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.

A112259 Expansion of x*(1+8*x)/((1-8*x)*(1+11*x+64*x^2)).

Original entry on oeis.org

1, 5, 9, 605, 961, 16245, 284089, 29645, 15046641, 101025125, 73222249, 9908816445, 23755748641, 204034140245, 5031349566489, 1965713970605, 219320727489361, 1965265930868805, 374345220088009, 158335559155140125
Offset: 1

Views

Author

Russell Walsmith, Aug 30 2005

Keywords

Comments

Previous name was: Let p = the golden mean = (1+sqrt(5))/2. A point in 3-space is identified by three numbers t = (a,b,c). f(t) is the product a*b*c. Let t = (-1/p,1,p): using the rules of 'triternion' multiplication, e.g., (1,2,3)*(1,2,3)= 1,2,3 + 6,2,4 + 6,9,3 = (13,13,10), then -f(t^n) gives the sequence.
Numbers in the sequence are alternatively products of squares or five times a product of squares.
If f(t) is the sum of a+b+c then a(n)=2^(n+1). - Robert G. Wilson v, May 16 2006

Examples

			t = (-0.618...,1,1.618...); t^2 = (3.618...,1.381...,-1). Hence -f(t^2) = 5

Links

Crossrefs

Cf. A001622, A112260, A112261.

Programs

  • Mathematica
    s = {-1/GoldenRatio, 1, GoldenRatio}; trit[lst_] := Block[{a, b, c, d, e, f}, {a, b, c} = lst[[1]]; {d, e, f} = lst[[2]]; {{a, b, c}, FullSimplify[{a*d + b*f + c*e, a*e + b*d + c*f, a*f + b*e + c*d}]}]; f[n_] := FullSimplify[ -Times @@ Nest[trit, {s, s}, n][[2]]]; Table[ f[n], {n, 0, 20}] (* Robert G. Wilson v, May 16 2006 *)
CoefficientList[Series[(1 + 8 x) / ((1 - 8 x) (1 + 11 x + 64 x^2)), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 04 2013 *)
LinearRecurrence[{-3,24,512},{1,5,9},20] (* Harvey P. Dale, Feb 28 2024 *)

Formula

t = (-1/p, 1, p). f(t) is the product 1/p*1*p. For t1 = (a, b, c) and t2 = (x, y, z), t1 - t2 = a(x, y, z) + b(z, x, y) + c(y, z, x) = (ax+bz+cy, ay+bx+cz, az+by+cx). -f(t^n) = the sequence.
G.f.: x*(1+8*x)/((1-8*x)*(1+11*x+64*x^2)). [Joerg Arndt, Aug 03 2013]
From G. C. Greubel, Sep 21 2020: (Start)
a(n) = 2^(3*n+1) * (1 - (-1)^n * T_{n}(11/16))/27, where T_{n}(x) is the Chebyshev polynomial.
a(n) = -3*a(n-1) + 24*a(n-2) + 512*a(n-3). (End)

Extensions

More terms from Robert G. Wilson v, May 16 2006
New name using g.f. from Joerg Arndt, Sep 20 2020

A112260 Expansion of -x*(8*x^2-4*x+1) / ((2*x-1)*(4*x^2-x+1)).

Original entry on oeis.org

1, -1, -1, 11, 31, 19, -41, 11, 431, 899, 199, -1349, 1951, 15539, 24119, -5269, -36209, 115939, 522919, 583451, -459649, -696301, 5336599, 16510411, 11941231, -20545981, -1202041, 215199611, 488443231, 164515699, -715515401, 773905451, 7930934351
Offset: 1

Views

Author

Russell Walsmith, Aug 30 2005

Keywords

Comments

Previous name was: Let p = the golden mean = (1+sqrt(5))/2, t = the ordered triple (-1/p,1,p). Using the rules of 'triternion' multiplication, e.g., (1,2,3)*(1,2,3) = 1,2,3 + 6,2,4 + 6,9,3 = (13,13,10), t^n gives a sequence of ordered triples, one of which is an integer = the n-th term of the sequence.
The signs in the pattern seems to cycle through period 12. The n-th term of this sequence is a factor of the n-th term of A112259.
Let M = [1, 1-p, p; p, 1, 1-p; 1-p, p, 1] a 3 X 3 matrix where p = (1 + sqrt(5))/2. All the numbers on the main diagonal of M^n are equal to a(n). - Philippe Deléham, Sep 19 2020

Examples

			t = (-0.618...,1,1.618...); t^2 = (3.618...,1.381...,-1). Hence a(2) = -1.

Links

Crossrefs

Cf. A112259, A112261 (first differences).

Programs

  • Magma
    I:=[1,-1,-1]; [n le 3 select I[n] else 3*Self(n-1)-6*Self(n-2)+8*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Nov 02 2014
  • Mathematica
    s = {-1/GoldenRatio, 1, GoldenRatio}; trit[lst_] := Block[{a, b, c, d, e, f}, {a, b, c} = lst[[1]]; {d, e, f} = lst[[2]]; {{a, b, c}, FullSimplify[{a*d + b*f + c*e, a*e + b*d + c*f, a*f + b*e + c*d}]}]; f[n_] := Select[ Nest[trit, {s, s}, n][[2]], IntegerQ@# &][[1]]; Table[ f[n], {n, 0, 26}]
CoefficientList[Series[(8 x^2 - 4 x + 1)/((1 - 2 x) (4 x^2 - x + 1)), {x, 0, 70}], x] (* Vincenzo Librandi, Nov 02 2014 *)
  • PARI
    Vec(-x*(8*x^2-4*x+1)/((2*x-1)*(4*x^2-x+1)) + O(x^100)) \\ Colin Barker, Nov 02 2014

Formula

t = (-1/p, 1, p). (a, b, c)^2 = a(a, b, c) + b(c, a, b) + c(b, c, a) = (a^2+2bc, c^2+2ab, b^2+2ac). The integer term in t^n is the n-th term.
From Colin Barker, Nov 02 2014: (Start)
G.f.: -x*(8*x^2-4*x+1) / ((2*x-1)*(4*x^2-x+1)).
a(n) = 3*a(n-1)-6*a(n-2)+8*a(n-3). (End)

Extensions

More terms from Robert G. Wilson v, May 16 2006
New name (using g.f. by Colin Barker) from Joerg Arndt, Nov 04 2014
Showing 1-2 of 2 results.

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