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 1-8 of 8 results.

A092490 a(n) = A058094(n) - 3*A058094(n-1) + A058094(n-2) for n >=4.

Original entry on oeis.org

0, 0, 0, 1, 5, 20, 75, 271, 957, 3337, 11559, 39896, 137423, 472808, 1625632, 5587228, 19198971, 65963978, 226623902, 778551761, 2674604282, 9188106871, 31563807424, 108430368827, 372487292867, 1279591674070, 4395730089428
Offset: 1

Views

Author

N. J. A. Sloane, Apr 04 2004

Keywords

Comments

A recurrence relation follows in a straightforward manner from the above formula and the recurrence relation for A058094.

Links

Crossrefs

Cf. A058094, A092489, A092491, A092492.
Cf. A058094.

Programs

  • Maple
    b[1]:=1:b[2]:=2:b[3]:=5:b[4]:=14:b[5]:=42:b[6]:=132: for n from 6 to 32 do b[n+1]:=6*b[n]-11*b[n-1]+9*b[n-2]-4*b[n-3]-4*b[n-4]+b[n-5] od:a[1]:=0:a[2]:=0:a[3]:=0:for n from 4 to 32 do a[n]:=b[n]-3*b[n-1]+b[n-2] od: seq(a[n],n=1..32); # Emeric Deutsch, Apr 12 2005
  • PARI
    concat([0,0,0], Vec(x^4*(1 - x + x^2 + x^3) / (1 - 6*x + 11*x^2 - 9*x^3 + 4*x^4 + 4*x^5 - x^6) + O(x^30))) \\ Colin Barker, Aug 21 2019

Formula

G.f.: x^4*(1 - x + x^2 + x^3) / (1 - 6*x + 11*x^2 - 9*x^3 + 4*x^4 + 4*x^5 - x^6). - R. J. Mathar, Dec 02 2007
a(n) = 6*a(n-1) - 11*a(n-2) + 9*a(n-3) - 4*a(n-4) - 4*a(n-5) + a(n-6) for n>7. - Colin Barker, Aug 21 2019

Extensions

Edited by Emeric Deutsch, Apr 12 2005

A092491 a(n) = 2*A058094(n-2) - 5*A058094(n-3) + A058094(n-4) + a(n-1) for n >=4.

Original entry on oeis.org

0, 0, 0, 0, 1, 6, 25, 93, 333, 1172, 4083, 14137, 48778, 167981, 577874, 1986747, 6828120, 23462470, 80611581, 276944893, 951422603, 3268470411, 11228209786, 38572124196, 132505812826, 455192771711, 1563706508759, 5371738013650
Offset: 1

Views

Author

N. J. A. Sloane, Apr 04 2004

Keywords

Comments

A recurrence relation follows in a straightforward manner from the above formula and the recurrence relation for A058094.

Links

Crossrefs

Cf. A058094, A092489, A092490, A092492.

Programs

  • Maple
    b[1]:=1:b[2]:=2:b[3]:=5:b[4]:=14:b[5]:=42:b[6]:=132: for n from 6 to 34 do b[n+1]:=6*b[n]-11*b[n-1]+9*b[n-2]-4*b[n-3]-4*b[n-4]+b[n-5] od: a[1]:=0:a[2]:=0:a[3]:=0:a[4]:=0: for n from 5 to 34 do a[n]:=2*b[n-2]-5*b[n-3]+b[n-4]+a[n-1] od: seq(a[n],n=1..34); # Emeric Deutsch, Apr 12 2005
  • Mathematica
    LinearRecurrence[{6, -11, 9, -4, -4, 1}, {0, 0, 0, 0, 1, 6}, 40] (* Vincenzo Librandi, Aug 15 2017 *)
  • PARI
    concat([0,0,0,0], Vec(x^5 / (1 - 6*x + 11*x^2 - 9*x^3 + 4*x^4 + 4*x^5 - x^6) + O(x^30))) \\ Colin Barker, Aug 21 2019

Formula

From Colin Barker, Aug 21 2019: (Start)
G.f.: x^5 / (1 - 6*x + 11*x^2 - 9*x^3 + 4*x^4 + 4*x^5 - x^6).
a(n) = 6*a(n-1) - 11*a(n-2) + 9*a(n-3) - 4*a(n-4) - 4*a(n-5) + a(n-6) for n>6.
(End)

Extensions

Edited by Emeric Deutsch, Apr 12 2005

A092489 Arises in enumeration of 321-hexagon-avoiding permutations.

