A192237 -id:A192237 - cp's OEIS frontend

A318605 Decimal expansion of geometric progression constant for Coxeter's Loxodromic Sequence of Tangent Circles.

2, 8, 9, 0, 0, 5, 3, 6, 3, 8, 2, 6, 3, 9, 6, 3, 8, 1, 2, 4, 5, 7, 0, 0, 9, 2, 9, 6, 1, 0, 3, 1, 2, 9, 6, 0, 9, 4, 3, 5, 9, 1, 7, 2, 2, 1, 6, 4, 5, 8, 5, 9, 1, 1, 0, 7, 5, 2, 0, 8, 9, 0, 0, 5, 2, 4, 4, 5, 5, 8, 0, 3, 8, 3, 5, 4, 9, 7, 0, 4, 6, 1, 5, 3, 7, 5, 9, 1, 4, 1, 9, 1, 7, 7, 8, 5, 1, 3, 9, 6, 0, 2, 3, 2, 6, 8

Offset: 1

A.H.M. Smeets, Sep 07 2018

This constant and its reciprocal are the real solutions of x^4 - 2*x^3 - 2*x^2 - 2*x + 1 = (x^2 - (sqrt(5)+1)*x + 1)*(x^2 + (sqrt(5)-1)*x + 1) = 0.
This constant and its reciprocal are the solutions of x^2 - (1+sqrt(5))*x + 1 = 0.
Decimal expansion of the largest x satisfying x^2 - (1+sqrt(5))*x + 1 = 0.
For sequences of type aa(n) = 2*(aa(n-1) + aa(n-2) + aa(n-3)) - aa(n-4) for arbitrary initial terms (except the trivial all zero), i.e., linear recurrence relations of order 4 with signature (2,2,2,-1), lim_{n -> infinity} aa(n)/aa(n-1) = this constant; see for instance A192234, A192237, A317973, A317974, A317975, A317976.

			2.8900536382639638124570092961031296094359...

A.H.M. Smeets, Table of n, a(n) for n = 1..9999 (terms 1..3000 from Muniru A Asiru)
Index entries for algebraic numbers, degree 4

Cf. A001622, A139339, A192234, A192237, A317973, A317974, A317975, A317976.

evalf[180]((1+sqrt(5))/2+sqrt((1+sqrt(5))/2)); # Muniru A Asiru, Nov 21 2018

RealDigits[GoldenRatio + Sqrt[GoldenRatio], 10 , 120][[1]] (* Amiram Eldar, Nov 22 2018 *)

((1+sqrt(5))/2 + sqrt((1+sqrt(5))/2)) \\ Michel Marcus, Nov 21 2018

Equals A001622 + A139339, i.e., phi + sqrt(phi) where phi is the golden ratio.

A192235 Constant term of the reduction of the n-th 2nd-kind Chebyshev polynomial by x^2 -> x+1.

Original entry on oeis.org

0, 3, 8, 21, 64, 183, 528, 1529, 4416, 12763, 36888, 106605, 308096, 890415, 2573344, 7437105, 21493632, 62117747, 179523624, 518832901, 1499454912, 4333505127, 12524062256, 36195211689, 104606103232, 302317249227, 873713066040

Offset: 1

Clark Kimberling, Jun 26 2011

nonn

See A192232.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,2,-1).

Cf. A192232, A192236, A192237.

a:=[0,3,8,21];; for n in [5..40] do a[n]:=2*a[n-1]+2*a[n-2]+ 2*a[n-3]-a[n-4]; od; a; # G. C. Greubel, Jul 30 2019

I:=[0, 3, 8, 21]; [n le 4 select I[n] else 2*Self(n-1) +2*Self(n-2) +2*Self(n-3) -Self(n-4): n in [1..40]]; // G. C. Greubel, Jul 30 2019

q[x_]:= x + 1;
reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, ChebyshevU[n, x]]]], {n, 1, 40}];
Table[Coefficient[Part[t, n], x, 0], {n, 1, 40}] (* A192235 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 40}] (* A192236 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 40}] (* A192237 *)
(* by Peter J. C. Moses, Jun 25 2011 *)
LinearRecurrence[{2,2,2,-1}, {0,3,8,21}, 40] (* G. C. Greubel, Jul 30 2019 *)

a(n)=my(t=polchebyshev(n,2));while(poldegree(t)>1, t=substpol(t, x^2,x+1));subst(t,x,0) \\ Charles R Greathouse IV, Feb 09 2012

m=40; v=concat([0, 3, 8, 21], vector(m-4)); for(n=5, m, v[n] = 2*v[n-1]+2*v[n-2]+2*v[n-3]-v[n-4]); v \\ G. C. Greubel, Jul 30 2019

