Nest[f, x, n] == Ceiling[(x - (a b)/(b - 1))/b^n + (a b)/(b - 1)] /; f[x] == a + Ceiling[x/b] && Element[a, Integers] && Element[b, Integers] && b > 0

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> Nest </mi> <mo> [ </mo> <mrow> <mi> f </mi> <mo> , </mo> <mi> x </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> ] </mo> </mrow> <mo> &#10869; </mo> <mrow> <mo> &#8968; </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> - </mo> <mfrac> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> b </mi> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </msup> </mrow> <mo> + </mo> <mfrac> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <mo> &#8969; </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> f </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mo> &#8968; </mo> <mfrac> <mi> x </mi> <mi> b </mi> </mfrac> <mo> &#8969; </mo> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> a </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> b </mi> <mo> &#8712; </mo> <semantics> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox[&quot;\[DoubleStruckCapitalN]&quot;, &quot;+&quot;], Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Nest </ci> <ci> f </ci> <ci> x </ci> <ci> n </ci> </apply> <apply> <ceiling /> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <ci> b </ci> <apply> <power /> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <times /> <ci> a </ci> <ci> b </ci> <apply> <power /> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> f </ci> <ci> x </ci> </apply> <apply> <plus /> <ci> a </ci> <apply> <ceiling /> <apply> <times /> <ci> x </ci> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> a </ci> <integers /> </apply> <apply> <in /> <ci> b </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29