A274975 -id:A274975 - cp's OEIS frontend

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

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

A275831 a(n) = (sqrt(7)csc(Pi/7)/2)^n + (-sqrt(7)csc(2Pi/7)/2)^n + (-sqrt(7)csc(4*Pi/7)/2)^n.

Original entry on oeis.org

3, 0, 14, 21, 98, 245, 833, 2401, 7546, 22638, 69629, 211288, 645869, 1966419, 6000099, 18286016, 55765626, 170002805, 518361494, 1580379017, 4818550093, 14691183577, 44792503770, 136568135690, 416385811429, 1269524476220, 3870677629833, 11801372013543, 35981414742371, 109704347503632, 334479507291398

Offset: 0

Kai Wang, Aug 11 2016

(sqrt(7)*csc(Pi/7)/2), (-sqrt(7)*csc(2*Pi/7)/2) and (-sqrt(7)*csc(4*Pi/7)/2) are the roots of the polynomial x^3 - 7*x - 7. - Corrected by Colin Barker, Aug 12 2016

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

Cf. A274975, A275195, A275830.

RecurrenceTable[{a[0] == 3, a[1] == 0, a[2] == 14, a[n] == 7 a[n - 2] + 7 a[n - 3]}, a, {n, 0, 30}] (* Bruno Berselli, Aug 11 2016 *)
LinearRecurrence[{0,7,7},{3,0,14},40] (* Harvey P. Dale, Jan 01 2022 *)

Vec((3-7*x^2)/(1-7*x^2-7*x^3) + O(x^30)) \\ Colin Barker, Aug 12 2016

G.f.: (3 - 7*x^2)/(1 - 7*x^2 - 7*x^3). - Bruno Berselli, Aug 11 2016

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

Name and comment corrected by Colin Barker, Aug 12 2016

A293312 Rectangular array read by antidiagonals: A(n,k) = tr((M_n)^k), k >= 0, where M_n is the n X n matrix M_1 = {{1}}, M_n = {{0,...,0,1},{0,...,0,1,1},...,{0,1,...,1},{1,...,1}}, n > 1, and tr(.) is the trace.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 3, 2, 4, 1, 4, 6, 2, 5, 1, 7, 11, 10, 3, 6, 1, 11, 26, 23, 15, 3, 7, 1, 18, 57, 70, 42, 21, 4, 8, 1, 29, 129, 197, 155, 69, 28, 4, 9, 1, 47, 289, 571, 533, 301, 106, 36, 5, 10, 1, 76, 650, 1640, 1884, 1223, 532, 154, 45, 5, 11

Offset: 1

L. Edson Jeffery, Oct 10 2017

Conjecture: For all n >= 1, for all k >= 2, A(n, k) = A293311(k, n); i.e., A(n, k) = number of magic labelings of the graph LOOP X C_k with magic sum n - 1.

			Array begins:
.   1 1  1   1    1     1      1       1       1        1         1
.   2 1  3   4    7    11     18      29      47       76       123
.   3 2  6  11   26    57    129     289     650     1460      3281
.   4 2 10  23   70   197    571    1640    4726    13604     39175
.   5 3 15  42  155   533   1884    6604   23219    81555    286555
.   6 3 21  69  301  1223   5103   21122   87677   363606   1508401
.   7 4 28 106  532  2494  11998   57271  274132  1310974   6271378
.   8 4 36 154  876  4654  25362  137155  743724  4029310  21836366
.   9 5 45 215 1365  8105  49347  298184 1806597 10936124  66220705
.  10 5 55 290 2035 13355  89848  599954 4016683 26868719 179784715
.  11 6 66 381 2926 21031 154935 1132942 8306078 60843972 445824731
.  ...

Cf. A293311.
Cf. A000012, A000032, A274975, A188128, A189237 (rows 1..5).
Cf. A000027, A000217, A019298, A006325, A244497, A244879, A244873, A244880, A293310, A293309 (columns k = 0,2..10 (conjectured)).

s[0, x_] := 1; s[1, x_] := x; s[k_, x_] := x*s[k - 1, x] - s[k - 2, x]; c[n_, j_] := 2 (-1)^(j - 1) Cos[j*Pi/(2 n + 1)]; a[n_, k_] := Round[Sum[s[n - 1, c[n, j]]^(k), {j, n}]];
(* Array: *)
Grid[Table[a[n, k], {n, 11}, {k, 0, 10}]]
(* Array antidiagonals flattened (gives this sequence): *)
Flatten[Table[a[n, k - n], {k, 11}, {n, k}]]

Let S(0, x) = 1, S(1, x) = x, S(k, x) = x*S(k - 1, x) - S(k - 2, x) (the S-polynomials of Wolfdieter Lang) and c(n, j) = 2*(-1)^(j - 1)*cos(j*Pi/(2*n + 1)). Then A(n, k) = Sum_{j=1..n} S(n - 1, c(n, j))^(k), n >= 1, k >= 0.

