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.

A169947 Third entry in row n of triangle in A169945.

Original entry on oeis.org

1, 3, 8, 16, 29, 49, 82, 130, 205, 305, 450, 654, 947, 1343, 1902, 2648, 3675, 5015, 6824, 9166, 12343, 16393, 21762, 28682, 37695, 49055, 63892, 82610, 106691, 136643, 174862, 222524, 283073, 357691, 451538, 567498, 712817, 890365, 1112040, 1382374, 1717497
Offset: 1

Views

Author

N. J. A. Sloane, Aug 01 2010

Keywords

Links

Crossrefs

Related to thickness: A169940-A169954, A061909.

Programs

  • Mathematica
    b[n_, s_] := Module[{sn, m}, If[n<1, 1, sn = Append[s, n]; m = Length[sn]; If[m*(m - 1)/2 == Length[Table[sn[[i]] - sn[[j]], {i, 1, m - 1}, {j, i+1, m}] // Flatten // Union], b[n - 1, sn], 0] + b[n - 1, s]]];
c[n_] := c[n] = b[n - 1, {n}] + If[n == 0, 0, c[n - 1]];
a[n_] := c[n + 1] - n - 2;
Table[Print[n, " ", a[n]]; a[n], {n, 1, 64}] (* Jean-François Alcover, Sep 02 2019, after Alois P. Heinz in A143823 *)

Formula

Wanted: a recurrence. Are any of A169940-A169954 related to any other entries in the OEIS?
a(n) = A143823(n+1) - n - 2. - Nathaniel Johnston, Nov 12 2010

Extensions

More terms from R. J. Mathar, Aug 02 2010
a(22)-a(28) from Nathaniel Johnston, Nov 12 2010
More terms from Alois P. Heinz, Sep 16 2011

A169948 Fourth entry in row n of triangle in A169945.

Original entry on oeis.org

1, 2, 6, 14, 29, 52, 96, 160, 277, 450, 712, 1086, 1657, 2448, 3636, 5280, 7635, 10840, 15392, 21372, 29655, 40580, 55282, 74620, 100651, 134232, 178922, 236488, 312019, 408550, 534288, 692978, 897931, 1156256, 1485650, 1897704, 2421635, 3071608, 3894042
Offset: 2

Views

Author

N. J. A. Sloane, Aug 01 2010

Keywords

Comments

Wanted: a recurrence. Are any of A169940-A169954 related to any other entries in the OEIS?

Links

Crossrefs

Related to thickness: A169940-A169954, A061909.
Cf. A143823, A196723.

Programs

  • Mathematica
    b[n_, s_] := b[n, s] = Module[{sn, m}, m = Length[s]; sn = Append[s, n]; If[n < 1, 1, b[n - 1, s] + If[m*(m + 1)/2 == Length[Union[Flatten[Table[ sn[[i]] + sn[[j]], {i, 1, m}, {j, i + 1, m + 1}]]]], b[n - 1, sn], 0]]];
A196723[n_] := A196723[n] = b[n - 1, {n}] + If[n == 0, 0, A196723[n - 1]];
c[n_, s_] := c[n, s] = Module[{sn, m}, If[n < 1, 1, sn = Append[s, n]; m = Length[sn]; If[m*(m - 1)/2 == Length[Table[sn[[i]] - sn[[j]], {i, 1, m - 1}, {j, i + 1, m}] // Flatten // Union], c[n - 1, sn], 0] + c[n-1, s]]];
A143823[n_] := A143823[n] = c[n - 1, {n}] + If[n == 0, 0, A143823[n - 1]];
a[n_] := a[n] = A196723[n + 1] - A143823[n + 1];
Table[Print[n, " ", a[n]]; a[n], {n, 2, 40}] (* Jean-François Alcover, Aug 27 2019, after Alois P. Heinz in A196723 and A143823 *)

Formula

a(n) = A196723(n+1) - A143823(n+1). - Andrew Howroyd, Jul 09 2017

Extensions

a(15)-a(28) from Nathaniel Johnston, Nov 12 2010
a(29)-a(40) from Andrew Howroyd, Jul 09 2017

A169946 Triangle read by rows: A169945 with rows reversed.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 2, 8, 4, 1, 1, 3, 6, 16, 5, 1, 1, 2, 11, 14, 29, 6, 1, 1, 3, 9, 29, 29, 49, 7, 1, 1, 2, 14, 22, 74, 52, 82, 8, 1, 1, 3, 12, 42, 58, 160, 96, 130, 9, 1, 1, 2, 17, 30, 126, 128, 344, 160, 205, 10, 1, 1, 3, 15, 53, 98, 314, 294, 676, 277, 305, 11
Offset: 0

Views

Author

N. J. A. Sloane, Aug 01 2010

Keywords

Comments

The heuristic formula for the third column (verified for 2<=n<=21) is 3*n/2+7*(1-(-1)^n)/4. [From R. J. Mathar, Aug 02 2010]

Examples

			Triangle begins:
[1, 1]
[1, 2, 1]
[1, 3, 3, 1]
[1, 2, 8, 4, 1]
[1, 3, 6, 16, 5, 1]
[1, 2, 11, 14, 29, 6, 1]
[1, 3, 9, 29, 29, 49, 7, 1]
[1, 2, 14, 22, 74, 52, 82, 8, 1]
[1, 3, 12, 42, 58, 160, 96, 130, 9, 1]
[1, 2, 17, 30, 126, 128, 344, 160, 205, 10, 1]
[1, 3, 15, 53, 98, 314, 294, 676, 277, 305, 11, 1]
[1, 2, 20, 38, 185, 232, 796, 576, 1333, 450, 450, 12, 1]
[1, 3, 18, 64, 147, 501, 628, 1796, 1177, 2477, 712, 654, 13, 1]
[1, 2, 23, 46, 251, 368, 1425, 1370, 4075, 2212, 4563, 1086, 947, 14, 1]
[1, 3, 21, 75, 205, 729, 1117, 3515, 3265, 8535, 4289, 7997, 1657, 1343, 15, 1]
...

Links

Crossrefs

Related to thickness: A169940-A169954, A061909.

Formula

Wanted: a recurrence. Are any of A169940-A169954 related to any other entries in the OEIS?

A169949 Maximal entry in row n of triangle in A169945.

Original entry on oeis.org

1, 2, 3, 8, 16, 29, 49, 82, 160, 344, 676, 1333, 2477, 4563, 8535, 17976, 35810, 71374, 135876, 258898, 476494, 981244, 1962600, 3942389, 7647545, 14879679, 28109033, 55334514, 111234624
Offset: 0

Views

Author

N. J. A. Sloane, Aug 01 2010

Keywords

Links

Crossrefs

Related to thickness: A169940-A169954, A061909.

Formula

Wanted: a recurrence. Are any of A169940-A169954 related to any other entries in the OEIS?

Extensions

a(15)-a(28) from Nathaniel Johnston, Nov 12 2010
Showing 1-4 of 4 results.

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

Content is available under The OEIS End-User License Agreement.