A198636 -id:A198636 - cp's OEIS frontend

A185095 Rectangular array read by antidiagonals: row q has generating function F_q(x) = sum_{r=0,...,q-1} ((q-r)(-1)^rbinomial(2q-r,r)x^r) / sum_{s=0,...,q} ((-1)^sbinomial(2q-s,s)*x^s), where q=1,2,....

1, 2, 1, 3, 3, 1, 4, 5, 7, 1, 5, 7, 13, 18, 1, 6, 9, 19, 38, 47, 1, 7, 11, 25, 58, 117, 123, 1, 8, 13, 31, 78, 187, 370, 322, 1, 9, 15, 37, 98, 257, 622, 1186, 843, 1, 10, 17, 43, 118, 327, 874, 2110, 3827, 2207, 1, 11, 19, 49, 138, 397, 1126, 3034, 7252, 12389, 5778, 1

Offset: 0

L. Edson Jeffery, Jan 23 2012

Row indices q begin with 1, column indices n begin with 0.

			Array begins as
1,  1,  1,  1,   1,    1, ...
2,  3,  7, 18,  47,  123, ...
3,  5, 13, 38, 117,  370, ...
4,  7, 19, 58, 187,  622, ...
5,  9, 25, 78, 257,  874, ...
6, 11, 31, 98, 327, 1126, ...
...

S. Barbero, Dickson Polynomials, Chebyshev Polynomials, and Some Conjectures of Jeffery, Journal of Integer Sequences, 17 (2014), #14.3.8.

Conjecture. Transpose of array A186740.
Conjecture. Rows 0,1,2 (up to an offset) are A000012, A005248, A198636 (proved, see the Barbero, et al., reference there).
Conjecture. Columns 0,1,2,3,4 (up to an offset) are A000027, A005408, A016921, A114698, A114646.
Cf. A209235.

Conjecture. The n-th entry in row q is given by R_q(n) = 2^(2*n)*(sum_{j=1,...,n+1} (cos(j*Pi/(2*q+1)))^(2*n)), q >= 1, n >= 0.
Conjecture. G.f. for column n is of the form G_n(x) = H_n(x)/(1-x)^2, where H_n(x) is a polynomial in x, n >= 0.
Conjecture. 2*A185095(q,n) = A198632(2*q,n), q >= 1, n >= 0. - L. Edson Jeffery, Nov 23 2013

A186740 Sequence read from antidiagonals of rectangular array with entry in row n and column q given by T(n,q) = 2^(2n)(Sum_{j=1..n+1} (cos(jPi/(2q+1)))^(2*n)), n >= 0, q >= 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 7, 5, 4, 1, 18, 13, 7, 5, 1, 47, 38, 19, 9, 6, 1, 123, 117, 58, 25, 11, 7, 1, 322, 370, 187, 78, 31, 13, 8, 1, 843, 1186, 622, 257, 98, 37, 15, 9, 1, 2207, 3827, 2110, 874, 327, 118, 43, 17, 10, 1, 5778, 12389, 7252, 3034, 1126, 397, 138, 49, 19, 11

Offset: 0

L. Edson Jeffery, Jan 21 2012

Row indices n begin with 0, column indices q begin with 1.

			Array begins:
1    2     3     4     5     6     7     8     9 ...
1    3     5     7     9    11    13    15    17 ...
1    7    13    19    25    31    37    43    49 ...
1   18    38    58    78    98   118   138   158 ...
1   47   117   187   257   327   397   467   537 ...
1  123   370   622   874  1126  1378  1630  1882 ...
1  322  1186  2110  3034  3958  4882  5806  6730 ...
1  843  3827  7252 10684 14116 17548 20980 24412 ...
1 2207 12389 25147 38017 50887 63757 76627 89497 ...
...
As a triangle:
1,
1,  2,
1,  3,  3,
1,  7,  5,  4,
1, 18, 13,  7, 5,
1, 47, 38, 19, 9, 6,
...

Andrew Howroyd, Table of n, a(n) for n = 0..1275
S. Barbero, Dickson Polynomials, Chebyshev Polynomials, and Some Conjectures of Jeffery, Journal of Integer Sequences, 17 (2014), #14.3.8.

Conjecture: Transpose of array A185095.
Conjecture: Columns 0,1,2 (up to an offset) are A000012, A005248, A198636 (proved, see the Barbero, et al., reference there).
Conjecture: Rows 0,1,2,3,4 (up to an offset) are A000027, A005408, A016921, A114698, A114646.
Cf. A209235.

Conjecture: G.f. for column q is F_q(x) = (Sum_{r=0..q-1} ((q-r)*(-1)^r*binomial(2*q-r,r)*x^r)) / (Sum_{s=0..q} ((-1)^s*binomial(2*q-s,s)*x^s)), q >= 1.

Conjecture: G.f. for n-th row is of the form G_n(x) = H_n(x)/(1-x)^2, where H_n(x) is a polynomial in x.