A320918 Sum of n-th powers of the roots of x^3 + 9x^2 + 20x - 1.

Original entry on oeis.org

3, -9, 41, -186, 845, -3844, 17510, -79865, 364741, -1667859, 7636046, -35002493, 160633658, -738017016, 3394477491, -15629323441, 72036344133, -332346150886, 1534759151873, -7093873005004, 32817327856690, -151943731458257, 704053152985509, -3264786419847751

Offset: 0

Kai Wang, Oct 24 2018

In general, for integer h, k let
  X = (sin^(h+k)(2*Pi/7))/(sin^(h)(4*Pi/7)*sin^(k)(8*Pi/7)),
  Y = (sin^(h+k)(4*Pi/7))/(sin^(h)(8*Pi/7)*sin^(k)(2*Pi/7)),
  Z = (sin^(h+k)(8*Pi/7))/(sin^(h)(2*Pi/7)*sin^(k)(4*Pi/7)).
then X, Y, Z are the roots of a monic equation
    t^3 + a*t^2 + b*t + c = 0
where a, b, c are integers and c = 1 or -1.
Then X^n + Y^n + Z^n, n = 0, 1, 2, ... is an integer sequence.
Instances of such sequences with (h,k) values:
  (-3,0), (0,3), (3,-3): gives A274663;
  (-3,3), (0,-3): give A274664;
  (-2,0), (0,2), (2,-2): give A198636;
  (-2,-3), (-1,-2), (2,-1), (3,-1): give A274032;
  (-1,-1), (-1,2): give A215076;
  (-1,0), (0,1), (1,-1): give A094648;
  (-1,1), (0,-1), (1,0): give A274975;
  (1,1), (-2,1), (1,-2): give A274220;
  (1,2), (-3,1), (2,-3): give A274075;
  (1,3): this sequence.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-9,-20,1).

Cf. A248417, A274032, A274075.

a := proc(n) option remember; if n < 3 then [3, -9, 41][n+1] else
-9*a(n-1) - 20*a(n-2) + a(n-3) fi end: seq(a(n), n=0..32); # Peter Luschny, Oct 25 2018

CoefficientList[Series[(3 + 18*x + 20*x^2) / (1 + 9*x + 20*x^2 - x^3) , {x, 0, 50}], x] (* Amiram Eldar, Dec 09 2018 *)
LinearRecurrence[{-9,-20,1},{3,-9,41},30] (* Harvey P. Dale, Dec 10 2023 *)

polsym(x^3 + 9*x^2 + 20*x - 1, 25) \\ Joerg Arndt, Oct 24 2018

Vec((3 + 18*x + 20*x^2) / (1 + 9*x + 20*x^2 - x^3) + O(x^30)) \\ Colin Barker, Dec 09 2018

a(n) = ((sin^4(2*Pi/7))/(sin(4*Pi/7)*sin^3(8*Pi/7)))^n
     + ((sin^4(4*Pi/7))/(sin(8*Pi/7)*sin^3(2*Pi/7)))^n
     + ((sin^4(8*Pi/7))/(sin(2*Pi/7)*sin^3(4*Pi/7)))^n.
a(n) = -9*a(n-1) - 20*a(n-2) + a(n-3) for n>2.
G.f.: (3 + 18*x + 20*x^2) / (1 + 9*x + 20*x^2 - x^3). - Colin Barker, Dec 09 2018

A321173 a(n) = -2*a(n-1) + a(n-2) + a(n-3), a(0) = -1, a(1) = 3, a(2) = -9.

