A214683 -id:A214683 - cp's OEIS frontend

A215100 a(n) = 3a(n-1) + 4a(n-2) + a(n-3) with a(0)=2, a(1)=5, a(2)=22.

2, 5, 22, 88, 357, 1445, 5851, 23690, 95919, 388368, 1572470, 6366801, 25778651, 104375627, 422608286, 1711106017, 6928126822, 28051412820, 113577851765, 459867333397, 1861964820071, 7538941645566, 30524551550379, 123591386053472, 500411306007498, 2026124013786761

Offset: 0

Roman Witula, Aug 03 2012

Ramanujan-type sequence number 4 for the argument 2*Pi/7. We have a(n)=bs(3n+2), where the sequence bs(n) and its two conjugate sequences as(n) and cs(n) are defined in the comments to A214683 (see also A215076, A120757, A006053). Since we also have as(3n+2)=cs(3n+2)=0 from the formula for S(n) (see Comments at A214683) we obtain the relation 7^(1/3)*a(n)= (c(1)/c(4))^(n + 2/3) + (c(4)/c(2))^(n + 2/3) + (c(2)/c(1))^(n + 2/3).

			From 4*a(2) = a(3) = 88 we get 88*7^(1/3) = 4*((c(1)/c(4))^(8/3) + (c(4)/c(2))^(8/3) + (c(2)/c(1))^(8/3))=(c(1)/c(4))^(11/3) + (c(4)/c(2))^(11/3) + (c(2)/c(1))^(11/3).

R. Witula, E. Hetmaniok and D. Slota, Sums of the powers of any order roots taken from the roots of a given polynomial, Proceedings of the Fifteenth International Conference on Fibonacci Numbers and Their Applications, Eger, Hungary, 2012.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Roman Witula, Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2*Pi/7, Journal of Integer Sequences, Vol. 12 (2009), Article 09.8.5.
Roman Witula, Full Description of Ramanujan Cubic Polynomials, Journal of Integer Sequences, Vol. 13 (2010), Article 10.5.7.
Roman Witula, Ramanujan Cubic Polynomials of the Second Kind, Journal of Integer Sequences, Vol. 13 (2010), Article 10.7.5.
Roman Witula, Ramanujan Type Trigonometric Formulae, Demonstratio Math. 45 (2012) 779-796.
Index entries for linear recurrences with constant coefficients, signature (3,4,1).

Cf. A006053, A214683, A215076, A120757.

I:=[2,5,22]; [n le 3 select I[n] else 3*Self(n-1) +4*Self(n-2) +Self(n-3): n in [1..41]]; // G. C. Greubel, Nov 25 2022

LinearRecurrence[{3, 4, 1}, {2, 5, 22}, 40]

Vec((2-x-x^2)/(1-3*x-4*x^2-x^3) + O(x^40)) \\ Michel Marcus, Apr 20 2016

@CachedFunction
def a(n): # a = A215100
    if (n<3): return (2,5,22)[n]
    else: return 3*a(n-1) + 4*a(n-2) + a(n-3)
[a(n) for n in range(41)] # G. C. Greubel, Nov 25 2022

G.f.: (2 - x - x^2)/(1 - 3*x - 4*x^2 - x^3).

More terms from Michel Marcus, Apr 20 2016

A215560 a(n) = 3a(n-1) + 46a(n-2) + a(n-3) with a(0)=a(1)=3, a(2)=101.

