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.

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A255107 T(n,k)=Number of length n+k 0..2 arrays with at most one downstep in every k consecutive neighbor pairs.

Original entry on oeis.org

9, 26, 27, 66, 75, 81, 147, 168, 216, 243, 294, 331, 441, 622, 729, 540, 597, 789, 1137, 1791, 2187, 927, 1008, 1302, 1905, 2907, 5157, 6561, 1507, 1616, 2032, 2951, 4429, 7498, 14849, 19683, 2343, 2484, 3042, 4338, 6582, 10125, 19338, 42756, 59049, 3510
Offset: 1

Views

Author

R. H. Hardin, Feb 14 2015

Keywords

Comments

Table starts
......9.....26.....66....147....294....540....927...1507...2343...3510...5096
.....27.....75....168....331....597...1008...1616...2484...3687...5313...7464
.....81....216....441....789...1302...2032...3042...4407...6215...8568..11583
....243....622...1137...1905...2951...4338...6141...8448..11361..14997..19489
....729...1791...2907...4429...6582...9297..12662..16779..21765..27753..34893
...2187...5157...7498..10125..14001..19263..25578..33063..41851..52092..63954
...6561..14849..19338..23463..29147..38010..49611..63075..78552..96210.116236
..19683..42756..49698..55246..61542..73278..91887.115470.142200.172264.205869
..59049.123111.127871.129480.133392.143045.166290.202716.247600.297597.352935
.177147.354484.329325.300432.292534.288057.303969.348070.415308.496188.585101

Examples

			Some solutions for n=4 k=4
..0....0....1....0....1....0....0....0....0....1....0....1....1....0....0....1
..0....1....2....0....2....2....0....1....1....0....0....2....2....1....1....1
..0....2....0....1....0....2....1....0....0....1....1....0....2....0....2....1
..0....0....0....2....0....0....2....0....0....1....1....0....2....0....2....2
..2....0....2....0....1....2....2....0....1....2....1....0....0....2....2....2
..2....0....2....1....2....2....2....2....1....1....1....1....1....2....0....0
..0....2....1....1....0....2....1....0....2....2....2....0....2....2....0....1
..0....0....1....1....0....0....2....1....2....2....1....2....2....2....1....1

Links

Crossrefs

Column 1 is A000244(n+1)
Column 2 is A018919(n+1)

Formula

Empirical for column k:
k=1: a(n) = 3*a(n-1)
k=2: a(n) = 3*a(n-1) -a(n-3)
k=3: a(n) = 3*a(n-1) -3*a(n-2) +8*a(n-3) -9*a(n-4) +3*a(n-5) -a(n-6)
k=4: a(n) = 3*a(n-1) -3*a(n-2) +a(n-3) +12*a(n-4) -18*a(n-5) +7*a(n-6) -3*a(n-8) +a(n-9)
k=5: a(n) = 3*a(n-1) -3*a(n-2) +a(n-3) +18*a(n-5) -29*a(n-6) +12*a(n-7) -6*a(n-10) +3*a(n-11)
k=6: a(n) = 3*a(n-1) -3*a(n-2) +a(n-3) +25*a(n-6) -42*a(n-7) +18*a(n-8) -10*a(n-12) +6*a(n-13)
k=7: a(n) = 3*a(n-1) -3*a(n-2) +a(n-3) +33*a(n-7) -57*a(n-8) +25*a(n-9) -15*a(n-14) +10*a(n-15)
Empirical for row n:
n=1: a(n) = (1/120)*n^5 + (1/6)*n^4 + (19/24)*n^3 + (11/6)*n^2 + (16/5)*n + 3
n=2: a(n) = (1/120)*n^5 + (5/24)*n^4 + (37/24)*n^3 + (175/24)*n^2 + (239/20)*n + 6
n=3: a(n) = (1/120)*n^5 + (1/4)*n^4 + (59/24)*n^3 + (93/4)*n^2 + (1321/30)*n + 11
n=4: a(n) = (1/120)*n^5 + (7/24)*n^4 + (85/24)*n^3 + (1505/24)*n^2 + (2809/20)*n + 30 for n>2
n=5: a(n) = (1/120)*n^5 + (1/3)*n^4 + (115/24)*n^3 + (889/6)*n^2 + (3867/10)*n + 111 for n>3
n=6: a(n) = (1/120)*n^5 + (3/8)*n^4 + (149/24)*n^3 + (2521/8)*n^2 + (56417/60)*n + 385 for n>4
n=7: a(n) = (1/120)*n^5 + (5/12)*n^4 + (187/24)*n^3 + (7393/12)*n^2 + (20667/10)*n + 1143 for n>5

A123941 The (1,2)-entry in the 3 X 3 matrix M^n, where M = {{2, 1, 1}, {1, 1, 0}, {1, 0, 0}}.