Original entry on oeis.org

-1, 3, -9, 20, -46, 103, -232, 521, -1171, 2631, -5912, 13284, -29849, 67070, -150705, 338631, -760897, 1709720, -3841706, 8632235, -19396456, 43583441, -97931103, 220049191, -494446044, 1111010176, -2496417205, 5609398542, -12604204113, 28321389563, -63637584697

Offset: 0

Kai Wang, Jan 10 2019

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).
Let k = Pi/7.
This sequence has (a, b, c) = (2, -1, -1), (u, v, w) = (2*cos(4k), 2*cos(8k), 2*cos(2k)).
A033304, A274975: (a, b, c) = (2, -1, -1), (u, v, w) = (2*cos(2k), 2*cos(4k), 2*cos(8k)).
A321174 : (a, b, c) = (2, -1, -1), (u, v, w) = (2*cos(8k), 2*cos(2k), 2*cos(4k)).
X = sin(2k)/sin(4k), Y = sin(4k)/sin(8k), Z = sin(8k)/sin(2k).

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

Cf. A321174, A033304, A274975.

LinearRecurrence[{-2,1,1},{-1,3,-9},50] (* Stefano Spezia, Jan 11 2019 *)

Vec(-(1 - x + 2*x^2) / (1 + 2*x - x^2 - x^3) + O(x^30)) \\ Colin Barker, Jan 11 2019

G.f.: -(1 - x + 2*x^2) / (1 + 2*x - x^2 - x^3). - Colin Barker, Jan 11 2019

A321174 a(n) = -2*a(n-1) + a(n-2) + a(n-3), a(0) = -1, a(1) = -4, a(2) = 5.

Original entry on oeis.org

-1, -4, 5, -15, 31, -72, 160, -361, 810, -1821, 4091, -9193, 20656, -46414, 104291, -234340, 526557, -1183163, 2658543, -5973692, 13422764, -30160677, 67770426, -152278765, 342167279, -768842897, 1727574308, -3881824234, 8722379879, -19599009684, 44038575013

Offset: 0

Kai Wang, Jan 10 2019

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).
Let k = Pi/7.
This sequence has (a, b, c) = (2, -1, -1), (u, v, w) = (2*cos(8k), 2*cos(2k), 2*cos(4k)).
A033304, A274975: (a, b, c) = (2, -1, -1), (u, v, w) = (2*cos(2k), 2*cos(4k), 2*cos(8k)).
A321173 : (a, b, c) = (2, -1, -1), (u, v, w) = (2*cos(4k), 2*cos(8k), 2*cos(2k)).
X = sin(2k)/sin(4k), Y = sin(4k)/sin(8k), Z = sin(8k)/sin(2k).

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

Cf. A321173, A033304, A274975.

LinearRecurrence[{-2,1,1},{-1,-4,5},50] (* Stefano Spezia, Jan 11 2019 *)

Vec(-(1 + 6*x + 2*x^2) / (1 + 2*x - x^2 - x^3) + O(x^30)) \\ Colin Barker, Jan 11 2019

G.f.: -(1 + 6*x + 2*x^2) / (1 + 2*x - x^2 - x^3). - Colin Barker, Jan 11 2019

A287396 a(n) = (7(csc(2Pi/7))^2)^n + (7(csc(4Pi/7))^2)^n + (7(csc(8Pi/7))^2)^n.

Original entry on oeis.org

3, 56, 1568, 53312, 1931776, 71300096, 2645479424, 98305622016, 3654656065536, 135885355483136, 5052615982317568, 187873377732526080, 6985794697679601664, 259756778648305139712, 9658687473893481906176, 359144636249686988029952, 13354285908291066433372160

Offset: 0

Kai Wang, May 24 2017

Colin Barker, Table of n, a(n) for n = 0..600
Roman Witula, Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2Pi/7, J. Integer Seq., 12 (2009), Article 09.8.5.
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 and Damian Slota, Quasi-Fibonacci Numbers of Order 11, Journal of Integer Sequences, Vol. 10 (2007), Article 07.8.5
Roman Witula, Damian Slota and Adam Warzynski, Quasi-Fibonacci Numbers of the Seventh Order, J. Integer Seq., 9 (2006), Article 06.4.3.
Index entries for linear recurrences with constant coefficients, signature (56,-784,3136).

Cf. A033304, A094648, A274220, A215076, A274075, A274032, A275195, A215575, A274975.