Original entry on oeis.org

3, 3, 101, 444, 5981, 38468, 390974, 2948431, 26868565, 216624495, 1888775906, 15657923053, 134074085330, 1124375492334, 9556192325235, 80523923708399, 682280993578341, 5760499663646612, 48746948619251921, 411906111379078256, 3483838470286469746, 29447943482916260935

Offset: 0

Roman Witula, Aug 16 2012

The Ramanujan-type sequence number 6 for the argument 2Pi/7 (see also A214683, A215112, A006053, A006054, A215076, A215100, A120757 for the numbers: 1, 1a, 2, 2a, 3, 4 and 5 respectively).
The sequence a(n) is one of the three special sequences (the remaining two are A215569 and A215572) connected with the following recurrence relation: T(n):=49^(1/3)*T(n-2)+T(n-3), with T(0)=3, T(1)=0, and T(2)=2*49^(1/3) - see the comments to A214683.
It can be proved that
T(n) = (c(1)^4/c(2))^(n/3) + (c(2)^4/c(4))^(n/3) + (c(4)^4/c(1))^(n/3), where c(j):=2*cos(2*Pi*j/7), and the following decomposition hold true:
T(n) = at(n) + bt(n)*7^(1/3) + ct(n)*49^(1/3), where sequences at(n), bt(n), and ct(n) satisfy the following system of recurrence equations: at(n)=7*bt(n-2)+at(n-3),
bt(n)=7*ct(n-2)+bt(n-3), ct(n)=at(n-2)+ct(n-3), with at(0)=3, at(1)=at(2)=bt(0)=bt(1)=bt(2)=ct(0)=ct(1)=0, ct(2)=2 - for details see the first Witula reference.
It follows that a(n)=at(3*n), bt(3*n)=ct(3*n)=0.
Every difference of the form a(n)-a(n-2)-a(n-3) is divisible by 3. Because the difference a(n+1)-a(n) is congruent to the difference a(n-4)-a(n-2) modulo 3 we easily deduce that a(6)-a(5) and a(7)-a(6)-2 are both divisible by 3.

R. Witula, E. Hetmaniok, D. Slota, Sums of the powers of any order roots taken from the roots of a given polynomial, Proceedings of the Fifteenth International Conference on Fibonacci Numbers and Their Applications, Eger, Hungary, 2012

Roman Witula, Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2*Pi/7, Journal of Integer Sequences, Vol. 12 (2009), Article 09.8.5.
Roman Witula, Full Description of Ramanujan Cubic Polynomials, Journal of Integer Sequences, Vol. 13 (2010), Article 10.5.7.
Roman Witula, Ramanujan Cubic Polynomials of the Second Kind, Journal of Integer Sequences, Vol. 13 (2010), Article 10.7.5.
Roman Witula, Ramanujan Type Trigonometric Formulae, Demonstratio Math. 45 (2012) 779-796.
Index entries for linear recurrences with constant coefficients, signature (3,46,1).

Cf. A214683, A215112, A006053, A006054, A215076, A215100, A120757, A214699.

LinearRecurrence[{3, 46, 1}, {3, 3, 101}, 50]

Vec((3-6*x-46*x^2)/(1-3*x-46*x^2-x^3) + O(x^40)) \\ Michel Marcus, Apr 20 2016

a(n) = (c(1)^4/c(2))^n + (c(2)^4/c(4))^n + (c(4)^4/c(1))^n, where c(j) = 2*cos(2*Pi*j/7).

G.f.: (3-6*x-46*x^2)/(1-3*x-46*x^2-x^3).

More terms from Michel Marcus, Apr 20 2016

A215572 a(n) = 3a(n-1) + 46a(n-2) + a(n-3) with a(0)=2, a(1)=5, a(2)=106.

Original entry on oeis.org

2, 5, 106, 550, 6531, 44999, 435973, 3384404, 30252969, 246877464, 2135653370, 17793576423, 151867661753, 1276243154087, 10832435479322, 91356359187721, 773637352766062, 6534137016412674, 55281085635664595, 467187197014742851, 3951025667301212597, 33398969150217473532