@cached_function
def a(n):
    if (n==0): return 0
    elif (1 <= n <= 3): return fibonacci(2*n+2)
    else: return 2*(a(n-1) + a(n-2) + a(n-3)) - a(n-4)
[a(n) for n in (0..40)] # G. C. Greubel, Jul 30 2019

Empirical G.f.: x^2*(3-x)*(1+x)/(1-2*x-2*x^2-2*x^3+x^4). - Colin Barker, Sep 11 2012

A192236 Coefficient of x in the reduction of the n-th 2nd-kind Chebyshev polynomial by x^2 -> x+1.

Original entry on oeis.org

2, 4, 12, 36, 102, 296, 856, 2472, 7146, 20652, 59684, 172492, 498510, 1440720, 4163760, 12033488, 34777426, 100508628, 290475324, 839489268, 2426169014, 7011758584, 20264358408, 58565082744, 169256230458, 489159584636, 1413697437268

Offset: 1

Clark Kimberling, Jun 26 2011

nonn

See A192232.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,2,-1).

Cf. A192232, A192235, A192237.

a:=[2,4,12,36];; for n in [5..40] do a[n]:=2*a[n-1]+2*a[n-2]+ 2*a[n-3]-a[n-4]; od; a; # G. C. Greubel, Jul 30 2019

I:=[2,4,12,36]; [n le 4 select I[n] else 2*Self(n-1) +2*Self(n-2) +2*Self(n-3) -Self(n-4): n in [1..40]]; // G. C. Greubel, Jul 30 2019

q[x_]:= x + 1; m:=40;
reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, ChebyshevU[n, x]]]], {n, m}];
Table[Coefficient[Part[t, n], x, 0], {n, m}] (* A192235 *)
Table[Coefficient[Part[t, n], x, 1], {n, m}] (* A192236 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n, m}] (* A192237 *)
(* Peter J. C. Moses, Jun 25 2011 *)
LinearRecurrence[{2,2,2,-1}, {2,4,12,36}, 40] (* G. C. Greubel, Jul 30 2019 *)

m=40; v=concat([2,4,12,36], vector(m-4)); for(n=5, m, v[n] = 2*v[n-1]+2*v[n-2]+2*v[n-3]-v[n-4]); v \\ G. C. Greubel, Jul 30 2019

def a(n):
    if (n==0): return 2
    elif (1 <= n <= 3): return 4*3^(n-1)
    else: return 2*(a(n-1) + a(n-2) + a(n-3)) - a(n-4)
[a(n) for n in (0..40)] # G. C. Greubel, Jul 30 2019

a(n) = 2*A192237(n+2).

G.f.: 2*x/(1-2*x-2*x^2-2*x^3+x^4). - Colin Barker, Sep 12 2012

A317976 a(n) = 2*(a(n-1)+a(n-2)+a(n-3))-a(n-4) for n >= 4, with initial terms 0,0,1,0.

Original entry on oeis.org

0, 0, 1, 0, 2, 6, 15, 46, 132, 380, 1101, 3180, 9190, 26562, 76763, 221850, 641160, 1852984, 5355225, 15476888, 44729034, 129269310, 373595239, 1079710278, 3120420620, 9018182964, 26063032485, 75323561860, 217689133998, 629133273722, 1818228906675, 5254779066930, 15186593360656, 43890069394800, 126844654738097

Offset: 0

N. J. A. Sloane, Sep 03 2018

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,2,-1).

Other sequences with this recurrence but different initial conditions: A192234, A192237, A317973, A317974, A317975.

Mathematica
```
(See A192235.)
```

concat(vector(2), Vec(x^2*(1 - 2*x) / (1 - 2*x - 2*x^2 - 2*x^3 + x^4) + O(x^40))) \\ Colin Barker, Sep 09 2018

G.f.: x^2*(1 - 2*x) / (1 - 2*x - 2*x^2 - 2*x^3 + x^4). - Colin Barker, Sep 09 2018

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A318605 Decimal expansion of geometric progression constant for Coxeter's Loxodromic Sequence of Tangent Circles.

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A192235 Constant term of the reduction of the n-th 2nd-kind Chebyshev polynomial by x^2 -> x+1.

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A192236 Coefficient of x in the reduction of the n-th 2nd-kind Chebyshev polynomial by x^2 -> x+1.

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A317976 a(n) = 2*(a(n-1)+a(n-2)+a(n-3))-a(n-4) for n >= 4, with initial terms 0,0,1,0.

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