Search: a329678 -id:a329678
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A130543
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Multiplicative persistence of n!.
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+0
2
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0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
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OFFSET
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0,1
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COMMENTS
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From 5! on all the factorials end with "zero" thus the persistence is equal to 1.
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LINKS
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FORMULA
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G.f.: x^4/(1-x).
a(n) = 1 for n >= 4.
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EXAMPLE
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0!=1; 1!=1; 2!=2; 3!=6 --> Persistence=0
4!=24 --> 2*4=8 --> Persistence=1
5!=120 --> 1*2*0=0 --> Persistence=1
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MAPLE
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P:=proc(n)local i, k, w, ok, cont; for i from 0 by 1 to n do w:=1; k:=i!; ok:=1; if k<10 then print(0); else cont:=1; while ok=1 do while k>0 do w:=w*(k-(trunc(k/10)*10)); k:=trunc(k/10); od; if w<10 then ok:=0; print(cont); else cont:=cont+1; k:=w; w:=1; fi; od; fi; od; end: P(100);
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CROSSREFS
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KEYWORD
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easy,base,nonn
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AUTHOR
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STATUS
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approved
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A329677
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Number of excursions of length n with Motzkin-steps consisting only of consecutive steps UH, HD, and DH.
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+0
2
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1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
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graph;
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listen;
history;
text;
internal format)
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OFFSET
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0
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COMMENTS
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The Motzkin step set is U=(1,1), H=(1,0) and D=(1,-1). An excursion is a path starting at (0,0), ending on the x-axis and never crossing the x-axis, i.e., staying at nonnegative altitude.
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LINKS
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FORMULA
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G.f.: 1 + t + t^3 + t^4.
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EXAMPLE
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We only have the following four excursions of this type: the empty walk, H, UHD and UHDH.
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MATHEMATICA
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PadRight[#, 105] &@ CoefficientList[Series[1 + x + x^3 + x^4, {x, 0, 105}], x] (* Michael De Vlieger, Dec 16 2019 *)
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CROSSREFS
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Cf. A329670, A329678, A329679 (other Motzkin excursions avoiding certain consecutive steps such that the sequence counting them has growth rate zero).
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KEYWORD
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nonn,walk,easy
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AUTHOR
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STATUS
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approved
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A329679
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Number of excursions of length n with Motzkin-steps consisting only of consecutive steps UH, UD, HD and DH.
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+0
2
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1, 1, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,4
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COMMENTS
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The Motzkin step set is U=(1,1), H=(1,0) and D=(1,-1). An excursion is a path starting at (0,0), ending on the x-axis and never crossing the x-axis, i.e., staying at nonnegative altitude.
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LINKS
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FORMULA
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G.f.: 1 + t + t^2 + 2t^3 + t^4.
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EXAMPLE
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We only have the following six excursions of this type: the empty walk, H, UD, UDH, UHD and UHDH.
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CROSSREFS
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Cf. A329670, A329677, A329678 (other Motzkin excursions avoiding certain consecutive steps such that the sequence counting them has growth rate zero).
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KEYWORD
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nonn,walk,easy
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AUTHOR
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STATUS
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approved
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