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 1-4 of 4 results.

A231181 Expansion of 1/(1 - x - 4*x^2 + 3*x^3 + 3*x^4 - x^5).

Original entry on oeis.org

1, 1, 5, 6, 20, 27, 75, 110, 275, 429, 1001, 1637, 3639, 6172, 13243, 23104, 48280, 86090, 176341, 319792, 645150, 1185305, 2363596, 4386331, 8669142, 16212913, 31825005, 59873834, 116914020, 220964744, 429737220, 815057639, 1580244061
Offset: 0

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Author

Wolfdieter Lang, Nov 05 2013

Keywords

Comments

This sequence is fundamental for the coefficient sequences for the nonnegative powers of rho(11) = 2*cos(Pi/n) (length ration (smallest diagonal)/side in the regular 11-gon (Hendecagon)) when written in the power basis of the degree 5 number field Q(rho(11)). See A187360 for the minimal polynomial of rho(11) which is C(11, x) = x^5 - x^4 - 4*x^3 + 3*x^2 + 3*x - 1. See A231182-5 for these coefficient sequences.

Links

Crossrefs

Cf. A231182, A231183, A231184, A231185.

Programs

  • Mathematica
    CoefficientList[Series[1/(1-x-4x^2+3x^3+3x^4-x^5),{x,0,50}],x] (* or *) LinearRecurrence[{1,4,-3,-3,1},{1,1,5,6,20},50] (* Harvey P. Dale, Nov 13 2013 *)

Formula

G.f.: 1/(1 - x - 4*x^2 + 3*x^3 + 3*x^4 - x^5).
a(n) = a(n-1) + 4*a(n-2) - 3*a(n-3) - 3*a(n-4) + a(n-5) for n>=0, with a(-5)=1, a(-4)=a(-3)=a(-2)=a(-1)=0.

A231182 Coefficients for the nonnegative powers of rho(11) = 2*cos(Pi/11) when written in the power basis of the degree 5 number field Q(rho(11)). Coefficients for the zeroth and fourth powers.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 5, 6, 20, 27, 75, 110, 275, 429, 1001, 1637, 3639, 6172, 13243, 23104, 48280, 86090, 176341, 319792, 645150, 1185305, 2363596, 4386331, 8669142, 16212913, 31825005, 59873834, 116914020, 220964744, 429737220, 815057639
Offset: 0

Views

Author

Wolfdieter Lang, Nov 05 2013

Keywords

Comments

The formula for rho(11)^n, with rho(11) = 2*cos(Pi/11) (the length ratio (smallest diagonal)/side in the regular 11-gon) written in the power basis of the number field Q(rho(11)) is: rho(11)^n = a(n)*1 - A231183(n)*rho(11) - A231184(n-2)* rho(11)^2 + A231185(n-3)*rho(11)^3 + a(n+1)*rho(11)^4, n >= 0.

Examples

			rho(11)^4 = 0*1 - 0*rho(11) - 0*rho(11)^2 + 0*rho(11)^3 + 1*rho(11)^4 (trivial).
rho(11)^5 = 1*1 - 3*rho(11) - 3*rho(11)^2 + 4*rho(11)^3 + 1*rho(11)^4. Approximately 26.02309649, with rho(11) approximately 1.918985947.

Links

Crossrefs

Cf. A231181, A231183, A231184, A231185.

Formula

G.f.: (1-x-x^2)*(1-3*x^2)/(1-x-4*x^2+3*x^3+3*x^4-x^5).
a(n) = a(n-1) + 4*a(n-2) - 3*a(n-3) - 3*a(n-4) + a(n-5) for n>= 5, with a(0)=1, a(1)=a(2)=a(3)=a(4)=0.
a(n) = b(n) - b(n-1) - 4*b(n-2) + 3*b(n-3) + 3*b(n-4) for n>=0, with b(n) = A231181(n).

A231183 Coefficients of the nonnegative powers of rho(11) = 2*cos(Pi/11) when written in the power basis of the degree 5 number field Q(rho(11)). Negative of the coefficients of the first power.

Original entry on oeis.org

0, -1, 0, 0, 0, 3, 2, 14, 13, 54, 61, 198, 255, 715, 1012, 2574, 3910, 9280, 14877, 33557, 56069, 121736, 209990, 442933, 783035, 1615658, 2910765, 5905483, 10795397, 21621095, 39969597, 79262102, 147796497, 290868226, 545980212, 1068246916
Offset: 0

Views

Author

Wolfdieter Lang, Nov 07 2013

Keywords

Comments

The formula for rho(11)^n is (see A231182): rho(11)^n = A231182(n)*1 - a(n)*rho(11) - A231184(n-2)*rho(11)^2 + A231185(n-3)*rho(11)^3 + A231182(n+1)*rho(11)^4, n >= 0.

Examples

			rho(11)^5 = 1*1 - 3*rho(11) - 3*rho(11)^2 + 4*rho(11)^3 + 1*rho(11)^4. Approximately 26.02309649, with rho(11) approximately 1.918985947.

Links

Crossrefs

Cf. A231181, A231182, A231184, A231185.

Formula

G.f.: x*(-1 + x + 4*x^2 -3*x^3)/(1-x-4*x^2+3*x^3+3*x^4-x^5).
a(n) = a(n-1) + 4*a(n-2) - 3*a(n-3) - 3*a(n-4) + a(n-5) for n >= 5, with a(0)=0, a(1)=-1, a(2)=a(3)=a(4)=0.
a(n) = -b(n-1) + b(n-2) + 4*b(n-3) - 3*b(n-4) for n>=0, with b(n) = A231181(n).

A231185 Coefficients of the nonnegative powers of rho(11) = 2*cos(Pi/11) when written in the power basis of the degree 5 number field Q(rho(11)). Coefficients of the third power.

Original entry on oeis.org

1, 0, 4, 1, 14, 7, 48, 35, 165, 154, 572, 636, 2002, 2533, 7071, 9861, 25176, 37810, 90251, 143451, 325358, 540155, 1178291, 2022735, 4282811, 7543771, 15612092, 28048829, 57040186, 104050724, 208772476, 385320419, 765186422, 1425038684
Offset: 0

Views

Author

Wolfdieter Lang, Nov 07 2013

Keywords

Comments

This sequence gives the first differences of A231181.
The formula for rho(11)^n is (see A231182): rho(11)^n = A231182(n)*1 - A231183(n)*rho(11) - A231184(n-2)*rho(11)^2 + a(n-3)*rho(11)^3 + A231182(n+1)*rho(11)^4, n >= 0.

Examples

			rho(11)^5 = 1*1 - 3*rho(11) - 3*rho(11)^2 + 4*rho(11)^3 + 1*rho(11)^4. Approximately 26.02309649, with rho(11) approximately 1.918985947.

Links

Crossrefs

Cf. A231181, A231182, A231183, A231184.

Programs

  • Mathematica
    LinearRecurrence[{1,4,-3,-3,1},{1,0,4,1,14},40] (* Harvey P. Dale, Aug 03 2023 *)

Formula

G.f.: (1 - x)/(1-x-4*x^2+3*x^3+3*x^4-x^5).
a(n) = a(n-1) + 4*a(n-2) - 3*a(n-3) - 3*a(n-4) + a(n-5) for n >= 0, with a(-5)=-2, a(-4)=-1 , a(-3)=a(-2)=a(-1)=0.
a(n) = b(n) - b(n-1) for n>=0, with b(n) = A231181(n) (first differences).
Showing 1-4 of 4 results.

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