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.

User: Cheyne Homberger

Cheyne Homberger's wiki page.

Cheyne Homberger has authored 8 sequences.

A230551 Number of involutions avoiding the pattern 2431.

Original entry on oeis.org

1, 2, 4, 10, 24, 62, 154, 396, 992, 2536, 6376, 16238, 40914, 103954, 262298, 665478, 1680726, 4260262, 10766470, 27274444
Offset: 1

Views

Author

Cheyne Homberger, Oct 23 2013

Keywords

Examples

			Of the 26 involutions of length 26, only 35142 and 52431 contain the pattern 2431, so a(5) = 24.

Links

Crossrefs

Cf. A001006, A121704, A230552-A230556.

A230556 Number of involutions avoiding the pattern 4231.

Original entry on oeis.org

1, 1, 2, 4, 9, 21, 51, 128, 327, 858, 2272, 6146, 16716, 46246, 128414, 361493, 1020506, 2913060, 8335405, 24067930, 69646035
Offset: 0

Views

Author

Cheyne Homberger, Oct 23 2013

Keywords

Examples

			Of the 26 involutions of length 5, only 53241, 42315, 52431, 52341, and 15342 contain the pattern 4231, so a(5) = 21.

Links

Crossrefs

Cf. A000085, A001006, A121704, A230551-A230555.

A230555 Number of involutions avoiding 3421.

Original entry on oeis.org

1, 1, 2, 4, 10, 25, 66, 173, 460, 1218, 3240, 8602, 22878, 60794, 161668, 429752, 1142758, 3038173, 8078606, 21479469, 57113888
Offset: 0

Views

Author

Cheyne Homberger, Oct 23 2013

Keywords

Examples

			Of the 26 involutions of length 5, only 45312 contains the pattern 3421, so a(5) = 25.

Links

Crossrefs

Cf. A000085, A001006, A121704, A230551-A230556.

A230554 Number of involutions avoiding the pattern 1324.

Original entry on oeis.org

1, 1, 2, 4, 9, 21, 51, 126, 321, 820, 2160, 5654, 15272, 40758, 112280, 304471, 852164, 2341980, 6640755, 18460066, 52915999
Offset: 0

Views

Author

Cheyne Homberger, Oct 23 2013

Keywords

Examples

			Of the 26 involutions of length 5, only 21435, 13254, 13245, 14325, and 12435 contain the pattern 1324, so a(5) = 21.

Links

Crossrefs

Cf. A000085, A001006, A121704, A230551-A230556.

A230553 Number of involutions avoiding the pattern 1342.

Original entry on oeis.org

1, 1, 2, 4, 10, 24, 62, 156, 406, 1040, 2714, 7012, 18322, 47560, 124358, 323708, 846766, 2208032, 5777330, 15082372, 39469786
Offset: 0

Views

Author

Cheyne Homberger, Oct 23 2013

Keywords

Examples

			Of the 26 involutions of length 5, only 14523 and 15342 contain the pattern 1342, so a(5) = 24.

Links

Crossrefs

Cf. A000085, A001006, A121704, A230551-A230556.

A230552 Number of involutions avoiding the pattern 2341.

Original entry on oeis.org

1, 2, 4, 10, 25, 66, 170, 441, 1124, 2870, 7273, 18477, 46825, 118917, 301734, 766525, 1946293, 4944614, 12557685
Offset: 1

Views

Author

Cheyne Homberger, Oct 23 2013

Keywords

Examples

			Of the 26 involutions of length 5, only 52341 contains the pattern 2341, so a(5) = 25.

Links

Crossrefs

Cf. A001006, A121704, A230551-A230556.

A217711 Total number of 321 patterns in the set of permutations avoiding 123.

Original entry on oeis.org

1, 16, 144, 1016, 6271, 35584, 190628, 979496, 4876530, 23686560, 112796176, 528495600, 2442949979, 11163970432, 50520351612, 226688100104, 1009648508590, 4467591809376, 19654294688768, 86018255452048, 374715017442966, 1625489878136576, 7024392489806344
Offset: 3

Views

Author

Cheyne Homberger, Mar 20 2013

Keywords

Comments

a(n) is the total number of occurrences of 321 patterns in the set of all 123-avoiding n-permutations.

Examples

			a(3) = 1 since there is only one 321 pattern in the set {132, 213, 231, 312, 321}.

Links

Formula

G.f.: 1/2*(32*x^4 - 88*x^3 + 52*x^2 + sqrt(-4*x + 1)*(36*x^3 - 34*x^2 + 10*x - 1) - 12*x + 1)/(64*x^4 - 48*x^3 + 12*x^2 - x).
Conjecture: -(n+1)*(25*n-3314)*a(n) -5*n*(5*n+9446)*a(n-1) +2*(594*n^2 +128863*n -142613)*a(n-2) +16*(-119*n^2-39230*n+87888)*a(n-3) -32*(2*n-7)*(53*n-8687)*a(n-4)=0. - R. J. Mathar, Oct 08 2016

A210064 Total number of 231 patterns in the set of permutations avoiding 123.

Original entry on oeis.org

0, 0, 1, 11, 81, 500, 2794, 14649, 73489, 356960, 1691790, 7864950, 36000186, 162697176, 727505972, 3223913365, 14176874193, 61926666824, 268931341414, 1161913686618, 4997204887550, 21404922261112, 91351116184716, 388581750349946, 1647982988377786
Offset: 1

Views

Author

Cheyne Homberger, Mar 16 2012

Keywords

Comments

a(n) is the total number of 231 (and also 312) patterns in the set of all 123 avoiding n-permutations. Also the number of 231 (or 213, or 312) patterns in the set of all 132 avoiding n-permutations.

Examples

			a(3) = 1 since there is only one 231 pattern in the set {132,213,231,312,321}.

Links

Crossrefs

Cf. A045720.

Programs

  • Mathematica
    Rest[CoefficientList[Series[x/(2*(1-4*x)^2) + (x-1)/(2*(1-4*x)^(3/2)) + 1/(2 - 8*x), {x, 0, 20}], x]] (* Vaclav Kotesovec, Mar 15 2014 *)
  • PARI
    x='x+O('x^50); concat([0,0], Vec(x/(2*(1-4*x)^2) + (x-1)/(2*(1-4*x)^(3/2)) + 1/(2 - 8*x))) \\ G. C. Greubel, May 31 2017

Formula

G.f.: x/(2*(1-4*x)^2) + (x-1)/(2*(1-4*x)^(3/2)) + 1/(2 - 8*x).
a(n) ~ n * 2^(2*n-3) * (1 - 6/sqrt(Pi*n)). - Vaclav Kotesovec, Mar 15 2014
Conjecture: n*(n-3)*a(n) +2*(-4*n^2+11*n-2)*a(n-1) +8*(n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Oct 08 2016

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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