offset: 1

Views

Author

Reinhard Zumkeller, May 24 2004

Keywords

Comments

Row sums give A000578.
Triangle sums give A000537.

Examples

			First few rows of the triangle are:
   1;
   4,  4;
   9,  9,  9;
  16, 16, 16, 16;
  25, 25, 25, 25, 25;
  36, 36, 36, 36, 36, 36;
  49, 49, 49, 49, 49, 49, 49;
  ...

Links

Crossrefs

Cf. A000290, A000537, A000578, A016742, A016754, A127733, A199332, A265645.

Programs

  • Haskell
    a093995 n k = a093995_tabl !! (n-1) !! (k-1)
a093995_row n = a093995_tabl !! (n-1)
a093995_tabl = zipWith replicate [1..] $ tail a000290_list
a093995_list = concat a093995_tabl
-- Reinhard Zumkeller, Nov 11 2012, Mar 18 2011, Oct 17 2010
  • Magma
    [n^2: k in [1..n], n in [1..13]]; // G. C. Greubel, Dec 27 2021
  • Maple
    seq(seq(n^2, i=1..n), n=1..20); # Ridouane Oudra, Jun 18 2019
  • Mathematica
    Flatten[Table[Table[n^2,{n}],{n,11}]]  (* Harvey P. Dale, Feb 18 2011 *)
Table[PadRight[{},n,n^2],{n,12}]//Flatten (* Harvey P. Dale, Jun 28 2023 *)
  • Python
    from math import isqrt
def A093995(n): return ((m:=isqrt(k:=n<<1))+(k>m*(m+1)))**2 # Chai Wah Wu, Nov 07 2024
  • Sage
    flatten([[n^2 for k in (1..n)] for n in (1..13)]) # G. C. Greubel, Dec 27 2021

Formula

T(n, k) = n^2, 1<=k<=n.
a(n) = floor(sqrt(2*n - 1) + 1/2)^2. - Ridouane Oudra, Jun 18 2019
From G. C. Greubel, Dec 27 2021: (Start)
T(n, n-k) = T(n, k).
Sum_{k=1..floor(n/2)} T(n, k) = [n=1] + A265645(n+1).
Sum_{k=1..floor(n/2)} T(n-k, k) = (1/48)*n*(n-1)*(7*(2*n-1) + 3*(-1)^n).
T(2*n-1, n) = A016754(n).
T(2*n, n) = A016742(n). (End)

Extensions

Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar
Definition clarified by N. J. A. Sloane, Nov 09 2024
Showing 1-4 of 4 results.

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

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