Original entry on oeis.org

0, 0, 1, 4, 14, 48, 165, 568, 1954, 6717, 23082, 79307, 272470, 936065, 3215741, 11047122, 37950140, 130369334, 447853808, 1538496047, 5285135093, 18155807539, 62369881206, 214256590058, 736026444181, 2528439830821
Offset: 1

Views

Author

N. J. A. Sloane, Apr 04 2004

Keywords

Links

Crossrefs

Cf. A058094, A092490, A092491, A092492.

Programs

  • Maple
    b[1]:=1: b[2]:=2: b[3]:=5: b[4]:=14: b[5]:=42: b[6]:=132: for n from 6 to 35 do b[n+1]:=6*b[n]-11*b[n-1]+9*b[n-2]-4*b[n-3]-4*b[n-4]+b[n-5] od: seq(b[n],n=1..35): a[1]:=0: a[2]:=0: for n from 3 to 35 do a[n]:=b[n]-2*b[n-1] od: seq(a[n],n=1..35); # here b[n]=A058094(n).
  • PARI
    concat([0,0], Vec(x^3*(1 - 2*x + x^2 - x^3 - x^4) / (1 - 6*x + 11*x^2 - 9*x^3 + 4*x^4 + 4*x^5 - x^6) + O(x^30))) \\ Colin Barker, Aug 20 2019

Formula

Stankova and West give an explicit recurrence.
a(n) = A058094(n) - A058094(n-1) for n >= 3. - Emeric Deutsch, May 04 2004
From Colin Barker, Aug 20 2019: (Start)
G.f.: x^3*(1 - 2*x + x^2 - x^3 - x^4) / (1 - 6*x + 11*x^2 - 9*x^3 + 4*x^4 + 4*x^5 - x^6).
a(n) = 6*a(n-1) - 11*a(n-2) + 9*a(n-3) - 4*a(n-4) - 4*a(n-5) + a(n-6) for n>7.
(End)

Extensions

More terms from Emeric Deutsch, May 04 2004

A092492 Arises in enumeration of 321-hexagon-avoiding permutations.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 5, 19, 68, 240, 839, 2911, 10054, 34641, 119203, 409893, 1408873, 4841373, 16634350, 57149111, 196333312, 674477710, 2317047808, 7959739375, 27343914410, 93933688630, 322686958885, 1108513737048, 3808031504891
Offset: 1

Views

Author

N. J. A. Sloane, Apr 04 2004

Keywords

Links

Crossrefs

Cf. A058094, A092489, A092490, A092491.

Programs

  • Maple
    b[1]:=1: b[2]:=2: b[3]:=5: b[4]:=14: b[5]:=42: b[6]:=132: for n from 6 to 45 do b[n+1]:=6*b[n]-11*b[n-1]+9*b[n-2]-4*b[n-3]-4*b[n-4]+b[n-5] od: a[1]:=0: a[2]:=0: a[3]:=0: a[4]:=0: a[5]:=0: for n from 6 to 40 do a[n]:=2*b[n-3]-5*b[n-4]+b[n-5] od: seq(a[n],n=1..40); # Emeric Deutsch, Jun 08 2004
  • Mathematica
    LinearRecurrence[{6,-11,9,-4,-4,1},{0,0,0,0,0,1,5},30] (* Harvey P. Dale, Jan 31 2025 *)
  • PARI
    concat([0,0,0,0,0], Vec(x^6*(1 - x) / (1 - 6*x + 11*x^2 - 9*x^3 + 4*x^4 + 4*x^5 - x^6) + O(x^30))) \\ Colin Barker, Aug 21 2019

