A215493 -id:A215493 - cp's OEIS frontend

A215817 a(n) is the rational part of A(n) = (6-sqrt(7))A(n-1) - (12-4sqrt(7))A(n-2) + (8-3sqrt(7))A(n-3) with A(0)=3, A(1)=6-sqrt(7), A(2)=19-4sqrt(7).

3, 6, 19, 66, 237, 866, 3202, 11948, 44917, 169914, 646134, 2467988, 9462498, 36398004, 140399901, 542894726, 2103745125, 8167514346, 31762430143, 123704647562, 482435457922, 1883712663668, 7363103647479, 28809291337986, 112820819490970, 442175629583316

Offset: 0

Roman Witula, Aug 25 2012

nonn

The Berndt-type sequence number 14 for the argument 2Pi/7 defined by requiring a(n) to be the rational part of the trigonometric sum A(n) := c(1)^(2*n) + c(2)^(2*n) + c(4)^(2*n), where c(j) := 2*cos(Pi/4 + 2*Pi*j/7) = 2*cos((7+8*j)*Pi/28). We note that (A(n)-a(n))/sqrt(7) = A215877(n) are all integers. We have A(n)=2^n*O(n;i/2), where O(n;d) denote the big omega function with index n for the argument d in C defined in comments to A215794 (see also Witula-Slota's paper - Section 6). From the respective recurrence relation for this function we generate the title recurrence for A(n).

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

Cf. A215493, A215494, A215143, A215510, A094429, A215794.

a(n) = rational part of c(1)^(2n) + c(2)^(2n) + c(4)^(2n) = (1-s(1))^n + (1-s(2))^n + (1-s(4))^n, where c(j) := 2*cos((7+8*j)/28) and s(j) := sin(2*Pi*j/7).

Empirical g.f.: -(2*x-1)*(6*x^4 -40*x^3 +58*x^2 -24*x +3) / (x^6 -24*x^5 +86*x^4 -104*x^3 +53*x^2 -12*x +1). - Colin Barker, Jun 01 2013

A215877 a(n) = (A(n) - A215817(n))/sqrt(7), where A(n) = (6-sqrt(7))A(n-1) - (12-4sqrt(7))A(n-2) + (8-3sqrt(7))A(n-3), with A(0)=3, A(1)=6-sqrt(7), and A(2)=19-4*sqrt(7).

Original entry on oeis.org

0, -1, -4, -16, -64, -254, -1000, -3913, -15248, -59263, -229996, -892033, -3459544, -13421784, -52104416, -202436819, -787231328, -3064347392, -11940020992, -46569416006, -181808493296, -710442293743, -2778591945620, -10876271461745, -42606078512048

Offset: 0

Roman Witula, Aug 25 2012

sign

The Berndt-type sequence number 15 for the argument 2Pi/7 defined by requiring sqrt(7)*a(n) to be the irrational part of the trigonometric sum A(n) := c(1)^(2*n) + c(2)^(2*n) + c(4)^(2*n), where c(j) := 2*cos(Pi/4 + 2*Pi*j/7) = 2*cos((7+8*j)*Pi/28).

We note that A(n)-sqrt(7)*a(n)= A215817(n). For more facts on A(n) - see comments to A215817.

			We have a(2)/a(1) = a(3)/a(2) = a(4)/a(3) = 4, but a(5)-4*a(4)=2 and a(6)=4*(a(5)-a(2)). Moreover it follows
the relations: 4*A(1)-A(2) = 5 = (3+s(1))*(1-s(1)) + (3+s(2))*(1-s(2)) + (3+s(4))*(1-s(4)), 4*A(2)-A(3) = 10 =  (3+s(1))*(1-s(1))^2 + (3+s(2))*(1-s(2))^2 + (3+s(4))*(1-s(4))^2, 4*A(3)-A(4) = 27 = (3+s(1))*(1-s(1))^3 + (3+s(2))*(1-s(2))^3 + (3+s(4))*(1-s(4))^3, whereas 4*A(4)-a(5) = 82-2*sqrt(7) = (3+s(1))*(1-s(1))^4 + (3+s(2))*(1-s(2))^4 + (3+s(4))*(1-s(4))^4.

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

Cf. A215493, A215494, A215143, A215510, A094429, A215817.

sqrt(7)*a(n) = to the irrational part of c(1)^(2*n) + c(2)^(2*n) + c(4)^(2*n) = (1-s(1))^n + (1-s(2))^n + (1-s(4))^n, where c(j) = 2*cos((7+8*j)*Pi/28) and s(j) := sin(2*Pi*j/7).

