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 1-6 of 6 results.

A251268 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with no 2X2 subblock having x11-x00 less than x10-x01.

Original entry on oeis.org

11, 26, 35, 57, 114, 108, 120, 313, 480, 337, 247, 772, 1667, 2058, 1049, 502, 1775, 4930, 9109, 8812, 3268, 1013, 3894, 13052, 32636, 49872, 37772, 10179, 2036, 8277, 31936, 100843, 217634, 273607, 161906, 31707, 4083, 17224, 73805, 279718, 790734
Offset: 1

Views

Author

R. H. Hardin, Dec 01 2014

Keywords

Comments

Table starts
.....11.......26........57........120.........247..........502.........1013
.....35......114.......313........772........1775.........3894.........8277
....108......480......1667.......4930.......13052........31936........73805
....337.....2058......9109......32636......100843.......279718.......715685
...1049.....8812.....49872.....217634......790734......2510004......7189937
...3268....37772....273607....1457326.....6247708.....22806904.....73607411
..10179...161906...1501739....9772880....49523566....208452452....760734085
..31707...694042...8244503...65582500...393172015...1910905110...7901650053
..98764..2975162..45265163..440223510..3123669457..17543333688..82288916360
.307641.12753740.248529844.2955392154.24825649060.161181383956.858174176431

Examples

			Some solutions for n=4 k=4
..0..0..0..0..0....0..1..1..1..1....0..0..0..0..1....0..0..1..0..1
..1..1..1..1..1....0..0..0..1..1....0..1..1..1..1....0..0..0..1..1
..1..1..1..1..1....1..1..1..1..1....0..0..0..0..0....0..1..1..1..1
..1..1..1..1..1....0..0..0..0..0....0..0..1..1..1....0..0..0..0..0
..0..0..0..0..1....0..0..1..1..1....0..1..0..1..1....0..0..1..1..1

Links

Crossrefs

Column 1 is A052550(n+2)
Row 1 is A000295(n+3)

Formula

Empirical for column k:
k=1: a(n) = 3*a(n-1) +a(n-2) -2*a(n-3)
k=2: a(n) = 5*a(n-1) -2*a(n-2) -5*a(n-3) +2*a(n-4)
k=3: [order 10]
k=4: [order 16]
k=5: [order 36]
k=6: [order 62]
Empirical for row n:
n=1: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3)
n=2: a(n) = 6*a(n-1) -14*a(n-2) +16*a(n-3) -9*a(n-4) +2*a(n-5)
n=3: [order 8]
n=4: [order 10]
n=5: [order 12]
n=6: [order 14]
n=7: [order 16]

A184207 T(n,k)=Number of (n+1)X(k+1) binary arrays with no 2X2 subblock containing fewer than two 1s.

Original entry on oeis.org

11, 35, 35, 108, 195, 108, 337, 1028, 1028, 337, 1049, 5529, 8969, 5529, 1049, 3268, 29593, 80812, 80812, 29593, 3268, 10179, 158648, 721939, 1232795, 721939, 158648, 10179, 31707, 850155, 6468501, 18588589, 18588589, 6468501, 850155, 31707, 98764
Offset: 1

Views

Author

R. H. Hardin Jan 10 2011

Keywords

Comments

Table starts
.....11........35.........108............337.............1049
.....35.......195........1028...........5529............29593
....108......1028........8969..........80812...........721939
....337......5529.......80812........1232795.........18588589
...1049.....29593......721939.......18588589........471487297
...3268....158648.....6468501......281420718......12021784217
..10179....850155....57909336.....4255603649.....306055720601
..31707...4556371...518574936....64377477751....7795714368919
..98764..24418812..4643443850...973770904216..198537902186962
.307641.130868377.41579555979.14729764992131.5056537169718019

Examples

			Some solutions for 4X3
..1..1..1....0..1..0....1..1..1....0..1..1....1..0..1....0..1..0....0..1..0
..0..1..0....1..1..1....0..1..1....0..1..1....1..1..0....0..1..0....1..1..0
..0..1..0....0..1..1....0..1..0....1..1..0....0..1..1....0..1..1....0..1..0
..1..1..1....0..1..0....0..1..1....1..1..1....1..1..0....0..1..0....0..1..1

Links

Crossrefs

Diagonal is A140314(n+1)
Column 1 is A052550(n+2)

A218823 T(n,k)=Unmatched value maps: number of nXk binary arrays indicating the locations of corresponding elements not equal to any king-move neighbor in a random 0..2 nXk array.