Offset: 0

Roman Witula, Aug 16 2012

The Ramanujan-type sequence number 8 for the argument 2Pi/7 (see also A214683, A215112, A006053, A006054, A215076, A215100, A120757, A215560, A215569 for the numbers: 1, 1a, 2, 2a, 3-7 respectively). The sequence a(n) is one of the three special sequences (the remaining two are A215560 and A215569) connected with the following recurrence relation:
(c(1)^4/c(2))^(n/3) + (c(2)^4/c(4))^(n/3) + (c(4)^4/c(1))^(n/3) = at(n) + bt(n)*7^(1/3) + ct(n)*49^(1/3), where c(j):=2*cos(2*Pi*j/7), and the sequences at(n), bt(n), and ct(n) are defined in comments to A215560 (see also A215569). It follows that a(n)=ct(3*n+2), at(3*n+2)=bt(3*n+2)=0, which implies the first formula below.
We note that if a(n), a(n+1) and a(n+2) are all odd for some n in N then a(n+3) is even, a(n+4) is odd, a(n+5) and a(n+6) are both even, and the numbers a(n+7), a(n+8), a(n+9) are all odd again. In consequence, this situation hold for every n of the form 7*k+4, k=0,1,..., in the other words cyclical through all sequence a(n), n=4,5,... (from n=1 whenever we start from odd-even-even sequence).

			From 4*a(1)+5*a(2)=a(3) we obtain 4*((c(1)^4/c(2))^(5/3) + (c(2)^4/c(4))^(5/3) + (c(4)^4/c(1))^(5/3)) + 5*((c(1)^4/c(2))^(8/3) + (c(2)^4/c(4))^(8/3) + (c(4)^4/c(1))^(8/3)) = (4 + 5*c(1)^4/c(2))*((c(1)^4/c(2))^(5/3) + (4 + 5*c(2)^4/c(4))*((c(2)^4/c(4))^(5/3) + (4 + 5*c(4)^4/c(1))*((c(4)^4/c(1))^(5/3) = (c(1)^4/c(2))^(11/3) + (c(2)^4/c(4))^(11/3) + (c(4)^4/c(1))^(11/3) = 550*49^(1/3).

R. Witula, E. Hetmaniok, D. Slota, Sums of the powers of any order roots taken from the roots of a given polynomial, Proceedings of the Fifteenth International Conference on Fibonacci Numbers and Their Applications, Eger, Hungary, 2012.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Roman Witula, Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2*Pi/7, Journal of Integer Sequences, Vol. 12 (2009), Article 09.8.5.
Roman Witula, Full Description of Ramanujan Cubic Polynomials, Journal of Integer Sequences, Vol. 13 (2010), Article 10.5.7.
Roman Witula, Ramanujan Cubic Polynomials of the Second Kind, Journal of Integer Sequences, Vol. 13 (2010), Article 10.7.5.
Roman Witula, Ramanujan Type Trigonometric Formulae, Demonstratio Math. 45 (2012) 779-796.
Index entries for linear recurrences with constant coefficients, signature (3,46,1).

Cf. A214683, A215112, A006053, A006054, A215076, A215100, A120757, A215560, A215569.

LinearRecurrence[{3,46,1}, {2,5,106}, 50]
CoefficientList[Series[(2 - x - x^2)/(1 - 3*x - 46*x^2 - x^3), {x,0,50}], x] (* G. C. Greubel, Apr 16 2017 *)

Vec((2-x-x^2)/(1-3*x-46*x^2-x^3) + O(x^40)) \\ Michel Marcus, Apr 20 2016

49^(1/3)*a(n) = (c(1)^4/c(2))^(n+2/3) + (c(2)^4/c(4))^(n+2/3) + (c(4)^4/c(1))^(n+2/3) = (c(1)*(c(1)/c(2))^(1/3))^(3*n+2) + (c(2)*(c(2)/c(4))^(1/3))^(3*n+2) + (c(4)*(c(4)/c(1))^(1/3))^(3*n+2).

G.f.: (2-x-x^2)/(1-3*x-46*x^2-x^3).

More terms from Michel Marcus, Apr 20 2016

A215635 a(n) = - 12a(n-1) - 54a(n-2) - 112a(n-3) - 105a(n-4) -36a(n-5) - 2a(n-6), with a(0)=3, a(1)=-6, a(2)=18, a(3)=-60, a(4)=210, a(5)=-756.

