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.

A095125 Expansion of -x*(-1-x+x^2) / ( 1-2*x-3*x^2+x^3 ).

Original entry on oeis.org

1, 3, 8, 24, 69, 202, 587, 1711, 4981, 14508, 42248, 123039, 358314, 1043497, 3038897, 8849971, 25773136, 75057288, 218584013, 636566754, 1853828259, 5398772767, 15722463557, 45787417156, 133343452216, 388326692343, 1130896324178, 3293429273169, 9591220826529
Offset: 1

Views

Author

Gary W. Adamson, May 29 2004

Keywords

Comments

A sequence generated from a rotated Stirling number of the second kind matrix.
a(n)/a(n-1) tends to the largest positive eigenvalue of the matrix, 2.9122291784..., a root of the characteristic polynomial x^3 - 2x^2 - 3x + 1; e.g., a(9)/a(8) = 4981/1711 = 2.91116... A095127 is generated from an inverse of M, while A095126 is generated from M.

Examples

			a(5) = 69 = 2*a(4) + 3*a(3) - a(2) = 2*24 + 3*8 - 3.
a(5) = 69 since M^5 * [1 1 1] = [202 316 69] = [a(6) A095126(a) a(5)].

References

  • R. Aldrovandi, "Special Matrices of Mathematical Physics," World Scientific, 2001, Section 13.3.1 "Inverting Bell Matrices", p. 171.

Links

Crossrefs

Cf. A095126, A095127, A095128.

Programs

  • Magma
    I:=[1,3,8]; [n le 3 select I[n]  else 2*Self(n-1)+3*Self(n-2)-Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jul 25 2015
  • Mathematica
    a[n_] := (MatrixPower[{{1, 1, 1}, {3, 1, 0}, {1, 0, 0}}, n].{{1}, {1}, {1}})[[3, 1]]; Table[ a[n], {n, 25}] (* Robert G. Wilson v, Jun 01 2004 *)
LinearRecurrence[{2,3,-1},{1,3,8},30] (* Harvey P. Dale, Nov 13 2011 *)
  • PARI
    Vec((1+x-x^2)/(1-2*x-3*x^2+x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 25 2012

Formula

a(n+3) = 2*a(n+2) + 3*a(n+1) - a(n), with a(1) = 1, a(2) = 3, a(3) = 8.
M = [1 1 1 / 3 1 0 / 1 0 0], a rotation of a Stirling number of the second kind matrix [1 0 0 / 1 1 0 / 1 3 1]; then M^n * [1 1 1] = [a(n+1), A095126(n) a(n)].

Extensions

Edited, corrected and extended by Robert G. Wilson v, Jun 01 2004
Definition corrected by Harvey P. Dale, Nov 13 2011
a(27)-a(29) from Vincenzo Librandi, Jul 25 2015

A095126 Expansion of x*(4+5*x-x^2)/ (1-2*x-3*x^2+x^3).

Original entry on oeis.org

4, 13, 37, 109, 316, 922, 2683, 7816, 22759, 66283, 193027, 562144, 1637086, 4767577, 13884268, 40434181, 117753589, 342925453, 998677492, 2908377754, 8469862531, 24666180832, 71833571503, 209195822971, 609226179619
Offset: 1

Views

Author

Gary W. Adamson, May 29 2004

Keywords

Comments

A sequence generated from a rotated Stirling number of the second kind matrix, companion to A095125.
a(n)/a(n-1) tends to 2.9122291784...an eigenvalue of M and a root of the characteristic polynomial x^3 - 2x^2 - 3x + 1. A095127 is generated from the same polynomial, with the reversal x^3 - 3x^2 - 2x + 1 being the characteristic polynomial of A095128.

Examples

			a(6) = 922 = 2*316 + 3*109 - 37 = 2*a(5) + 3*a(4) - a(3).
a(5) = 316 since M^5 * [1 1 1] = [202 316 69] = [A095125(6), a(n), A095125(5)]

References

  • R. Aldrovandi, "Special Matrices of Mathematical Physics", World Scientific, 2001, Section 13.3.1, "Inverting Bell Matrices", p. 171.

Links

Crossrefs

Cf. A095125, A095127, A095128.

Programs

  • Mathematica
    a[n_] := (MatrixPower[{{1, 1, 1}, {3, 1, 0}, {1, 0, 0}}, n].{{1}, {1}, {1}})[[2, 1]]; Table[ a[n], {n, 26}] (* Robert G. Wilson v, Jun 01 2004 *)
LinearRecurrence[{2,3,-1},{4,13,37},30] (* Harvey P. Dale, Jan 18 2016 *)
  • PARI
    Vec((4+5*x-x^2)/(1-2*x-3*x^2+x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 25 2012

Formula

a(n+3) = 2*a(n+2) + 3*a(n+1) - a(n); with a(1) = 4, a(2) = 13, a(3) = 37.
Let M = a rotated Stirling number of the second kind matrix [1 1 1 / 3 1 0 / 1 0 0] (a rotation of [1 0 0 / 1 1 0 / 1 3 1]). Then M^n * [1 1 1] = [A095125(n+1), a(n), A095125(n)].

Extensions

Edited, corrected and extended by Robert G. Wilson v, Jun 01 2004

A095127 a(n+3) = 2*a(n+2) + 3*a(n+1) - a(n); with a(1) = 1, a(2) = 4, a(3) = 10.

Original entry on oeis.org

1, 4, 10, 31, 88, 259, 751, 2191, 6376, 18574, 54085, 157516, 458713, 1335889, 3890401, 11329756, 32994826, 96088519, 279831760, 814934251, 2373275263, 6911521519, 20127934576, 58617158446, 170706599101, 497136738964
Offset: 1

Views

Author

Gary W. Adamson, May 29 2004

Keywords

Comments

A sequence generated from the characteristic polynomial of A095125 and A095126.
a(n)/a(n-1) tends to a 2.9122291784..., a root of the polynomial x^3 - 2x^2 - 3x + 1; e.g. a(16)/a(15) = 11329756/3890401 = 2.912233...

Examples

			a(7) = 751 = 2*a(6) + 3*a(5) - a(4) = 2*259 + 3*88 - 31.
a(4) = 31 = center term in M^4 * [1 1 1] = [10 31 88].

Links

Crossrefs

Cf. A095125, A095126, A095128.

Programs

  • Mathematica
    a[1] = 1; a[2] = 4; a[3] = 10; a[n_] := a[n] = 2a[n - 1] + 3a[n - 2] - a[n - 3]; Table[ a[n], {n, 25}] (* Robert G. Wilson v, Jun 01 2004 *)
nxt[{a_,b_,c_}]:={b,c,2c+3b-a}; NestList[nxt,{1,4,10},30][[All,1]] (* or *) LinearRecurrence[{2,3,-1},{1,4,10},30] (* Harvey P. Dale, Feb 08 2022 *)
  • PARI
    Vec(x*(1 + 2*x - x^2) / (1 - 2*x - 3*x^2 + x^3) + O(x^30)) \\ Colin Barker, Aug 31 2019

Formula

M = a matrix having the same eigenvalues as the roots of the characteristic polynomial of A095125 and A095126: (x^3 - 2x^2 - 3x + 1). Then M^n * [1 1 1] = [p q r] where q = a(n) and p, r, are offset members of the same sequence.
G.f.: x*(1 + 2*x - x^2) / (1 - 2*x - 3*x^2 + x^3). - Colin Barker, Aug 31 2019

Extensions

Edited, corrected and extended by Robert G. Wilson v, Jun 01 2004
Showing 1-3 of 3 results.

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