Original entry on oeis.org

1, 2, 2, 4, 11, 4, 7, 35, 35, 7, 12, 108, 195, 108, 12, 21, 337, 1028, 1028, 337, 21, 37, 1049, 5529, 8961, 5529, 1049, 37, 65, 3268, 29593, 80692, 80692, 29593, 3268, 65, 114, 10179, 158648, 720305, 1229783, 720305, 158648, 10179, 114, 200, 31707, 850155
Offset: 1

Views

Author

R. H. Hardin Nov 06 2012

Keywords

Comments

Table starts
...1......2........4.........7.........12..........21..........37...........65
...2.....11.......35.......108........337........1049........3268........10179
...4.....35......195......1028.......5529.......29593......158648.......850155
...7....108.....1028......8961......80692......720305.....6449353.....57696032
..12....337.....5529.....80692....1229783....18519049...280040904...4229571209
..21...1049....29593....720305...18519049...468783431.11931365309.303182474797
..37...3268...158648...6449353..280040904.11931365309
..65..10179...850155..57696032.4229571209
.114..31707..4556371.516297052
.200..98764.24418812
.351.307641
.616

Examples

			Some solutions for n=3 k=4
..0..0..0..0....0..0..0..1....0..0..1..0....0..0..1..0....1..0..0..0
..0..0..0..0....1..0..0..0....0..0..1..0....1..0..0..1....1..0..0..0
..0..1..1..0....1..0..0..0....1..0..1..0....1..0..1..0....1..0..0..0

Links

Crossrefs

Column 1 is A005251(n+2)
Column 2 is A052550(n+1)
Column 3 is A184200(n-1)

A100058 Expansion of 1 / (1 - 3x - x^2 + 2x^3).

Original entry on oeis.org

1, 3, 10, 31, 97, 302, 941, 2931, 9130, 28439, 88585, 275934, 859509, 2677291, 8339514, 25976815, 80915377, 252043918, 785093501, 2445493667, 7617486666, 23727766663, 73909799321, 230222191294, 717120839877, 2233765112283
Offset: 0

Views

Author

Gary W. Adamson, Oct 31 2004

Keywords

Comments

a(n)/a(n-1) tends to 3.1149075414..., which is an eigenvalue of the matrix M and a root of the characteristic polynomial x^3 - 3x^2 - x + 2.

Examples

			a(5) = 97, center term in M^5 * [1 0 0]: [205 97 66].

References

  • Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.

Links

Crossrefs

Partial sums of A052911. Cf. A019481, A052550, A052939, A100059, A058071.

Programs

  • Mathematica
    CoefficientList[Series[1/(1 - 3x - x^2 + 2x^3), {x, 0, 25}], x] (* Or *)
Table[(MatrixPower[{{2, 1, 2}, {1, 1, 0}, {1, 0, 0}}, n].{1, 0, 0})[[2]], {n, 26}] (* Robert G. Wilson v, Nov 04 2004 *)
LinearRecurrence[{3,1,-2},{1,3,10},30] (* Harvey P. Dale, Mar 28 2012 *)
  • PARI
    Vec(1/(1-3*x-x^2+2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Formula

Recurrence: a(0) = 1, a(1) = 3, a(2) = 10; a(n) = 3*a(n-1) + a(n-2) - 2*a(n-3).
Given Hosoya's triangle: 1; 1, 1; 2, 1, 2; considered as an upper triangular 3 X 3 matrix M: [2 1 2 / 1 1 0 / 1 0 0]; a(n) = center term in M^n * [1 0 0].

