cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A208727 T(n,k)=Number of 2n-bead necklaces labeled with numbers 1..k not allowing reversal, with neighbors differing by exactly 1.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 5, 4, 1, 0, 5, 7, 8, 6, 1, 0, 6, 9, 12, 15, 8, 1, 0, 7, 11, 16, 25, 27, 14, 1, 0, 8, 13, 20, 35, 52, 60, 20, 1, 0, 9, 15, 24, 45, 78, 131, 123, 36, 1, 0, 10, 17, 28, 55, 104, 210, 316, 285, 60, 1, 0, 11, 19, 32, 65, 130, 290, 551, 835, 648, 108, 1, 0, 12, 21, 36
Offset: 1

Views

Author

R. H. Hardin, Mar 01 2012

Keywords

Comments

Table starts
.0.1..2...3...4....5....6....7....8....9...10...11...12..13..14..15.16.17.18
.0.1..3...5...7....9...11...13...15...17...19...21...23..25..27..29.31.33
.0.1..4...8..12...16...20...24...28...32...36...40...44..48..52..56.60
.0.1..6..15..25...35...45...55...65...75...85...95..105.115.125.135
.0.1..8..27..52...78..104..130..156..182..208..234..260.286.312
.0.1.14..60.131..210..290..370..450..530..610..690..770.850
.0.1.20.123.316..551..796.1042.1288.1534.1780.2026.2272
.0.1.36.285.835.1569.2366.3175.3985.4795.5605.6415

Examples

			All solutions for n=5, k=3:
..2....1....1....1....1....1....1....1
..3....2....2....2....2....2....2....2
..2....3....1....1....1....3....1....1
..3....2....2....2....2....2....2....2
..2....1....1....3....1....3....3....1
..3....2....2....2....2....2....2....2
..2....3....1....3....3....3....1....1
..3....2....2....2....2....2....2....2
..2....3....3....3....3....3....3....1
..3....2....2....2....2....2....2....2

Links

Crossrefs

Column 3 is A000031.
Cf. A198632, A232441, A186740.

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,....

Original entry on oeis.org

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

Views

Author

L. Edson Jeffery, Jan 23 2012

Keywords

Comments

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

Examples

			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, ...
...

Links

Crossrefs

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.

Formula

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

A209235 Rectangular array read by antidiagonals, with entry k in row n given by T(n,k) = 2^{k-1}*Sum_{j=1..n} (cos((2*j-1)*Pi/(2*n+1)))^{k-1}.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 4, 1, 3, 1, 5, 1, 5, 4, 1, 6, 1, 7, 4, 7, 1, 8, 1, 11, 4, 19, 16, 18, 1, 9, 1, 13, 4, 25, 16, 38, 29, 1, 10, 1, 15, 4, 31, 16, 58, 57, 47, 1, 11, 1, 17, 4, 37, 16, 78, 64, 117, 76, 1, 12, 1, 19, 4, 43, 16, 98, 64, 187, 193, 123, 1
Offset: 1

Views

Author

L. Edson Jeffery, Jan 12 2013

Keywords

Comments

Antidiagonal sums: {1,3,5,9,16,26,46,78,136,...}.

Examples

			Array begins as
.1..1...1..1...1...1
.2..1...3..4...7..11
.3..1...5..4..13..16
.4..1...7..4..19..16
.5..1...9..4..25..16
.6..1..11..4..31..16

Crossrefs

Rows: cf. A000012, A000032, A094649, A189234, A216605, etc.
Cf. A185095, A186740.

Formula

T(n,k) = 2^{k-1}*Sum_{j=1..n} (cos((2*j-1)*Pi/(2*n+1)))^{k-1}.
Empirical g.f. for row n: F(x) = (Sum_{u=0..n-1} A122765(n,n-1-u)*x^u)/(Sum_{v=0..n} A108299(n,v)*x^v).
Empirical: odd column first differences tend to A000984 = {1, 2, 6, 20, 70, 252, ...} (central binomial coefficients).

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.

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

Views

Author

L. Edson Jeffery, Nov 23 2013

Keywords

Comments

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

Examples

			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,...

Crossrefs

Cf. A186740, A185095, A198632, A198636.

Programs

  • Mathematica
    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 *)

Formula

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.
Showing 1-4 of 4 results.

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