 | ECS #12: Words with at most 4 consecutive b-letters |
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1. Description
- Words with at most 4 consecutive b-letters
- Pentanacci numbers
2. Specification
This unlabelled structure is specified as S in
\displaystyle
\left\{ S={\rm Prod} \left( X,{\rm Sequence} \left( {\rm Prod} \left(
a,X \right) \right) \right) ,X={\rm Sequence} \left( b,{\rm card}
\leq 4 \right) ,a={\rm Atom},b={\rm Atom} \right\}
other formats
3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ f \left( n \right) +f \left( n+1 \right) +f \left( n+2
\right) +f \left( n+3 \right) +f \left( n+4 \right) -f \left( n+5
\right) =0,f \left( 0 \right) =1,f \left( 1 \right) =2,f \left( 2
\right) =4,f \left( 3 \right) =8,f \left( 4 \right) =16 \right\}
other formats
3.3. Closed form
\displaystyle
\sum _{\alpha={\rm RootOf} \left( -1+{\rm \_Z}+{{\rm \_Z}}^{2}+{{\rm
\_Z}}^{3}+{{\rm \_Z}}^{4}+{{\rm \_Z}}^{5} \right) }{\frac {1}{1198}}\,
\left( 408+245\,{\alpha}^{2}+200\,{\alpha}^{3}+125\,{\alpha}^{4}+272\,
\alpha \right) {\alpha}^{-1-n}
other formats
3.4. Asymptotics
4. Ordinary generating function
\displaystyle
-{\frac {1+x+{x}^{2}+{x}^{3}+{x}^{4}}{-1+x+{x}^{2}+{x}^{3}+{x}^{4}+{x}^
{5}}}
other formats
5. References
6. Random structure
Search a combinatorial structure by: (firstTerms should be a sequence of integers, separated by commas).