Empirical g.f.: -x * (2*x-1)^2 * (x^2-4*x+1) / (x^6 -24*x^5 +86*x^4 -104*x^3 +53*x^2 -12*x +1). - Colin Barker, Jun 01 2013

A275830 a(n) = (2sqrt(7)sin(Pi/7))^n + (-2sqrt(7)sin(2Pi/7))^n + (-2sqrt(7)sin(4Pi/7))^n.

Original entry on oeis.org

3, -7, 49, -196, 1029, -4802, 24010, -117649, 588245, -2941225, 14823774, -74942413, 380476866, -1936973136, 9886633715, -50563069571, 259029803333, -1328763571296, 6823754590093, -35073821767334, 180407337377834, -928487386730281, 4780794440512601, -24625601552074341, 126883328914736618

Offset: 0

Kai Wang, Aug 11 2016

2*sqrt(7)*sin(Pi/7), -2*sqrt(7)*sin(2*Pi/7) and -2*sqrt(7)*sin(4*Pi/7) are roots of polynomial x^3 + 7*x^2 - 49.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-7,0,49).

Cf. A108716, A215794, A275195, A274975, A275831.

RecurrenceTable[{a[0] == 3, a[1] == -7, a[2] == 49, a[n] == -7 a[n - 1] + 49 a[n - 3]}, a, {n, 0, 30}] (* Bruno Berselli, Aug 11 2016 *)

Vec((3 + 14*x)/(1 + 7*x - 49*x^3) + O(x^30)) \\ Colin Barker, Aug 30 2016

G.f.: (3 + 14*x)/(1 + 7*x - 49*x^3). - Bruno Berselli, Aug 11 2016
a(n) = -7*a(n-1) + 49*a(n-3) with n>2, a(0)=3, a(1)=-7, a(2)=49.
a(2*n-1) = 7^n*A215493(n). - Kai Wang, May 25 2017

A217444 a(n) = A(n)7^(-floor(n+1)/3), where A(n) = 7A(n-1) - 14A(n-2) + 7A(n-3) with A(0)=0, A(1)=1, A(2)=7.

Original entry on oeis.org

0, 1, 1, 5, 22, 13, 52, 204, 113, 435, 1667, 910, 3471, 13224, 7192, 27367, 104105, 56563, 215098, 817909, 444276, 1689212, 6422529, 3488381, 13262821, 50424942, 27387681, 104126704, 395884336, 215018609, 817488295, 3108041875, 1688083894, 6417991803, 24400809980

Offset: 0

Roman Witula, Oct 03 2012

The Berndt-type sequence number 18a for the argument 2Pi/7, which is closely connected with the sequence A217274. Definitions other Berndt-type sequences for the argument 2Pi/7 like A215575, A215877, A033304 in sequences from Crossrefs are given.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,10,0,0,-17,0,0,1).

Cf. A215007, A215008, A215143, A215493, A215494, A215510, A215512, A215575, A215694, A215695, A108716, A215794, A215828, A215817, A215877, A094429, A094430, A217274, A094648, A033304.

i:=35; I:=[0, 1, 7]; A:=[m le 3 select I[m] else 7*Self(m-1)-14*Self(m-2)+7*Self(m-3): m in [1..i]]; [7^(-Floor(n/3))*A[n]: n in [1..i]]; // Bruno Berselli, Oct 03 2012

CoefficientList[Series[x*(1+x+5*x^2+12*x^3+3*x^4+2*x^5+x^6)/(1 - 10*x^3 + 17*x^6 - x^9), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 15 2012 *)
LinearRecurrence[{0,0,10,0,0,-17,0,0,1}, {0, 1, 1, 5, 22, 13, 52, 204, 113}, 50] (* G. C. Greubel, Apr 23 2018 *)

x='x+O('x^30); concat([0], Vec(x*(1+x+5*x^2+12*x^3+3*x^4 +2*x^5 +x^6)/(1- 10*x^3+17*x^6-x^9))) \\ G. C. Greubel, Apr 23 2018

G.f.: x*(1+x+5*x^2+12*x^3+3*x^4+2*x^5+x^6)/(1-10*x^3+17*x^6-x^9). - Bruno Berselli, Oct 03 2012

a(n) = 10*a(n-3) - 17*a(n-6) + a(n-9). - G. C. Greubel, Apr 23 2018

A319512 a(n) = 7a(n-1) - 14a(n-2) + 7*a(n-3), a(0) = 1, a(1) = 3, a(2) = 11.