Original entry on oeis.org

3, -6, 18, -60, 210, -756, 2772, -10296, 38610, -145860, 554268, -2116296, 8112462, -31201644, 120347532, -465328200, 1803025410, -6999149124, 27213719148, -105960069864, 413078158350, -1612098272460, 6297409350492, -24620247483624, 96324799842498, -377102656201956, 1477141800784668

Offset: 0

Roman Witula, Aug 18 2012

The Berndt-type sequence number 3 for the argument 2*Pi/9 defined by the relation: X(n) = a(n) + b(n)*sqrt(2), where X(n) := ((cos(Pi/24))^(2*n) + (cos(7*Pi/24))^(2*n) + (cos(3*Pi/8))^(2*n))*(-4)^n. We have b(n) = A215636(n).
We note that above formula is the Binet form of the following recurrence sequence: X(n+3) + 6*X(n+2) + 9*X(n+1) + (2 + sqrt(2))*X(n) = 0, which is a special type of the sequence X(n)=X(n;g) defined in the comments to A215634 for g:=Pi/24. The sequences a(n) and b(n) satisfy the following system of recurrence equations: a(n) = -b(n+3)-6*b(n+2)-9*b(n+1)-2*b(n), 2*b(n) =  -a(n+3)-6*a(n+2)-9*a(n+1)-2*a(n).
There exists an amazing relation: (-1)^n*a(n)=3*A000984(n) for every n=0,1,...,11 and 3*A000984(12)-a(12)=6.

Roman Witula and D. Slota, On modified Chebyshev polynomials, J. Math. Anal. Appl., 324 (2006), 321-343.
Index entries for linear recurrences with constant coefficients, signature (-12,-54,-112,-105,-36,-2).

Cf. A215455, A215634, A215636, A215007, A214699, A214683.

LinearRecurrence[{-12,-54,-112,-105,-36,-2}, {3,-6,18,-60,210,-756}, 50]

Vec((3+30*x+108*x^2+168*x^3+105*x^4+18*x^5) /(1+12*x+54*x^2+112*x^3+105*x^4+36*x^5+2*x^6)+O(x^99)) \\ Charles R Greathouse IV, Oct 01 2012

G.f.: (3+30*x+108*x^2+168*x^3+105*x^4+18*x^5) / (1+12*x+54*x^2+112*x^3+105*x^4+36*x^5+2*x^6).

A215112 a(n) = -2*a(n-1) + a(n-2) + a(n-3) with a(0)=a(1)=-1, a(2)=1.

Original entry on oeis.org

-1, -1, 1, -4, 8, -19, 42, -95, 213, -479, 1076, -2418, 5433, -12208, 27431, -61637, 138497, -311200, 699260, -1571223, 3530506, -7932975, 17825233, -40052935, 89998128, -202223958, 454393109, -1021012048, 2294193247, -5155005433, 11583192065

Offset: 0

Roman Witula, Aug 03 2012

We call this sequence the Ramanujan-type sequence number 1a for the argument 2Pi/7 because it forms the negative part of A214683 (i.e. for nonpositive indices). It is interesting that the same Ramanujan-type formula (with negative powers - see comments in A214683) is connected with a(n). Indeed, we have 7^(1/3)*a(n) = (c(1)/c(2))^(1/3)*(2c(1))^(-n) + (c(2)/c(4))^(1/3)*(2c(2))^(-n) + (c(4)/c(1))^(1/3)*(2c(4))^(-n) = (c(1)/c(2))^(1/3)*(2c(2))^(-n+1) + (c(2)/c(4))^(1/3)*(2c(4))^(-n+1) + (c(4)/c(1))^(1/3)*(2c(1))^(-n+1), where c(j) := Cos(2Pi*j/7). This relation follows from the following identity: (2*c(j))^(-n-1) = (2*c(2j)+2*c(j))*(2*c(j))^(-n) =((2*c(j))^2+2*c(j)-2)*(2*c(j))^(-n) whenever j is not divided by 7 since 8*c(j)*c(2j)*c(4j)=1.

