Search: a018919 -id:a018919
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A255107
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T(n,k)=Number of length n+k 0..2 arrays with at most one downstep in every k consecutive neighbor pairs
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+10
13
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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FORMULA
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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
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EXAMPLE
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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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A123941
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The (1,2)-entry in the 3 X 3 matrix M^n, where M = {{2, 1, 1}, {1, 1, 0}, {1, 0, 0}}.
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+10
1
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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
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OFFSET
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0,3
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COMMENTS
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REFERENCES
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Rosenblum and Rovnyak, Hardy Classes and Operator Theory, Dover, New York, 1985, page 26
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LINKS
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FORMULA
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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).
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MAPLE
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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);
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MATHEMATICA
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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 *)
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PROG
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(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
(PARI) my(x='x+O('x^30)); concat([0], Vec(x/(1-3*x+x^3))) \\ G. C. Greubel, Aug 05 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( 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
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CROSSREFS
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KEYWORD
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nonn,easy,less
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A069005
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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).
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+10
0
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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
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OFFSET
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1,2
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COMMENTS
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LINKS
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FORMULA
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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]
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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