Original entry on oeis.org

1, 3, 11, 42, 161, 616, 2352, 8967, 34153, 129997, 494606, 1881355, 7154980, 27208132, 103456689, 393367835, 1495638123, 5686513994, 21620239081, 82199944512, 312521862408, 1188195487255, 4517461948657, 17175149855885, 65298950120782, 248262786503683

Offset: 0

Kai Wang, Dec 10 2018

Let {X,Y,Z} be the roots of the cubic equation
  t^3 + at^2 + bt + c = 0
where {a, b, c} are integers. Let {u, v, w} be three numbers such that {u + v + w, u*X + v*Y + w*Z, u*X^2 + v*Y^2 + w*Z^2} are integers. Then
{p(n) = u*X^n + v*Y^n + w*Z^n | n = 0, 1, 2, ...}
is an integer sequence with the recurrence relation:
p(n) = -a*p(n-1) - b*p(n-2) - c*p(n-3).
This sequence has (a, b, c) = (-7, 14, -7), (u, v, w) = (1/(sqrt(7)*tan(4*(Pi/7))), 1/(sqrt(7)*tan(8*(Pi/7))), 1/(sqrt(7)*tan(2*(Pi/7)))).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-14,7)

Cf. A215007, A215008, A215143, A215493, A215494, A215510, A217274, A217444, A094430.

LinearRecurrence[{7, -14, 7}, {1, 3, 11}, 30] (* Amiram Eldar, Dec 10 2018 *)
CoefficientList[Series[(1-2x)^2/(1-7x+14x^2-7x^3),{x,0,30}],x] (* Harvey P. Dale, Oct 08 2023 *)

Vec((1 - 2*x)^2 / (1 - 7*x + 14*x^2 - 7*x^3) + O(x^40)) \\ Colin Barker, Dec 11 2018

(X, Y, Z) = (4*sin^2(2*(Pi/7)), 4*sin^2(4*(Pi/7)), 4*sin^2(8*(Pi/7)));
a(n) = 7*a(n-1) - 14*a(n-2) + 7*a(n-3), a(0) = 1, a(1) = 3, a(2) = 11.
G.f.: (1 - 2*x)^2 / (1 - 7*x + 14*x^2 - 7*x^3). - Colin Barker, Dec 11 2018

More terms from Felix Fröhlich, Dec 10 2018

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

sign

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

A215817 a(n) is the rational part of A(n) = (6-sqrt(7))*A(n-1) - (12-4*sqrt(7))*A(n-2) + (8-3*sqrt(7))*A(n-3) with A(0)=3, A(1)=6-sqrt(7), A(2)=19-4*sqrt(7).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A215877 a(n) = (A(n) - A215817(n))/sqrt(7), where A(n) = (6-sqrt(7))A(n-1) - (12-4*sqrt(7))A(n-2) + (8-3*sqrt(7))A(n-3), with A(0)=3, A(1)=6-sqrt(7), and A(2)=19-4*sqrt(7).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A275830 a(n) = (2*sqrt(7)*sin(Pi/7))^n + (-2*sqrt(7)*sin(2*Pi/7))^n + (-2*sqrt(7)*sin(4*Pi/7))^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A217444 a(n) = A(n)*7^(-floor(n+1)/3), where A(n) = 7*A(n-1) - 14*A(n-2) + 7*A(n-3) with A(0)=0, A(1)=1, A(2)=7.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A319512 a(n) = 7*a(n-1) - 14*a(n-2) + 7*a(n-3), a(0) = 1, a(1) = 3, a(2) = 11.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

References

Crossrefs

Formula

A215817 a(n) is the rational part of A(n) = (6-sqrt(7))A(n-1) - (12-4sqrt(7))A(n-2) + (8-3sqrt(7))A(n-3) with A(0)=3, A(1)=6-sqrt(7), A(2)=19-4sqrt(7).

A215877 a(n) = (A(n) - A215817(n))/sqrt(7), where A(n) = (6-sqrt(7))A(n-1) - (12-4sqrt(7))A(n-2) + (8-3sqrt(7))A(n-3), with A(0)=3, A(1)=6-sqrt(7), and A(2)=19-4*sqrt(7).

A275830 a(n) = (2sqrt(7)sin(Pi/7))^n + (-2sqrt(7)sin(2Pi/7))^n + (-2sqrt(7)sin(4Pi/7))^n.

A217444 a(n) = A(n)7^(-floor(n+1)/3), where A(n) = 7A(n-1) - 14A(n-2) + 7A(n-3) with A(0)=0, A(1)=1, A(2)=7.

A319512 a(n) = 7a(n-1) - 14a(n-2) + 7*a(n-3), a(0) = 1, a(1) = 3, a(2) = 11.