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 1-3 of 3 results.

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

A095003 a(n) = 9*a(n-1) - 9*a(n-2) + a(n-3).

Original entry on oeis.org

1, 6, 45, 352, 2769, 21798, 171613, 1351104, 10637217, 83746630, 659335821, 5190939936, 40868183665, 321754529382, 2533168051389, 19943589881728, 157015551002433, 1236180818137734, 9732430994099437, 76623267134657760, 603253706083162641, 4749406381530643366
Offset: 1

Views

Author

Gary W. Adamson, May 27 2004

Keywords

Comments

a(n)/a(n-1) tends to 7.87298... = 4 + sqrt(15) = C (having the property that C + 1/C = 8). Eigenvalues of M are C, 1/C, 1; being roots of x^3 - 9x^2 + 9x - 1.

Examples

			a(4) = 352 since M^4 * [1 0 0] = [145, 352, 640].

Links

Crossrefs

Cf. A095002, 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}})[[2, 1]]; Table[ a[n], {n, 20}]; (* Robert G. Wilson v, May 29 2004 *)
LinearRecurrence[{9,-9,1},{1,6,45},30] (* Harvey P. Dale, Nov 12 2022 *)

Formula

a(n+3) = 9*a(n+2) - 9*a(n+1) + a(n); given a(1) = 1, a(2) = 6, a(3) = 45.
Let M be the 3 X 3 matrix [1 1 1 / 1 2 3 / 1 3 6]. M^n * [1 0 0] = [A095002(n) a(n) A095004(n)].

Extensions

Edited and extended by Robert G. Wilson v, May 29 2004
Definition corrected and edited by Georg Fischer, Jun 06 2021

A356836 Coordination sequence of the {5,3,4} hyperbolic honeycomb.

Original entry on oeis.org

1, 12, 102, 812, 6402, 50412, 396902, 3124812, 24601602, 193688012, 1524902502, 12005532012, 94519353602, 744149296812, 5858675020902, 46125250870412, 363143331942402, 2859021404668812, 22509027905408102, 177213201838596012, 1395196586803360002, 10984359492588284012
Offset: 0

Views

Author

Eryk Kopczynski, Aug 31 2022

Keywords

Comments

a(n) is the number of cells n steps from an (arbitrarily chosen) central cell in the {5,3,4} honeycomb.

Examples

			Each dodecahedral cell has 12 neighbors, so a(1) = 12.

Links

Crossrefs

Cf. A076765, A095004, A356835.

Formula

It appears thata(n) = 10*A095004(n) + 2. - Hugo Pfoertner, Aug 30 2022
Showing 1-3 of 3 results.

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