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Explicit formula for recursive equation

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0

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I would like to find the explicit formula for $f(a)$, where the recursive formula is $f(a)=(a^2+r^2)p-q^2-f(r)$, $p=lfloor 1+log_2 a rfloor$, $q = 2^p$, and $r=q-a$. The base case is $f(1) = -1$. So my question is: what is the closed formula for $f(k)$?

Note: $a$ is always odd.

This is related to the integral of the Blancmange curve.

recursion fractals

share|cite|improve this question

edited Mar 29 at 1:17

LittleKnownMathematician

asked Mar 28 at 22:54

LittleKnownMathematicianLittleKnownMathematician

808

$endgroup$

add a comment | 

0

$begingroup$

I would like to find the explicit formula for $f(a)$, where the recursive formula is $f(a)=(a^2+r^2)p-q^2-f(r)$, $p=lfloor 1+log_2 a rfloor$, $q = 2^p$, and $r=q-a$. The base case is $f(1) = -1$. So my question is: what is the closed formula for $f(k)$?

Note: $a$ is always odd.

This is related to the integral of the Blancmange curve.

recursion fractals

share|cite|improve this question

edited Mar 29 at 1:17

LittleKnownMathematician

asked Mar 28 at 22:54

LittleKnownMathematicianLittleKnownMathematician

808

$endgroup$

add a comment | 

$begingroup$

I would like to find the explicit formula for $f(a)$, where the recursive formula is $f(a)=(a^2+r^2)p-q^2-f(r)$, $p=lfloor 1+log_2 a rfloor$, $q = 2^p$, and $r=q-a$. The base case is $f(1) = -1$. So my question is: what is the closed formula for $f(k)$?

Note: $a$ is always odd.

This is related to the integral of the Blancmange curve.

recursion fractals

share|cite|improve this question

edited Mar 29 at 1:17

LittleKnownMathematician

asked Mar 28 at 22:54

LittleKnownMathematicianLittleKnownMathematician

808

$endgroup$

share|cite|improve this question

edited Mar 29 at 1:17

LittleKnownMathematician

asked Mar 28 at 22:54

LittleKnownMathematicianLittleKnownMathematician

808

share|cite|improve this question

edited Mar 29 at 1:17

LittleKnownMathematician

asked Mar 28 at 22:54

LittleKnownMathematicianLittleKnownMathematician

808

asked Mar 28 at 22:54

LittleKnownMathematicianLittleKnownMathematician

808

asked Mar 28 at 22:54

LittleKnownMathematicianLittleKnownMathematician

808