Extensions

Edited by Ralf Stephan, Nov 02 2004
Corrected and extended by Robert G. Wilson v, Nov 04 2004

A100059 First differences of A052911.

Original entry on oeis.org

1, 5, 14, 45, 139, 434, 1351, 4209, 13110, 40837, 127203, 396226, 1234207, 3844441, 11975078, 37301261, 116189979, 361921042, 1127350583, 3511592833, 10938286998, 34071752661, 106130359315, 330586256610
Offset: 1

Views

Author

Gary W. Adamson, Oct 31 2004

Keywords

Comments

a(n)/a(n-1) tends to 3.11490754148...an eigenvalue of M and a root of the characteristic polynomial x^3 - 3x^2 - x + 2.

Examples

			a(5) = 139 = rightmost term in M^5 * [1 1 1] which is [434 205 139]. 434 = a(6), while 205 = A052911(5).
a(6) = 434 = 3*a(5) + a(4) - 2*a(3) = 3*139 + 45 - 2*14.

References

  • Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.

Links

Crossrefs

Cf. A019481, A052550, A052939, A100058, A058071.

Programs

  • Mathematica
    LinearRecurrence[{3,1,-2},{1,5,14},30] (* Harvey P. Dale, Apr 21 2016 *)

Formula

G.f.: (2*x^2-2*x-1)*x / (-2*x^3+x^2+3*x-1).
Recurrence: a(n) = 3*a(n-1) + a(n-2) - 2*a(n-3).
a(n) = rightmost term in M^5 * [1 1 1], where M = the 3 X 3 upper triangular matrix [2 1 2 / 1 1 0 / 1 0 0].
INVERT transform of (1, 4, 5, 6, 7, 8, 9, ...) with offset 0.

Extensions

Edited by Ralf Stephan, Nov 02 2004

A052669 Expansion of e.g.f. (1-2*x)/(1-3*x-x^2+2*x^3).

Original entry on oeis.org

1, 1, 8, 66, 840, 12960, 242640, 5286960, 131765760, 3693755520, 115058361600, 3942342835200, 147360531225600, 5967185903078400, 260221271108198400, 12158477739023616000, 605960547270414336000, 32087688283562655744000
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Links

Crossrefs

Cf. A000142, A052550.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( (1-2*x)/(1-3*x-x^2+2*x^3) ))); // G. C. Greubel, Jun 14 2022
  • Maple
    spec := [S,{S=Sequence(Prod(Z,Union(Z,Sequence(Union(Z,Z)))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
  • Mathematica
    b[n_]:= b[n]= If[n<3, 1+3*Floor[n/2], 3*b[n-1] +b[n-2] -2*b[n-3]];
A052669[n_] := n!*b[n]; (* b = A052550 *)
Table[A052669[n], {n, 0, 40}] (* G. C. Greubel, Jun 14 2022 *)
  • SageMath
    @CachedFunction
def b(n): # b = A052550
    if (n<3): return 1 + 3*(n//2)
    else: return 3*b(n-1) +b(n-2) -2*b(n-3)
def A052669(n): return factorial(n)*b(n)
[A052669(n) for n in (0..40)] # G. C. Greubel, Jun 14 2022

Formula

E.g.f.: (1 - 2*x)/(1 - 3*x - x^2 + 2*x^3).
Recurrence: a(0)=1, a(1)=1, a(2)=8, a(n) = 3*n*a(n-1) + n*(n-1)*a(n-2) - 2*n*(n-1)*(n-2)*a(n-3).
a(n) = (n!/229)*Sum_{alpha=RootOf(1 - 3*Z - Z^2 + 2*Z^3)} (5 + 74*alpha - 16*alpha^2)*alpha^(-1-n).
a(n) = n!*A052550(n). - R. J. Mathar, Nov 27 2011
Showing 1-6 of 6 results.

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