R. Witula, E. Hetmaniok, D. Slota, Sums of the powers of any order roots taken from the roots of a given polynomial, Proceedings of the Fifteenth International Conference on Fibonacci Numbers and Their Applications, Eger, Hungary, 2012.

Roman Witula, Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2*Pi/7, Journal of Integer Sequences, Vol. 12 (2009), Article 09.8.5.
Roman Witula, Full Description of Ramanujan Cubic Polynomials, Journal of Integer Sequences, Vol. 13 (2010), Article 10.5.7.
Roman Witula, Ramanujan Cubic Polynomials of the Second Kind, Journal of Integer Sequences, Vol. 13 (2010), Article 10.7.5.
Roman Witula, Ramanujan Type Trigonometric Formulae, Demonstratio Math. 45 (2012) 779-796.
Index entries for linear recurrences with constant coefficients, signature (-2,1,1).

Cf. A214683.

LinearRecurrence[{-2, 1, 1}, {-1, -1, 1}, 40]

G.f.: (1+3*x)/(-1-2*x+x^2+x^3).

A215569 a(n) = 3a(n-1) + 46a(n-2) + a(n-3) with a(0)=0, a(1)=14, a(2)=49.

Original entry on oeis.org

0, 14, 49, 791, 4641, 50358, 365351, 3417162, 27107990, 238878773, 1967021021, 16916594611, 141471629572, 1204545261843, 10138247340452, 85965295695706, 725459810009753, 6140921279372187, 51879880394260905, 438847479843913070, 3709157858947113027

Offset: 0

Roman Witula, Aug 16 2012

The Ramanujan-type sequence number 7 for the argument 2Pi/7 (see also A214683, A215112, A006053, A006054, A215076, A215100, A120757, A215560 for the numbers: 1, 1a, 2, 2a, 3-6 respectively). The sequence a(n) is one of the three special sequences (the remaining two are A215560 and A215572) connected with the following recurrence relation:
(c(1)^4/c(2))^(n/3) + (c(2)^4/c(4))^(n/3) + (c(4)^4/c(1))^(n/3) = at(n) + bt(n)*7^(1/3) + ct(n)*49^(1/3), where c(j):=2*cos(2*Pi*j/7), and the sequences at(n), bt(n), and ct(n) satisfy the following system of recurrence equations: at(n)=7*bt(n-2)+at(n-3),
bt(n)=7*ct(n-2)+bt(n-3), ct(n)=at(n-2)+ct(n-3), with at(0)=3, at(1)=at(2)=bt(0)=bt(1)=bt(2)=ct(0)=ct(1)=0, ct(2)=2 - for details see the Witula's first paper (see also A215560). It follows that a(n)=bt(3*n+1), at(3*n+1)=ct(3*n+1)=0, which implies the first formula below.
We note that all numbers a(n) are divided by 7.

			We have  (c(1)^4/c(2))^(4/3) + (c(2)^4/c(4))^(4/3) + (c(4)^4/c(1))^(4/3) = (2/7)*(c(1)^4/c(2))^(7/3) + (c(2)^4/c(4))^(7/3) + (c(4)^4/c(1))^(7/3)).

R. Witula, E. Hetmaniok, D. Slota, Sums of the powers of any order roots taken from the roots of a given polynomial, Proceedings of the Fifteenth International Conference on Fibonacci Numbers and Their Applications, Eger, Hungary, 2012.

Roman Witula, Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2*Pi/7, Journal of Integer Sequences, Vol. 12 (2009), Article 09.8.5.
Roman Witula, Full Description of Ramanujan Cubic Polynomials, Journal of Integer Sequences, Vol. 13 (2010), Article 10.5.7.
Roman Witula, Ramanujan Cubic Polynomials of the Second Kind, Journal of Integer Sequences, Vol. 13 (2010), Article 10.7.5.
Roman Witula, Ramanujan Type Trigonometric Formulae, Demonstratio Math. 45 (2012) 779-796.
Index entries for linear recurrences with constant coefficients, signature (3,46,1).