Formula

a(n) = 2*A058094(n-3) - 5*A058094(n-4) + A058094(n-5) for n >= 6. - Emeric Deutsch, Jun 08 2004
From Colin Barker, Aug 21 2019: (Start)
G.f.: x^6*(1 - x) / (1 - 6*x + 11*x^2 - 9*x^3 + 4*x^4 + 4*x^5 - x^6).
a(n) = 6*a(n-1) - 11*a(n-2) + 9*a(n-3) - 4*a(n-4) - 4*a(n-5) + a(n-6) for n>7.
(End)

Extensions

More terms from Emeric Deutsch, Jun 08 2004

A092493 a(n) = 4a(n-1) - 4a(n-2) + 3a(n-3) + a(n-4) - a(n-5).

Original entry on oeis.org

1, 2, 5, 14, 42, 128, 389, 1179, 3572, 10825, 32810, 99446, 301412, 913547, 2768863, 8392136, 25435699, 77092976, 233660832, 708201794, 2146486339, 6505777953, 19718339694, 59764246943, 181139247400, 549014312524, 1664005563066
Offset: 1

Views

Author

N. J. A. Sloane, Apr 04 2004

Keywords

Comments

Arises in enumeration of certain pattern-avoiding permutations.

Links

Crossrefs

Cf. A058094, A092489-A092492.

Programs

  • Maple
    a[1]:=1: a[2]:=2: a[3]:=5: a[4]:=14: a[5]:=42: for n from 6 to 32 do a[n]:=4*a[n-1]-4*a[n-2]+3*a[n-3]+a[n-4]-a[n-5] od: seq(a[j],j=1..32); # Emeric Deutsch, Apr 12 2005
  • Mathematica
    LinearRecurrence[{4,-4,3,1,-1},{1,2,5,14,42},40] (* Harvey P. Dale, Jul 14 2024 *)

Formula

G.f.: x*(1 - 2*x + x^2 - x^3 - x^4)/(1 - 4*x + 4*x^2 - 3*x^3 - x^4 + x^5). [Maksym Voznyy (voznyy(AT)mail.ru), Aug 12 2009; corrected by R. J. Mathar, Sep 16 2009]

Extensions

Edited by Emeric Deutsch, Apr 12 2005

A129776 Number of maximally-clustered hexagon-avoiding permutations in S_n; the maximally-clustered hexagon-avoiding permutations are those that avoid 3421, 4312, 4321, 46718235, 46781235, 56718234, 56781234.

Original entry on oeis.org

1, 1, 2, 6, 21, 78, 298, 1157, 4535, 17872, 70644, 279704, 1108462, 4395045, 17431206, 69144643, 274300461, 1088215370, 4317321235, 17128527716, 67956202025, 269612504850, 1069675361622, 4243893926396, 16837490364983, 66802139457897, 265035151393777
Offset: 0

Views

Author

Brant Jones (brant(AT)math.washington.edu), May 17 2007

Keywords

Comments

If w is maximally-clustered and hexagon-avoiding, there are simple explicit formulas for all the Kazhdan-Lusztig polynomials P_{x,w}.

Examples

			a(8)=4535 because there are 4535 permutations of size 8 that avoid 3421, 4312, 4321, 46718235, 46781235, 56718234 and 56781234.

References

  • Jozsef Losonczy, Maximally clustered elements and Schubert varieties, Ann. Comb. 11 (2007), no. 2, 195-212.

Links

Crossrefs

Cf. A058094, A108600.

