A112260 -id:A112260 - cp's OEIS frontend

A112261 a(n) = A112260(n+1) - A112260(n).

-2, 0, 12, 20, -12, -60, 52, 420, 468, -700, -1548, 3300, 13588, 8580, -29388, -30940, 152148, 406980, 60532, -1043100, -236652, 6032900, 11173812, -4569180, -32487212, 19343940, 216401652, 273243620, -323927532, -880031100, 1489420852, 7157028900

Offset: 1

Russell Walsmith, Aug 30 2005

sign

			a(4) = A112260(5) - A112260(4) = 31-11 = 20.

Index entries for linear recurrences with constant coefficients, signature (3,-6,8).

Cf. A112259, A112260.

a(n) = 3*a(n-1)-6*a(n-2)+8*a(n-3); G.f.: -2*(3*x-1) / ((2*x-1)*(4*x^2-x+1)). - Colin Barker, Nov 02 2014

Name corrected and more terms from Colin Barker, Nov 02 2014

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

Russell Walsmith, Aug 30 2005

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

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (-3,24,512).

Cf. A001622, A112260, A112261.

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 *)

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)

More terms from Robert G. Wilson v, May 16 2006

New name using g.f. from Joerg Arndt, Sep 20 2020

A335840 Expansion of x(1+2x)/((1-2x)(1-x+4*x^2)).

Original entry on oeis.org

1, 5, 9, 5, 1, 45, 169, 245, 81, 125, 1849, 5445, 6241, 845, 8649, 70805, 167281, 146205, 1369, 465125, 2556801, 4890605, 3052009, 266805, 21613201, 87654845, 135419769, 53235845, 48427681, 909226125, 2862999049, 3521061845, 659000241, 3754622045

Offset: 1

Philippe Deléham, Sep 18 2020

Numbers in the sequence are alternatively products of squares or five times a product of squares.

			a(11) = 43^2, a(12) = 5*3^2*11^2.

Index entries for linear recurrences with constant coefficients, signature (3,-6,8).

Cf. A112259, A112260.

LinearRecurrence[{3,-6,8},{1,5,9},34] (* Stefano Spezia, Sep 19 2020 *)

G.f.: x*(1+2*x)/((1-2*x)*(1-x+4*x^2)).
a(n) = 3*a(n-1) - 6*a(n-2) + 8*a(n-3) for n > 3.
a(n) = A112259(n)/(A112260(n))^2.
3*a(n) = 2^(n+1) - A272931(n). - R. J. Mathar, Aug 19 2022

cp's OEIS Frontend

A112261 a(n) = A112260(n+1) - A112260(n).

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

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A335840 Expansion of x(1+2x)/((1-2x)(1-x+4*x^2)).

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cp's OEIS Frontend

A112261 a(n) = A112260(n+1) - A112260(n).

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Author

Keywords

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Links

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Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A335840 Expansion of x*(1+2*x)/((1-2*x)*(1-x+4*x^2)).

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Author

Keywords

Comments

Examples

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Programs

Mathematica

Formula

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

A335840 Expansion of x(1+2x)/((1-2x)(1-x+4*x^2)).