Cf. A214683, A215112, A006053, A006054, A215076, A215100, A120757, A215560.

LinearRecurrence[{3,46,1},{0,14,49},30] (* Harvey P. Dale, Jan 12 2015 *)

7^(1/3)*a(n) = (c(1)^4/c(2))^(n+1/3) + (c(2)^4/c(4))^(n+1/3) + (c(4)^4/c(1))^(n+1/3) = (c(1)*(c(1)/c(2))^(1/3))^(3*n+1) + (c(2)*(c(2)/c(4))^(1/3))^(3*n+1) + (c(4)*(c(4)/c(1))^(1/3))^(3*n+1).

G.f.: (14*x+7*x^2)/(1-3*x-46*x^2-x^3).

More terms from Harvey P. Dale, Jan 12 2015

A215139 a(n) = (a(n-1) - a(n-3))*7^((1+(-1)^n)/2) with a(6)=5, a(7)=4, a(8)=22.

Original entry on oeis.org

5, 4, 22, 17, 91, 69, 364, 273, 1428, 1064, 5537, 4109, 21315, 15778, 81683, 60368, 312130, 230447, 1190553, 878423, 4535832, 3345279, 17267992, 12732160, 65708167, 48440175, 249956105, 184247938, 950654341, 700698236, 3615152086, 2664497745, 13746596563, 10131444477

Offset: 6

Roman Witula, Aug 04 2012

The Ramanujan-type sequence the number 9 for the argument 2*Pi/7. The sequence is connecting with the following decomposition: (s(4)/s(1))^(1/3)*s(1)^n + (s(1)/s(2))^(1/3)*s(2)^n + (s(2)/s(4))^(1/3)*s(4)^n = x(n)*(4-3*7^(1/3))^(1/3) + y(n)*(11-3*49^(1/3))^(1/3), where s(j) := sin(2*Pi*j/7), x(0)=1, x(1)=-7^(1/6)/2, x(2)=y(0)=y(1)=0, y(2)=7^(1/3)/4 and X(n)=sqrt(7)*(X(n-1)-X(n-3)) for every n=3,4,..., and X=x or X=y. It could be deduced the formula 4*y(n) = a(n)*7^(1/3 + (3+(-1)^n)/4), which implies a(0)=0, a(1)= 0, a(2)= 1/7, a(3)=1/7, a(4)=1, a(5)=6/7, i.e., A163260(n)=7*a(n) for every n=0,1,...,5. The sequence a(n) is discussed in third Witula paper.

			From values of x(2),y(2) and the identity 2*sin(t)^2=1-cos(2*t) we obtain (s(4)/s(1))^(1/3)*c(1) + (s(1)/s(2))^(1/3)*c(4) + (s(2)/s(4))^(1/3)*c(1) = (4-3*7^(1/3))^(1/3) - (1/2)*(7*(11-3*49^(1/3)))^(1/3), where c(j):=cos(2*Pi*j/7). Further, from values of x(1),x(3),y(1),y(3) and the identity 4*sin(t)^3=3*sin(t)-sin(3*t) we obtain (s(4)/s(1))^(1/3)*s(4) + (s(1)/s(2))^(1/3)*s(1) + (s(2)/s(4))^(1/3)*s(2) = (-3*7^(1/6)/2 +4*7^(1/2))*(4-3*7^(1/3))^(1/3) - 7^(5/6)*(11-3*49^(1/3))^(1/3).

R. Witula, E. Hetmaniok and D. Slota, Sums of the powers of any order roots taken from the roots of a given polynomial, Proceedings of the Fifteenth International Conference on Fibonacci Numbers and Their Applications, Eger, Hungary, 2012.

