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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A259874 Array read by antidiagonals upwards: Davenport-Schinzel numbers T(n,k), n >= 1, k >= 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 4, 1, 1, 5, 7, 8, 5, 1, 1, 6, 9, 12, 10, 6, 1, 1, 7, 11, 17, 16, 14, 7, 1, 1, 8, 13, 22, 22, 23, 16, 8, 1, 1, 9, 15, 27, 29, 34, 28, 20, 9, 1, 1, 10, 17, 32
Offset: 1

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Author

N. J. A. Sloane, Jul 09 2015

Keywords

Comments

Named after the English mathematician Harold Davenport (1907-1969) and the Polish mathematician Andrzej Schinzel (1937-2021). - Amiram Eldar, Jun 06 2021

Examples

			First few antidiagonals:
  1;
  1,  1;
  1,  2,  1;
  1,  3,  3,  1;
  1,  4,  5,  4,  1;
  1,  5,  7,  8,  5,  1;
  1,  6,  9, 12, 10,  6,  1;
  1,  7, 11, 17, 16, 14,  7,  1;
  1,  8, 13, 22, 22, 23, 16,  8,  1;
  ...
First few rows:
  1,  1,  1,  1,  1,  1, ...
  1,  2,  3,  4,  5,  6,  7,  8, ...
  1,  3,  5,  8, 10, 14, 16, 20, 22, 26, ...
  1,  4,  7, 12, 16, 23, 28, 35, 40, 47, ...
  1,  5,  9, 17, 22, 34, 41, 53, 61, 73, ...
  ...

References

  • R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section E20.
  • R. G. Stanton and P. H. Dirksen, Davenport-Schinzel sequences, Ars. Combin., 1 (1976), 43-51.

Links

Crossrefs

Rows and columns include A005004, A005005, A005006, A002004.

Extensions

More terms from Sean A. Irvine, Feb 21 2016

A095002 a(n) = 9*a(n-1) - 9*a(n-2) + a(n-3); given a(1) = 1, a(2) = 3, a(3) = 19.

Original entry on oeis.org

1, 3, 19, 145, 1137, 8947, 70435, 554529, 4365793, 34371811, 270608691, 2130497713, 16773373009, 132056486355, 1039678517827, 8185371656257, 64443294732225, 507360986201539, 3994444594880083, 31448195772839121, 247591121587832881, 1949280776929823923
Offset: 1

Views

Author

Gary W. Adamson, May 27 2004

Keywords

Comments

A companion to A095003, A005004; a(n)/a(n-1) tending to 4 + sqrt(15).
a(n)/a(n-1) tends to C = 4 + sqrt(15); C having the property that C + 1/C = 8. Eigenvalues of M (1, C, 1/C) are roots to x^3 - 9x^2 + 9x - 1.

Examples

			a(4) = 145 = 9*19 - 9*3 + 1.
a(4) = 145, leftmost term in M^4 * [1 0 0] = [145 352 640].

Links

Crossrefs

Cf. A095003, A095004, A076765.

Programs

  • Maple
    a:= n-> (<<1|1|1>, <1|2|3>, <1|3|6>>^n)[1$2]:
seq(a(n), n=1..23);  # Alois P. Heinz, Jun 06 2021
  • Mathematica
    a[n_] := (MatrixPower[{{1, 1, 1}, {1, 2, 3}, {1, 3, 6}}, n].{{1}, {0},
{0}})[[1, 1]]; Table[ a[n], {n, 20}]; (* Robert G. Wilson v, May 29 2004 *)
nxt[{a_,b_,c_}]:={b,c,9c-9b+a}; NestList[nxt,{1,3,19},30][[All,1]] (* Harvey P. Dale, Sep 02 2022 *)
  • PARI
    Vec(x*(1-6*x+x^2)/((1-x)*(1-8*x+x^2)) + O(x^20)) \\ Michel Marcus, Mar 21 2015

Formula

Let M be the 3 X 3 matrix [1 1 1 / 1 2 3 / 1 3 6]. M^n * [1 0 0] = [a(n) A095003(n) A095004(n)].
From R. J. Mathar, Aug 22 2008: (Start)
O.g.f.: x*(1-6x+x^2)/((1-x)*(1-8x+x^2)).
a(n) = (2 + A001090(n+1) - 7*A001090(n))/3. (End)

Extensions

Edited and extended by Robert G. Wilson v, May 29 2004
Edited by Georg Fischer, Jun 06 2021
Showing 1-2 of 2 results.

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

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