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 1-2 of 2 results.

A185780 Array T(n,k) = k*(n*k-n+1), by antidiagonals.

Original entry on oeis.org

1, 4, 1, 9, 6, 1, 16, 15, 8, 1, 25, 28, 21, 10, 1, 36, 45, 40, 27, 12, 1, 49, 66, 65, 52, 33, 14, 1, 64, 91, 96, 85, 64, 39, 16, 1, 81, 120, 133, 126, 105, 76, 45, 18, 1, 100, 153, 176, 175, 156, 125, 88, 51, 20, 1, 121, 190, 225, 232, 217, 186, 145, 100, 57, 22, 1, 144, 231, 280, 297, 288, 259, 216, 165, 112, 63, 24, 1, 169, 276, 341, 370, 369, 344, 301, 246, 185, 124, 69, 26, 1, 196, 325, 408, 451, 460, 441, 400, 343, 276, 205, 136, 75, 28, 1
Offset: 1

Views

Author

Clark Kimberling, Feb 03 2011

Keywords

Comments

This is the accumulation array of A185781, the weight array of A185782, and second weight array of A185783. See A144112 for definitions of accumulation array and weight array.

Examples

			Northwest corner:
  1....4....9....16....25....36
  1....6....15...28....45....66
  1....8....21...40....65....96
  1....10...27...52....85....126

Links

Crossrefs

Cf. A144112, A185781, A185782, A185783.

Programs

  • Mathematica
    (* This code yields arrays A185780, A185781, and A185782. *)
f[n_,0]:=0;f[0,k_]:=0;  (* Used to make weight array A185782 *)
f[n_,k_]:=k(n*k-n+1);
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* this array *)
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* acc array of {f(n,k)} *)
FullSimplify[s[n,k]]
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]]  (* A185781 *)
Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
w[m_,n_]:=f[m,n]+f[m-1,n-1]-f[m,n-1]-f[m-1,n]/;Or[m>0,n>0];
TableForm[Table[w[n,k],{n,1,10},{k,1,15}]] (* array A185782 *)
Table[w[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten (* seq A185782 *)

Formula

T(n,k) = k*(n*k - n + 1), k>=1, n>=1.

A302576 Numbers k such that k/10 + 1 is a square.

Original entry on oeis.org

-10, 0, 30, 80, 150, 240, 350, 480, 630, 800, 990, 1200, 1430, 1680, 1950, 2240, 2550, 2880, 3230, 3600, 3990, 4400, 4830, 5280, 5750, 6240, 6750, 7280, 7830, 8400, 8990, 9600, 10230, 10880, 11550, 12240, 12950, 13680, 14430, 15200, 15990, 16800, 17630, 18480, 19350, 20240
Offset: 1

Views

Author

Bruno Berselli, Apr 10 2018

Keywords

Comments

Equivalently, numbers k such that (k + 10)*10 is a square.
The positive terms belong to the fourth column of the array in A185781.

Links

Crossrefs

Cf. A000290, A008592, A033583, A185781.
After -10, subsequence of A174133 because a(n) = ((n-1)^2-1)*(3^2+1).
Similar lists of k for which k/j + 1 is a square: A067998 (j=1), A054000 (j=2), A067725 (j=3), A134582 (j=4), A067724 (j=5), A067726 (j=6), A067727 (j=7), second bisection of A067728 (j=8), A147651 (j=9), this sequence (j=10), A067705 (j=11), second bisection of A067707 (j=12).

Programs

  • GAP
    List([1..50], n -> 10*n*(n-2));
  • Julia
    [10*n*(n-2) for n in 1:50] |> println
  • Magma
    [10*n*(n-2): n in [1..50]];
  • Mathematica
    Table[10 n (n - 2), {n, 1, 50}]
  • Maxima
    makelist(10*n*(n-2), n, 1, 50);
  • PARI
    vector(50, n, nn; 10*n*(n-2))
  • Python
    [10*n*(n-2) for n in range(1, 50)]
  • Sage
    [10*n*(n-2) for n in (1..50)]

Formula

O.g.f.: -10*x*(1 - 3*x)/(1 - x)^3.
E.g.f.: -10*x*(1 - x)*exp(x).
a(n) = a(2-n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 10*n*(n - 2) = 10*A067998(n).
a(n) = A033583(n-1) - 10. - Altug Alkan, Apr 10 2018
Showing 1-2 of 2 results.

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