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.

A095004 a(n) = 9*a(n-1) - 9*a(n-2) + a(n-3); given a(1) = 1, a(2) = 10, a(3) = 81.

Original entry on oeis.org

1, 10, 81, 640, 5041, 39690, 312481, 2460160, 19368801, 152490250, 1200553201, 9451935360, 74414929681, 585867502090, 4612525087041, 36314333194240, 285902140466881, 2250902790540810, 17721320183859601, 139519658680336000, 1098435949258828401, 8647967935390291210
Offset: 1

Views

Author

Gary W. Adamson, May 27 2004

Keywords

Comments

A sequence derived from A076765, with a(n)/a(n-1) tending to 4 + sqrt(15).
a(n)/a(n-1) tends to C = 4 + sqrt(15) = 7.87298334... (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.
This is the r=10 member of the r-family of sequences S_r(n), n>=1, defined in A092184, where more information can be found.

Examples

			a(4) = 640 = 568 + 72 = A076765(3) + A076765(2).
a(4) = 640 = 9*81 - 9*10 + 1.
a(4) = 640, rightmost term in M^4 * [1 0 0]: [145 352 640] = [A095002(4) A095003(4) A095004(4)].

Links

Crossrefs

Cf. A095002, A095003, A076765.

Programs

  • Maple
    a:= n-> (<<1|1|1>, <1|2|3>, <1|3|6>>^n)[1, 3]:
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}})[[3, 1]]; Table[ a[n], {n, 20}]; (* Robert G. Wilson v, May 29 2004 *)

Formula

a(n) = A076765(n-1) + A076765(n-2).
Let M be the 3 X 3 matrix [1 1 1 / 1 2 3 / 1 3 6]; then M^n * [1 0 0] = [A095002(n) A095003(n) a(n)].
a(n)= (T(n, 4)-1)/3 with Chebyshev's polynomials of the first kind evaluated at x=4: T(n, 4)=A001091(n). a(0):=0. - Wolfdieter Lang, Oct 18 2004
G.f.: x*(1+x)/((1-x)*(1-8*x+x^2)) = x*(1+x)/(1-9*x+9*x^2-x^3).

Extensions

Edited and extended by Robert G. Wilson v, May 29 2004
Definition aligned with A095002, A095003 by Georg Fischer, Jun 06 2021

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.

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