LinearRecurrence[{56,-784,3136},{3,56,1568},30] (* Harvey P. Dale, Aug 08 2017 *)

Vec((3 - 28*x)*(1 - 28*x) / (1 - 56*x + 784*x^2 - 3136*x^3) + O(x^30)) \\ Colin Barker, May 25 2017

polsym(x^3 - 56*x^2 + 784* x - 3136, 20) \\ Joerg Arndt, May 26 2017

a(n) = x1^n + x2^n + x3^n, where x1, x2, x3 are the roots of x^3 - 56*x^2 + 784* x - 3136, x1 = 7*(csc(2*Pi/7))^2, x2 = 7*(csc(4*Pi/7))^2, x3 = 7*(csc(8*Pi/7))^2.
a(n) = 56*a(n-1) - 784*a(n-2) + 3136*a(n-3) for n>2, a(0) = 3, a(1) = 56, a(2) = 1568.
G.f.: (3 - 28*x)*(1 - 28*x) / (1 - 56*x + 784*x^2 - 3136*x^3). - Colin Barker, May 25 2017

A287405 a(n) = (7(cot(1Pi/7))^2)^n + (7(cot(2Pi/7))^2)^n + (7(cot(4Pi/7))^2)^n.

Original entry on oeis.org

3, 35, 931, 27587, 830403, 25054435, 756187747, 22824258947, 688917131651, 20793986742179, 627637106311971, 18944339609269571, 571808137046942019, 17259221092289630307, 520945214725090792931, 15723995613526902256387, 474606601742375424297731

Offset: 0

Kai Wang, May 24 2017

a(n) = x1^n + x2^n + x3^n, where x1, x2, x3 are the roots of x^3 - 35*x^2 + 147*x - 49, x1 = 7*(cot(1*Pi/7))^2, x2 = 7*(cot(2*Pi/7))^2, x3 = 7*(cot(4*Pi/7))^2.

Colin Barker, Table of n, a(n) for n = 0..650
Roman Witula, Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2Pi/7, J. Integer Seq., 12 (2009), Article 09.8.5.
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.
Index entries for linear recurrences with constant coefficients, signature (35,-147,49).

Cf. A108716, A033304, A094648, A274220, A215076, A274075 A274032, A275195, A215575, A274975.

LinearRecurrence[{35,-147,49},{3,35,931},30] (* Harvey P. Dale, Mar 15 2018 *)

Vec((3 - 7*x)*(1 - 21*x) / (1 - 35*x + 147*x^2 - 49*x^3) + O(x^30)) \\ Colin Barker, May 26 2017

polsym(x^3 - 35*x^2 + 147*x - 49, 20) \\ Joerg Arndt, May 26 2017

a(n) = 35*a(n-1) - 147*a(n-2) + 49*a(n-3), a(0) = 3, a(1) = 35, a(2) = 931.
Bisection of A215575: a(n) = A215575(2*n).
G.f.: (3 - 7*x)*(1 - 21*x) / (1 - 35*x + 147*x^2 - 49*x^3). - Colin Barker, May 26 2017

cp's OEIS Frontend

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

A275831 a(n) = (sqrt(7)*csc(Pi/7)/2)^n + (-sqrt(7)*csc(2*Pi/7)/2)^n + (-sqrt(7)*csc(4*Pi/7)/2)^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A293312 Rectangular array read by antidiagonals: A(n,k) = tr((M_n)^k), k >= 0, where M_n is the n X n matrix M_1 = {{1}}, M_n = {{0,...,0,1},{0,...,0,1,1},...,{0,1,...,1},{1,...,1}}, n > 1, and tr(.) is the trace.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A320918 Sum of n-th powers of the roots of x^3 + 9*x^2 + 20*x - 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A321173 a(n) = -2*a(n-1) + a(n-2) + a(n-3), a(0) = -1, a(1) = 3, a(2) = -9.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A321174 a(n) = -2*a(n-1) + a(n-2) + a(n-3), a(0) = -1, a(1) = -4, a(2) = 5.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A287396 a(n) = (7*(csc(2*Pi/7))^2)^n + (7*(csc(4*Pi/7))^2)^n + (7*(csc(8*Pi/7))^2)^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

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

A275831 a(n) = (sqrt(7)csc(Pi/7)/2)^n + (-sqrt(7)csc(2Pi/7)/2)^n + (-sqrt(7)csc(4*Pi/7)/2)^n.

A320918 Sum of n-th powers of the roots of x^3 + 9x^2 + 20x - 1.

A287396 a(n) = (7(csc(2Pi/7))^2)^n + (7(csc(4Pi/7))^2)^n + (7(csc(8Pi/7))^2)^n.

A287405 a(n) = (7(cot(1Pi/7))^2)^n + (7(cot(2Pi/7))^2)^n + (7(cot(4Pi/7))^2)^n.