G. C. Greubel, Table of n, a(n) for n = 6..1005
Roman Witula and Damian Slota, New Ramanujan-Type Formulas and Quasi-Fibonacci Numbers of Order 7, Journal of Integer Sequences, Vol. 10 (2007), Article 07.5.6.
Roman Witula, Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2*Pi/7, Journal of Integer Sequences, Vol. 12 (2009), Article 09.8.5.
Roman Witula, Full Description of Ramanujan Cubic Polynomials, Journal of Integer Sequences, Vol. 13 (2010), Article 10.5.7.
Roman Witula, Ramanujan Cubic Polynomials of the Second Kind, Journal of Integer Sequences, Vol. 13 (2010), Article 10.7.5.
Roman Witula, Ramanujan Type Trigonometric Formulae, Demonstratio Math. 45 (2012) 779-796.
Index entries for linear recurrences with constant coefficients, signature (0,7,0,-14,0,7).

Cf. A214683, A215112, A006053, A006054, A215076, A215100, A120757, A215560, A215569, A215572, A214699.

I:=[5,4,22,17,91,69]; [n le 6 select I[n] else 7*Self(n-2) - 14*Self(n-4) + 7*Self(n-6): n in [1..30]]; // G. C. Greubel, Apr 19 2018

LinearRecurrence[{0,7,0,-14,0,7}, {5,4,22,17,91,69}, {1,50}] (* G. C. Greubel, Apr 19 2018 *)

Vec(-x*(1+x)*(6*x^4+x^3-12*x^2-x+5)/(-1+7*x^2-14*x^4+7*x^6) + O(x^50)) \\ Michel Marcus, Apr 20 2016

G.f.: -x*(1+x)*(6*x^4+x^3-12*x^2-x+5) / ( -1+7*x^2-14*x^4+7*x^6 ). - R. J. Mathar, Sep 14 2012

More terms from Michel Marcus, Apr 20 2016

A218664 Coefficients of cubic polynomials p(x+n), where p(x) = x^3 + x^2 - 2*x - 1.

Original entry on oeis.org

1, 1, -2, -1, 1, 4, 3, -1, 1, 7, 14, 7, 1, 10, 31, 29, 1, 13, 54, 71, 1, 16, 83, 139, 1, 19, 118, 239, 1, 22, 159, 377, 1, 25, 206, 559, 1, 28, 259, 791, 1, 31, 318, 1079, 1, 34, 383, 1429, 1, 37, 454, 1847, 1, 40, 531, 2339, 1, 43, 614, 2911, 1, 46, 703, 3569, 1, 49, 798, 4319

Offset: 0

Roman Witula, Nov 04 2012

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We have p(x) = (x - c(1))*(x - c(2))*(x - c(4)), where c(j) := 2*cos(2*Pi*j/7). We note that c(4) = c(3) = -c(1/2), c(1) = s(3) and c(2) = -s(1), where s(j) := 2*sin(Pi*j/14). Moreover we obtain -p(-x) = x^3 - x^2 - 2*x + 1 = (x + c(1))*(x + c(2))*(x + c(4)), q(x) := -x^3*p(1/x) = x^3 + 2*x^2 + x - 1 = (x - c(1)^(-1))*(x - c(2)^(-1))*(x - c(4)^(-1)), and -q(-x) = x^3 - 2*x^2 + x + 1 = (x + c(1)^(-1))*(x + c(2)^(-1))*(x + c(4)^(-1)).
We also have p(x+2) =  x^3 + 7*x^2 + 14*x + 7 = (x + s(2)^2)*(x + s(4)^2)*(x + s(6)^2). The polynomial -p(-x-2) = x^3 - 7*x^2 + 14*x - 7 = (x - s(2)^2)*(x - s(4)^2)*(x - s(6)^2) is known as Johannes Kepler's cubic polynomial (see Witula's book).
Let us set r(x) := p(x+1). It can be verified that -x^3*r(1/x) = x^3 - 3*x^2 - 4*x - 1 = (x - c(1)/c(4))*(x - c(4)/c(2))*(x - c(2)/c(1)); for example, we have c(1)^3 + c(1)^2 - 2*c(1) - 1 = 0 which implies that c(1)^2 + 2*c(1) = 1/(c(1) - 1), and then c(1)^2 + 2*c(1) = c(4)/c(2) since c(4)/c(2) = (c(1)^4 - 4*c(1)^2 + 2)/(c(1)^2 - 2).
The polynomials p(x+n) and the ones obtained as above (i.e., after simple algebraic transformations) are the characteristic polynomials of many sequences in the OEIS; see crossrefs.

