A231181 -id:A231181 - cp's OEIS frontend

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.

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

Wolfdieter Lang, Nov 05 2013

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.

			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.

Index entries for linear recurrences with constant coefficients, signature (1,4,-3,-3,1).

Cf. A231181, A231183, A231184, A231185.

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

Wolfdieter Lang, Nov 07 2013

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.

			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.

Index entries for linear recurrences with constant coefficients, signature (1,4,-3,-3,1).

Cf. A231181, A231182, A231184, A231185.

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).

A231184 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 second power.

Original entry on oeis.org

-1, 0, 0, 3, 6, 17, 32, 73, 135, 286, 528, 1080, 2002, 4015, 7485, 14827, 27796, 54606, 102869, 200909, 380006, 739013, 1402309, 2718485, 5171573, 10001553, 19064476, 36802823, 70259834, 135444612, 258883604, 498538557, 953762458

Offset: 0

Wolfdieter Lang, Nov 07 2013

The formula for rho(11)^n is (see A231182): rho(11)^n = A231182(n)*1 - A231183(n)*rho(11) - a(n-2)*rho(11)^2 + A231185(n-3)*rho(11)^3 + A231182(n+1)*rho(11)^4, n >= 0.

			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.

Index entries for linear recurrences with constant coefficients, signature (1,4,-3,-3,1).

Cf. A231181, A231182, A231183, A231185.

LinearRecurrence[{1,4,-3,-3,1},{-1,0,0,3,6},40] (* Harvey P. Dale, Apr 26 2019 *)

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

Wolfdieter Lang, Nov 07 2013

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.

			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.

Index entries for linear recurrences with constant coefficients, signature (1,4,-3,-3,1).

Cf. A231181, A231182, A231183, A231184.

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

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).

A309896 Generalized Fibonacci numbers. Square array read by ascending antidiagonals. F(n,k) for n >= 0 and k >= 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 3, 3, 1, 0, 1, 1, 4, 4, 5, 1, 0, 1, 1, 5, 5, 9, 8, 1, 0, 1, 1, 6, 6, 14, 14, 13, 1, 0, 1, 1, 7, 7, 20, 20, 28, 21, 1, 0, 1, 1, 8, 8, 27, 27, 48, 47, 34, 1, 0, 1, 1, 9, 9, 35, 35, 75, 75, 89, 55, 1, 0

Offset: 0

Peter Luschny, Aug 21 2019

			Array starts:
[0] 1, 0, 0,  0,  0,  0,   0,   0,    0,    0,    0,    0, ...
[1] 1, 1, 1,  1,  1,  1,   1,   1,    1,    1,    1,    1, ...
[2] 1, 1, 2,  3,  5,  8,  13,  21,   34,   55,   89,  144, ...
[3] 1, 1, 3,  4,  9, 14,  28,  47,   89,  155,  286,  507, ...
[4] 1, 1, 4,  5, 14, 20,  48,  75,  165,  274,  571,  988, ...
[5] 1, 1, 5,  6, 20, 27,  75, 110,  275,  429, 1001, 1637, ...
[6] 1, 1, 6,  7, 27, 35, 110, 154,  429,  637, 1638, 2548, ...
[7] 1, 1, 7,  8, 35, 44, 154, 208,  637,  910, 2548, 3808, ...
[8] 1, 1, 8,  9, 44, 54, 208, 273,  910, 1260, 3808, 5508, ...
[9] 1, 1, 9, 10, 54, 65, 273, 350, 1260, 1700, 5508, 7752, ...

Genki Shibukawa, New identities for some symmetric polynomials and their applications, arXiv:1907.00334 [math.CA], 2019.

Cf. A000007 (n=0), A000012 (n=1), A000045 (n=2), A006053 (n=3), A188021 (n=4), A231181 (n=5).

@cached_function
def F(n, k):
    if k <  0: return 0
    if k == 0: return 1
    a = sum((-1)^j*binomial(n-1-j,j  )*F(n,k-1-2*j) for j in (0..(n-1)/2))
    b = sum((-1)^j*binomial(n-1-j,j+1)*F(n,k-2-2*j) for j in (0..(n-2)/2))
    return a + b
print([F(n-k, k) for n in (0..11) for k in (0..n)])

F(n, k) = Sum_{j=0..(n-1)/2} (-1)^j*binomial(n-1-j,j)*F(n, k-1-2*j) + Sum_{j=0..(n-2)/2} (-1)^j*binomial(n-1-j, j+1)*F(n, k-2-2*j) for k > 0; F(n, 0) = 1 and F(n, k) = 0 if k < 0.

cp's OEIS Frontend

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.

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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.

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A231184 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 second power.

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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.

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A309896 Generalized Fibonacci numbers. Square array read by ascending antidiagonals. F(n,k) for n >= 0 and k >= 0.

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SageMath

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