07/05/2011, 12:28 PM
this forum is mainly dedicated to (continu) iterations in particular of e^x.
but some people have already suggested a subtread about the somewhat logical generalization of iterations ; recursions.
i think and hope there is alot to be learned and discovered about recursions and i wonder how you guys feel about it.
i dont know if it well be relevant to tetration , but in a way i consider it of equal style , since just like tetration , there is not much know about ( certain types of ) recursions imho ... however that might be my subjective and ingnorant opinion as a nonexpert ...
i have many ideas about recursion and perhaps many of you have them too.
in particular im intrested in integer recursion such as fibonacci , tribonacci , 4-somos , 5-somos , 6-somos , 7-somos ,...
but also u_n = (u_(n-1)^k + u_(n-1)) /u_(n-2) for e.g. k = 2 or 3.
( u_(n-1)^2 + u_(n-1) ) / u_(n-2) generates an integer sequence for u_1 = u_2 = 1 despite the division in the formula )
u_n = (u_(n-1)^2 + k) / u_(n-2) is another integer sequence for many initial conditions.
also many recursions exist for taylor series coefficients of certain functions.
some combinatorial recursions exist as well though often of the type above.
but beyond those , studied recursions get rare i think.
and of course the innevitable question ; are there integer recursions that have limit growth rate between exponential and polynomial such as floor(exp^[1/3](x)) ?
( im also aware of recursion related to differential equations and ofcourse ackermann type but i find those less intresting )
it seems almost all recursions are combinatorics or grow like exponential , polynomial , factorial or those combinations.
but some people have already suggested a subtread about the somewhat logical generalization of iterations ; recursions.
i think and hope there is alot to be learned and discovered about recursions and i wonder how you guys feel about it.
i dont know if it well be relevant to tetration , but in a way i consider it of equal style , since just like tetration , there is not much know about ( certain types of ) recursions imho ... however that might be my subjective and ingnorant opinion as a nonexpert ...
i have many ideas about recursion and perhaps many of you have them too.
in particular im intrested in integer recursion such as fibonacci , tribonacci , 4-somos , 5-somos , 6-somos , 7-somos ,...
but also u_n = (u_(n-1)^k + u_(n-1)) /u_(n-2) for e.g. k = 2 or 3.
( u_(n-1)^2 + u_(n-1) ) / u_(n-2) generates an integer sequence for u_1 = u_2 = 1 despite the division in the formula )
u_n = (u_(n-1)^2 + k) / u_(n-2) is another integer sequence for many initial conditions.
also many recursions exist for taylor series coefficients of certain functions.
some combinatorial recursions exist as well though often of the type above.
but beyond those , studied recursions get rare i think.
and of course the innevitable question ; are there integer recursions that have limit growth rate between exponential and polynomial such as floor(exp^[1/3](x)) ?
( im also aware of recursion related to differential equations and ofcourse ackermann type but i find those less intresting )
it seems almost all recursions are combinatorics or grow like exponential , polynomial , factorial or those combinations.