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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A327259 Array T(n,k) = 2*n*k - A319929(n,k), n >= 1, k >= 1, read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 3, 8, 3, 4, 10, 10, 4, 5, 16, 13, 16, 5, 6, 18, 20, 20, 18, 6, 7, 24, 23, 32, 23, 24, 7, 8, 26, 30, 36, 36, 30, 26, 8, 9, 32, 33, 48, 41, 48, 33, 32, 9, 10, 34, 40, 52, 54, 54, 52, 40, 34, 10, 11, 40, 43, 64, 59, 72, 59, 64, 43, 40, 11, 12, 42, 50, 68, 72, 78, 78, 72, 68, 50, 42, 12
Offset: 1

Views

Author

David Lovler, Aug 27 2019

Keywords

Comments

Associative multiplication-like table whose values depend on whether n and k are odd or even.
Associativity is proved by checking the formula with eight cases of three odd and even arguments. T(n,k) is distributive as long as partitioning an even number into two odd numbers is not allowed.
T(n,k) has the same group structure as A319929, A322630 and A322744. For those arrays, position (3,3) is 5, 7 and 11 respectively. T(3,3) = 13. If we didn't have the formula for these arrays, their entries could be computed knowing one position and applying the arithmetic rules.

Examples

			Array T(n,k) begins:
   1   2   3   4   5   6   7   8   9  10
   2   8  10  16  18  24  26  32  34  40
   3  10  13  20  23  30  33  40  43  50
   4  16  20  32  36  48  52  64  68  80
   5  18  23  36  41  54  59  72  77  90
   6  24  30  48  54  72  78  96 102 120
   7  26  33  52  59  78  85 104 111 130
   8  32  40  64  72  96 104 128 136 160
   9  34  43  68  77 102 111 136 145 170
  10  40  50  80  90 120 130 160 170 200

Links

Crossrefs

Equals A322744 + A322630 - A319929.
Equals 4*A322630 - 3*A319929.
Cf. A327260, A327261, A327263.
0 and diagonal is A354595.

Programs

  • Mathematica
    T[n_,k_]:=2n*k-If[Mod[n,2]==1,If[Mod[k,2]==1,n+k-1,k],If[Mod[k,2]==1,n,0]]; MatrixForm[Table[T[n,k],{n,1,10},{k,1,10}]] (* Stefano Spezia, Sep 05 2019 *)
  • PARI
    T(n,k) = 2*n*k - if (n%2, if (k%2, n+k-1, k), if (k%2, n, 0));
matrix(8, 8, n, k, T(n,k)) \\ Michel Marcus, Sep 04 2019

Formula

T(n,k) = 2*n*k - n - k + 1 if n is odd and k is odd;
T(n,k) = 2*n*k - n if n is even and k is odd;
T(n,k) = 2*n*k - k if n is odd and k is even;
T(n,k) = 2*n*k if n is even and k is even.
T(n,k) = 8*floor(n/2)*floor(k/2) + A319929(n,k).
T(n,n) = A354595(n). - David Lovler, Jul 09 2022
Writing T(n,k) as (4*n*k - 2*A319929(n,k))/2 shows that the array is U(4;n,k) of A327263. - David Lovler, Jan 15 2022

A354594 a(n) = n^2 + 2*floor(n/2)^2.

Original entry on oeis.org

0, 1, 6, 11, 24, 33, 54, 67, 96, 113, 150, 171, 216, 241, 294, 323, 384, 417, 486, 523, 600, 641, 726, 771, 864, 913, 1014, 1067, 1176, 1233, 1350, 1411, 1536, 1601, 1734, 1803, 1944, 2017, 2166, 2243, 2400, 2481, 2646, 2731, 2904
Offset: 0

Views

Author

David Lovler, Jun 01 2022

Keywords

Comments

The first bisection is A033581, the second bisection is A080859. - Bernard Schott, Jun 07 2022

Links

Crossrefs

Cf. A000290, A008794, A033581, A080859, A247375.
Cf. A213037, A322744 (diagonal), A354595, A354596.

Programs

  • Mathematica
    a[n_] := n^2 + 2 Floor[n/2]^2
Table[a[n], {n, 0, 90}]    (* A354594 *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 6, 11, 24}, 60]
  • PARI
    a(n) = n^2 + 2*(n\2)^2;

Formula

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5), n >= 5.
a(n) = A000290(n) + 2*A008794(n).
G.f.: x*(1 + 5*x + 3*x^2 + 3*x^3)/((1 - x)^3*(1 + x)^2).
E.g.f.: (x*(1 + 3*x)*cosh(x) + (1 + 3*x + 3*x^2)*sinh(x))/2. - Stefano Spezia, Jun 07 2022

A354596 Array T(n,k) = k^2 + (2n-4)*floor(k/2)^2, n >= 0, k >= 0, read by descending antidiagonals.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 5, 2, 1, 0, 0, 7, 4, 1, 0, 9, 8, 9, 6, 1, 0, 0, 17, 16, 11, 8, 1, 0, 13, 18, 25, 24, 13, 10, 1, 0, 0, 31, 36, 33, 32, 15, 12, 1, 0, 17, 32, 49, 54, 41, 40, 17, 14, 1, 0, 0, 49, 64, 67, 72, 49, 48, 19, 16, 1, 0, 21, 50, 81, 96, 85, 90, 57, 56, 21, 18, 1, 0
Offset: 0

Views

Author

David Lovler, Jun 01 2022

Keywords

Comments

Column k is an arithmetic progression with difference 2*A008794(k).
Odd rows of A133728 triangle are contained in row 0.
For i = 0 through 4, row i is 0 and the diagonal of A319929, A322630 = A213037, A003991, A322744, and A327259, respectively. In general, row i is 0 and the diagonal of array U(i;n,k) described in A327263.

Examples

			T(n,k) begins:
  0,   1,   0,   5,   0,   9,   0,  13, ...
  0,   1,   2,   7,   8,  17,  18,  31, ...
  0,   1,   4,   9,  16,  25,  36,  49, ...
  0,   1,   6,  11,  24,  33,  54,  67, ...
  0,   1,   8,  13,  32,  41,  72,  85, ...
  0,   1,  10,  15,  40,  49,  90, 103, ...
  0,   1,  12,  17,  48,  57, 108, 121, ...
  ...

Links

Crossrefs

Cf. A000290, A008794, A133728, A213037, A247375.
Cf. A266222, A266439.
Cf. A319929, A322630, A322744, A327259, A327263, A354594, A354595.

Programs

  • Mathematica
    T[n_, k_] := k^2 + (2*n - 4)*Floor[k/2]^2; Table[T[n - k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Amiram Eldar, Jun 20 2022 *)
  • PARI
    T(n,k) = k^2 + (2*n-4)*(k\2)^2;

Formula

T(n,k) = U(n;k,k) (see A327263).
For each row, T(n,k) = T(n,k-1) + 2*T(n,k-2) - 2*T(n,k-3) - T(n,k-4) + T(n,k-5), k >= 5.
G.f. for row n: x*(1 + (2*n-1)*x + 3*x^2 + (2*n-3)*x^3)/((1 - x)^3*(1 + x)^2). When n = 2, this reduces to x*(1 + x)/(1 - x)^3.
E.g.f. for row n: (((4-n)*x + n*x^2)*cosh(x) + (n-2 + n*x + n*x^2)*sinh(x))/2. When n = 2, this reduces to (x + x^2)*cosh(x) + (x + x^2)*sinh(x) = (x + x^2)*exp(x).
Showing 1-3 of 3 results.

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