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: Vincent Delecroix

Vincent Delecroix's wiki page.

Vincent Delecroix has authored 3 sequences.

A300901 Number of closed meanders with 2n crossings and 5 digons.

Original entry on oeis.org

16, 40, 168, 280, 544, 1152, 1560, 2640, 3504, 5824, 6552, 12000, 11456, 19176, 18648, 31312, 30640, 50064, 43736, 71392, 62304, 104800, 87672, 141048, 121968, 191632, 154200, 255192, 209536, 327360, 265880, 435960, 328176, 533688, 419064, 649272, 525280
Offset: 4

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Author

Vincent Delecroix, Mar 14 2018

Keywords

Comments

A meander together with the horizontal line separates the plane into several connected components. Each component has a given number of edges which is always an even number. The digons (or bigons) are the faces with least number of edges, that is 2. Equivalently, the number of digons is the number of arches between adjacent sites ("minimal arches") where the two extremal ones are considered adjacent.

Links

Crossrefs

A002618 is the number of closed meanders with 4 digons. A301940 is the number of meanders with 6 digons. A005315 is the total number of closed meanders.

Formula

Known asymptotics: Sum_{n <= N} a(n) ~ 16 N^5/(3 Pi^4).

A301940 Number of closed meanders with 2n crossings and 6 digons.

Original entry on oeis.org

2, 16, 110, 416, 1470, 4128, 9102, 20240, 40106, 71312, 127426, 203056, 336070, 491392, 790126, 1067160, 1650530, 2086720, 3180030, 3878952, 5768170, 6771680, 9871350, 11231064, 16241094, 17936352, 25665290, 27729640, 39210350, 41583104, 58341778, 60751880, 84510650
Offset: 3

Views

Author

Vincent Delecroix, Mar 29 2018

Keywords

Comments

A meander together with the horizontal line separates the plane into several connected components. Each component has a given number of edges which is always an even number. The digons (or bigons) are the faces with least number of edges, that is 2. Equivalently, the number of digons is the number of arches between adjacent sites ("minimal arches") where the two extremal ones are considered adjacent.

Links

Crossrefs

A002618 is the number of closed meanders with 4 digons. A300901 is the number of closed meanders with 5 digons. A005315 is the total number of closed meanders.

Formula

Known asymptotics: Sum_{n <= N} a(n) ~ 70 N^7/(9 Pi^6).

A162698 Numbers n such that the incidence matrix of the grid n X n has -1 as eigenvalue.

Original entry on oeis.org

4, 5, 9, 11, 14, 17, 19, 23, 24, 29, 34, 35, 39, 41, 44, 47, 49, 53, 54, 59, 64, 65, 69, 71, 74, 77, 79, 83, 84, 89, 94, 95, 99, 101, 104, 107, 109, 113, 114, 119, 124, 125, 129, 131, 134, 137, 139, 143, 144, 149, 154, 155, 159, 161, 164, 167, 169, 173, 174, 179, 184, 185, 189, 191, 194, 197, 199
Offset: 1

Views

Author

Vincent Delecroix, Jul 11 2009

Keywords

Comments

Numbers n such that n+1 is a multiple of 5 or 6. - Tom Edgar, Dec 15 2017

Links

Crossrefs

Cf. A093509, A117870.

Programs

  • Mathematica
    With[{nn=40},Select[Union[Join[5*Range[nn],6*Range[nn]]]-1,#<=5nn&]] (* Harvey P. Dale, Sep 04 2023 *)
  • PARI
    for(n=1,100, if( matdet(matrix(n^2,n^2,i,j, (abs((i-1)\n - (j-1)\n) + abs((i-1)%n - (j-1)%n)==1) + (i==j) ))==0, print1(n,", ") ) ) \\ Max Alekseyev, Apr 23 2010
  • PARI
    Vec(x*(x^9+4*x^8-3*x^7+7*x^6-5*x^5+8*x^4-5*x^3+7*x^2-3*x+4) / ((x-1)^2*(x^4-x^3+x^2-x+1)*(x^4+x^3+x^2+x+1)) + O(x^100)) \\ Colin Barker, Dec 15 2017
  • Sage
    [n for n in [1..200] if (n+1)%5==0 or (n+1)%6==0] # Tom Edgar, Dec 15 2017

Formula

G.f.: x*(x^9+4*x^8-3*x^7+7*x^6-5*x^5+8*x^4-5*x^3+7*x^2-3*x+4) / ((x-1)^2*(x^4-x^3+x^2-x+1)*(x^4+x^3+x^2+x+1)). - Colin Barker, Dec 03 2012 ["Empirical" removed after Tom Edgar's comment by Andrey Zabolotskiy, Dec 15 2017]
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - 2*a(n-4) + 2*a(n-5) - 2*a(n-6) + 2*a(n-7) - 2*a(n-8) + 2*a(n-9) - a(n-10) for n>10.

Extensions

Twelve more terms from Max Alekseyev, Apr 23 2010
a(33)-a(40) from Max Alekseyev, Feb 15 2013
More terms from Tom Edgar, Dec 15 2017

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