Programs

  • PARI
    lista(nt) = { my(x = 'x + 'x*O('x^nt) ); P = (3*x^6+x^5-5*x^4+7*x^3-5*x^2+x) / (-3*x^6+4*x^5+8*x^4-14*x^3+15*x^2-7*x+1); print(Vec(P));} \\ Michel Marcus, Mar 17 2013

Formula

G.f.: 1+(3x^6+x^5-5x^4+7x^3-5x^2+x) / (-3x^6+4x^5+8x^4-14x^3+15x^2-7x+1).

Extensions

More terms from Michel Marcus, Mar 17 2013
a(0)=1 prepended by Alois P. Heinz, Jan 12 2025

A129777 Number of freely-braided hexagon-avoiding permutations in S_n; the freely-braided hexagon-avoiding permutations are those that avoid 3421, 4231, 4312, 4321, 46718235, 46781235, 56718234 and 56781234.

Original entry on oeis.org

1, 1, 2, 6, 20, 71, 260, 971, 3670, 13968, 53369, 204352, 783408, 3005284, 11533014, 44267854, 169935041, 652385639, 2504613713, 9615798516, 36917689075, 141737959416, 544175811783, 2089262741393, 8021347093432, 30796530585417, 118237818141689, 453953210838465
Offset: 0

Views

Author

Brant Jones (brant(AT)math.washington.edu), May 17 2007

Keywords

Comments

If w is freely-braided and hexagon-avoiding, there are simple explicit formulas for all the Kazhdan-Lusztig polynomials P_{x,w}.

Examples

			a(8)=3670 because there are 3670 permutations of size 8 that avoid 3421, 4231, 4312, 4321, 46718235, 46781235, 56718234 and 56781234.

References

  • Jozsef Losonczy, Maximally clustered elements and Schubert varieties, Ann. Comb. 11 (2007), no. 2, 195-212.

Links

Crossrefs

Cf. A058094, A108600.

Programs

  • Mathematica
    LinearRecurrence[{6, -9, 3, 1, -8, -1, 1}, {1, 2, 6, 20, 71, 260, 971}, 27] (* Jean-François Alcover, Feb 02 2019 *)
  • PARI
    lista(nt) = { my(x = 'x + 'x*O('x^nt) ); P = (-x^7-2*x^6+2*x^5+x^4-3*x^3+4*x^2-x) / (x^7-x^6-8*x^5+x^4+3*x^3-9*x^2+6*x-1); print(Vec(P));}  \\ Michel Marcus, Mar 17 2013

Formula

G.f.: 1+(-x^7-2x^6+2x^5+x^4-3x^3+4x^2-x) / (x^7-x^6-8x^5+x^4+3x^3-9x^2+6x-1).

Extensions

More terms from Michel Marcus, Mar 17 2013
a(0)=1 prepended by Alois P. Heinz, Jan 12 2025

A129778 Number of Deodhar elements in the finite Weyl group D_n.

Original entry on oeis.org

2, 5, 14, 48, 167, 575, 1976, 6791
Offset: 1

Views

Author

Brant Jones (brant(AT)math.washington.edu), May 17 2007

Keywords

Comments

The Deodhar elements are a subset of the fully commutative elements. If w is Deodhar, there are simple explicit formulas for all the Kazhdan-Lusztig polynomials P_{x,w} and the Kazhdan-Lusztig basis element C'w is the product of C'{s_i}'s corresponding to any reduced expression for w.

Examples

			a(4)=48 because there are 48 fully commutative elements in D_4 and since the first non-Deodhar fully-commutative element does not appear until D_6, these are all of the Deodhar elements in D_4.

References

  • S. Billey and G. S. Warrington, Kazhdan-Lusztig polynomials for 321-hexagon-avoiding permutations, J. Algebraic Combin., 13(2):111-136, 2001.
  • V. Deodhar, A combinatorial setting for questions in Kazhdan-Lusztig theory, Geom. Dedicata, 36(1): 95-119, 1990.

Links

Crossrefs

Cf. A058094.
Showing 1-8 of 8 results.

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

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