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.

A126566 a(0)=1; a(1)=2; a(2)=5; a(3)=14; for n>3: a(n) = 8*a(n-1)-20*a(n-2)+16*a(n-3)-a(n-4).

Original entry on oeis.org

1, 2, 5, 14, 43, 142, 495, 1794, 6681, 25346, 97357, 377038, 1468019, 5735758, 22460935, 88083586, 345754097, 1358000258, 5335796565, 20970349326, 82429113339, 324040664974, 1273932845663, 5008544929922, 19691924052361
Offset: 0

Views

Author

Gary W. Adamson, Dec 28 2006

Keywords

Comments

a(n) is also the first entry of the vector M^n*[1,0,0,0], where M is the type F_4 Cartan matrix [[2,-1,0,0]; [ -1,2,-2,0]; [0,-1,2,-1]; [0,0,-1,2]].

Crossrefs

Cf. A126569.

Programs

  • Mathematica
    a = {1,2,5,14}; Do[AppendTo[a, 8*a[[ -1]]-20*a[[ -2]]+16*a[[ -3]]-a[[ -4]]],{24}]; a

Formula

G.f.: -(2*x-1)*(x^2-4*x+1)/(1-8*x+20*x^2-16*x^3+x^4). - R. J. Mathar, Nov 14 2007

Extensions

Edited and extended by Stefan Steinerberger, Aug 12 2007

A126567 Sequence generated from the E6 Cartan matrix.

Original entry on oeis.org

1, 2, 5, 14, 42, 132, 430, 1444, 4981, 17594, 63441, 232806, 866870, 3266460, 12426210, 47629020, 183638729, 711285170, 2764753405, 10775740030, 42086252770, 164635420788, 644811687734, 2527808259668, 9916569410301, 38923511495402, 152841133694345, 600349070362454
Offset: 0

Views

Author

Gary W. Adamson, Dec 28 2006

Keywords

Links

Crossrefs

Cf. A126566, A126568, A126569.

Programs

  • Mathematica
    f[n_] := (MatrixPower[{{2, -1, 0, 0, 0, 0}, {-1, 2, -1, 0, 0, 0}, {0, -1, 2, -1, 0, -1}, {0, 0, -1, 2, -1, 0}, {0, 0, 0, -1, 2, 0}, {0, 0, -1, 0, 0, 2}}, n].{1, 0, 0, 0, 0, 0})[[1]]; Table[ f@n, {n, 0, 25}] (* Robert G. Wilson v, Aug 07 2007 *)
  • PARI
    a(n) = ([2,-1,0,0,0,0; -1,2,-1,0,0,0; 0,-1,2,-1,0,-1; 0,0,-1,2,-1,0; 0,0,0,-1,2,0; 0,0,-1,0,0,2]^n)[1,1]; \\ Michel Marcus, Jan 30 2023

Formula

Let M = [2,-1,0,0,0,0; -1,2,-1,0,0,0; 0,-1,2,-1,0,-1; 0,0,-1,2,-1,0; 0,0,0,-1,2,0; 0,0,-1,0,0,2] then a(n) is the upper left term in M^n.
G.f.: -(2*x-1)*(2*x^4-16*x^3+20*x^2-8*x+1) / ((x-1)*(3*x-1)*(x^4-16*x^3+20*x^2-8*x+1)). - Colin Barker, May 25 2013
a(n) ~ c*(2 + sqrt(2 + sqrt(3)))^n, where c = (3 - sqrt(3))/24. - Stefano Spezia, Jan 29 2023
a(n) = (3^n + 1)/4 + ((3 + sqrt(3))*((2 - sqrt(2 - sqrt(3)))^n + (2 + sqrt(2 - sqrt(3)))^n) + (3 - sqrt(3))*((2 - sqrt(2 + sqrt(3)))^n + (2 + sqrt(2 + sqrt(3)))^n))/24. - Vaclav Kotesovec, Jan 30 2023

Extensions

More terms from Robert G. Wilson v, Aug 07 2007

A125501 The (1,1)-entry in the matrix M^n, where M is the 7 X 7 Cartan matrix [2,-1,0,0,0,0,0; -1,2,-1,0,0,0,0; 0,-1,2,-1,0,0,-1; 0,0,-1,2,-1,0,0; 0,0,0,-1,2,-1,0; 0,0,0,0,-1,2,0; 0,0,-1,0,0,0,2].

Original entry on oeis.org

1, 2, 5, 14, 42, 132, 430, 1444, 4981, 17594, 63442, 232828, 867145, 3269034, 12446307, 47767466, 184508963, 716386598, 2793067210, 10926148172, 42857189054, 168471757292, 663434825367, 2616336659586, 10329939578230
Offset: 0

Views

Author

Gary W. Adamson, Dec 28 2006

Keywords

Examples

			a(6) = 430 = leftmost term in M^6 * [1,0,0,0,0,0,0].

Links

Crossrefs

Cf. A126566, A126567, A126569.

Programs

  • Maple
    with(linalg): M[1]:=matrix(7,7,[2,-1,0,0,0,0,0,-1,2,-1,0,0,0,0,0,-1,2,-1,0,0,-1,0,0,-1,2,-1,0,0,0,0,0,-1,2,-1,0,0,0,0,0,-1,2,0,0,0,-1,0,0,0,2]): for n from 2 to 30 do M[n]:=multiply(M[1],M[n-1]) od:1, seq(M[n][1,1],n=1..30); # Emeric Deutsch, Jan 20 2007
  • PARI
    {a(n)=local(E7=[2,-1,0,0,0,0,0; -1,2,-1,0,0,0,0; 0,-1,2,-1,0,0,-1; 0,0,-1,2,-1,0,0; 0,0,0,-1,2,-1,0; 0,0,0,0,-1,2,0; 0,0,-1,0,0,0,2]); (E7^n)[1,1]} \\ Paul D. Hanna, Jan 02 2007

Formula

G.f.: -(2*x-1)*(x^4 - 12*x^3 + 19*x^2 - 8*x + 1) / (x^6 - 36*x^5 + 105*x^4 - 112*x^3 + 54*x^2 - 12*x + 1). - Colin Barker, May 25 2013

Extensions

More terms from Paul D. Hanna, Jan 02 2007
Showing 1-3 of 3 results.

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