Original entry on oeis.org

0, 1, 3, 9, 26, 75, 216, 622, 1791, 5157, 14849, 42756, 123111, 354484, 1020696, 2938977, 8462447, 24366645, 70160958, 202020427, 581694636, 1674922950, 4822748423, 13886550633, 39984728949, 115131438424, 331507764639, 954538564968, 2748484256480
Offset: 0

Views

Author

Roger L. Bagula and Gary W. Adamson, Oct 25 2006

Keywords

Comments

Essentially the same as A076264. - Tom Edgar, May 12 2015

References

  • Rosenblum and Rovnyak, Hardy Classes and Operator Theory, Dover, New York, 1985, page 26

Links

Crossrefs

Cf. A018919, A122099, A122100.

Programs

  • GAP
    a:=[0,1,3];; for n in [4..30] do a[n]:=3*a[n-1]-a[n-3]; od; a; # Muniru A Asiru, Oct 28 2018
  • Magma
    R:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( x/(1-3*x+x^3) )); // G. C. Greubel, Aug 05 2019
  • Maple
    with(linalg): M[1]:=matrix(3,3,[2,1,1,1,1,0,1,0,0]): for n from 2 to 30 do M[n]:=multiply(M[1],M[n-1]) od: 0,seq(M[n][1,2], n=1..30);
a[0]:=0: a[1]:=1: a[2]:=3: for n from 3 to 30 do a[n]:=3*a[n-1]-a[n-3] od: seq(a[n], n=0..30);
  • Mathematica
    M = {{2,1,1}, {1,1,0}, {1,0,0}}; v[1] = {0,0,1}; v[n_]:= v[n] =M.v[n-1];Table[v[n][[2]], {n, 30}]
LinearRecurrence[{3,0,-1}, {0,1,3}, 30] (* G. C. Greubel, Aug 05 2019 *)
  • PARI
    my(x='x+O('x^30)); concat([0], Vec(x/(1-3*x+x^3))) \\ G. C. Greubel, Aug 05 2019
  • Sage
    (x/(1-3*x+x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 05 2019

Formula

a(n) = 3*a(n-1) - a(n-3), a(0)=0, a(1)=1, a(2)=3 (follows from the minimal polynomial x^3-3x^2+1 of the matrix M).
a(n) = A076264(n-1). - R. J. Mathar, Jun 18 2008
G.f.: x/(1 - 3*x + x^3). - Arkadiusz Wesolowski, Oct 29 2012
a(n) = A018919(n-2) for n >= 2. - Georg Fischer, Oct 28 2018

Extensions

Edited by N. J. A. Sloane, Nov 07 2006

A069005 Let M = 4 X 4 matrix with rows /1,1,1,1/1,1,1,0/1,1,0,0/1,0,0,0/ and A(n) = vector (x(n),y(n),z(n),t(n)) = M^n*A where A is the vector (1,1,1,1); then a(n)=z(n).

Original entry on oeis.org

1, 7, 19, 56, 160, 462, 1329, 3828, 11021, 31735, 91376, 263108, 757588, 2181389, 6281058, 18085587, 52075371, 149945056, 431749580, 1243173370, 3579575053, 10306975580, 29677753369, 85453685055, 246054079584, 708484485384
Offset: 1

Views

Author

Benoit Cloitre, Apr 02 2002

Keywords

Comments

a(n) = A091024(n+1) for n > 1. - Georg Fischer, Oct 19 2018

Links

Crossrefs

Cf. A018919, A091024.

Programs

  • Mathematica
    CoefficientList[Series[x (-x^4 - 2 x^3 + 2 x^2 + 5 x + 1)/((1 + x) (1 - 3 x + x^3)), {x, 0, 40}], x] (* Georg Fischer, May 24 2019 *)

Formula

G.f.: x*(-x^4-2*x^3+2*x^2+5*x+1)/((1+x)*(1-3*x+x^3)). [Corrected by Georg Fischer, May 24 2019]
Showing 1-3 of 3 results.

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