R. Witula, Complex Numbers, Polynomials and Partial Fraction Decomposition, Part 3, Wydawnictwo Politechniki Slaskiej, Gliwice 2010 (Silesian Technical University publishers).

Cf. A094648, A096975, A033304, A214683, A006053, A215076, A215100, A215007, A215008, A215143, A215493, A215494, A215510.

We have a(4*k) = 1, a(4*k + 1) = 3*k + 1, a(4*k + 2) = 3*k^2 + 2*k - 2, a(4*k + 3) = k^3 + k^2 - 2*k - 1. Further, the following relations hold true: b(k+1) = b(k) + 3, c(k+1) = 2*b(k) -2*c(k) + 3, d(k+1) = b(k) - 2*c(k) - d(k) + 1, where p(x + k) = x^3 + b(k)*x^2 + c(k)*x + d(k).

Empirical g.f.: -(x^15 - x^14 - 2*x^13 + x^12 - 5*x^11 + 10*x^10 + 3*x^9 - 3*x^8 - 3*x^7 - 11*x^6 + 3*x^4 + x^3 + 2*x^2 - x - 1) / ((x-1)^4*(x+1)^4*(x^2+1)^4). - Colin Barker, May 17 2013

cp's OEIS Frontend

A215100 a(n) = 3*a(n-1) + 4*a(n-2) + a(n-3) with a(0)=2, a(1)=5, a(2)=22.

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A215560 a(n) = 3*a(n-1) + 46*a(n-2) + a(n-3) with a(0)=a(1)=3, a(2)=101.

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A215572 a(n) = 3*a(n-1) + 46*a(n-2) + a(n-3) with a(0)=2, a(1)=5, a(2)=106.

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A215635 a(n) = - 12*a(n-1) - 54*a(n-2) - 112*a(n-3) - 105*a(n-4) -36*a(n-5) - 2*a(n-6), with a(0)=3, a(1)=-6, a(2)=18, a(3)=-60, a(4)=210, a(5)=-756.

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A215112 a(n) = -2*a(n-1) + a(n-2) + a(n-3) with a(0)=a(1)=-1, a(2)=1.

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A215569 a(n) = 3*a(n-1) + 46*a(n-2) + a(n-3) with a(0)=0, a(1)=14, a(2)=49.

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A215139 a(n) = (a(n-1) - a(n-3))*7^((1+(-1)^n)/2) with a(6)=5, a(7)=4, a(8)=22.

A215100 a(n) = 3a(n-1) + 4a(n-2) + a(n-3) with a(0)=2, a(1)=5, a(2)=22.

A215560 a(n) = 3a(n-1) + 46a(n-2) + a(n-3) with a(0)=a(1)=3, a(2)=101.

A215572 a(n) = 3a(n-1) + 46a(n-2) + a(n-3) with a(0)=2, a(1)=5, a(2)=106.

A215635 a(n) = - 12a(n-1) - 54a(n-2) - 112a(n-3) - 105a(n-4) -36a(n-5) - 2a(n-6), with a(0)=3, a(1)=-6, a(2)=18, a(3)=-60, a(4)=210, a(5)=-756.

A215569 a(n) = 3a(n-1) + 46a(n-2) + a(n-3) with a(0)=0, a(1)=14, a(2)=49.