A208725 Number of 2n-bead necklaces labeled with numbers 1..6 not allowing reversal, with neighbors differing by exactly 1.

Original entry on oeis.org

5, 9, 16, 35, 78, 210, 551, 1569, 4475, 13078, 38465, 114584, 343026, 1034471, 3134135, 9540969, 29154478, 89407073, 275016292, 848329872, 2623322133, 8130714643, 25252366057, 78577560856, 244933963301, 764707458720, 2391026407058, 7486342546939

Offset: 1

R. H. Hardin, Mar 01 2012

nonn

			All solutions for n=3:
..3....2....1....3....5....2....4....1....2....3....4....4....1....2....1....3
..4....3....2....4....6....3....5....2....3....4....5....5....2....3....2....4
..5....4....3....5....5....2....6....1....4....3....4....4....3....2....1....3
..4....3....4....6....6....3....5....2....5....4....5....5....2....3....2....4
..5....4....3....5....5....2....6....1....4....5....4....6....3....4....3....3
..4....3....2....4....6....3....5....2....3....4....5....5....2....3....2....4

Andrew Howroyd, Table of n, a(n) for n = 1..100

Column 6 of A208727.

a(n) = (1/n) * Sum_{d | n} totient(n/d) * A198636(d). - Andrew Howroyd, Mar 18 2017

a(17)-a(28) from Andrew Howroyd, Mar 18 2017

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

A232441 Sequence read from antidiagonals of rectangular array given by A(n,k) = 2^(2k)(Sum_{j=1..n-floor(n/2)-1} (cos(jPi/n))^(2k)), rows n >= 3, columns k >= 0.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 3, 4, 1, 3, 4, 7, 8, 1, 3, 5, 10, 18, 16, 1, 4, 6, 13, 28, 47, 32, 1, 4, 7, 16, 38, 82, 123, 64, 1, 5, 8, 19, 48, 117, 244, 322, 128, 1, 5, 9, 22, 58, 152, 370, 730, 843, 256, 1, 6, 10, 25, 68

Offset: 3

L. Edson Jeffery, Nov 23 2013

Row indices n begin with 3, column indices k begin with 0.

			1,    1,    1,    1,    1,    1,    1,    1,    1,    1,    1,...
1,    2,    4,    8,   16,   32,   64,  128,  256,  512, 1024,...
2,    3,    7,   18,   47,  123,  322,  843, 2207, 5778,15127,...
2,    4,   10,   28,   82,  244,  730, 2188, 6562,19684,59050,...

Cf. A186740, A185095, A198632, A198636.

Table[Function[m, FullSimplify[2^(2 k)*Sum[Cos[j*Pi/m]^(2 k), {j, m - Floor[m/2] - 1}]]][n - k + 1], {n, 3, 12}, {k, 0, n - 2}] // Flatten (* Michael De Vlieger, Mar 18 2017 *)

A(2*m+1,k) = A186740(m,k), m = 1,2,....

Conjecture: A(n,k) = floor(A198632(n-1,k)/2), n >= 3, k >= 0.

cp's OEIS Frontend

A185095 Rectangular array read by antidiagonals: row q has generating function F_q(x) = sum_{r=0,...,q-1} ((q-r)*(-1)^r*binomial(2*q-r,r)*x^r) / sum_{s=0,...,q} ((-1)^s*binomial(2*q-s,s)*x^s), where q=1,2,....

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A186740 Sequence read from antidiagonals of rectangular array with entry in row n and column q given by T(n,q) = 2^(2*n)*(Sum_{j=1..n+1} (cos(j*Pi/(2*q+1)))^(2*n)), n >= 0, q >= 1.

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A208725 Number of 2n-bead necklaces labeled with numbers 1..6 not allowing reversal, with neighbors differing by exactly 1.

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

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A232441 Sequence read from antidiagonals of rectangular array given by A(n,k) = 2^(2*k)*(Sum_{j=1..n-floor(n/2)-1} (cos(j*Pi/n))^(2*k)), rows n >= 3, columns k >= 0.

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Mathematica

Formula

A185095 Rectangular array read by antidiagonals: row q has generating function F_q(x) = sum_{r=0,...,q-1} ((q-r)(-1)^rbinomial(2q-r,r)x^r) / sum_{s=0,...,q} ((-1)^sbinomial(2q-s,s)*x^s), where q=1,2,....

A186740 Sequence read from antidiagonals of rectangular array with entry in row n and column q given by T(n,q) = 2^(2n)(Sum_{j=1..n+1} (cos(jPi/(2q+1)))^(2*n)), n >= 0, q >= 1.

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

A232441 Sequence read from antidiagonals of rectangular array given by A(n,k) = 2^(2k)(Sum_{j=1..n-floor(n/2)-1} (cos(jPi/n))^(2k)), rows n >= 3, columns k >= 0.