The Keith sequence of an $$n$$-digit ($$n > 1$$) integer $$k$$ ($$k > 9$$) is an integer sequence whose first $$n$$ integers are the $$n$$ digits of $$k$$. So the first three integers in the Keith sequence of 197 are 1, 9 and 7. Each subsequent integer in the sequence is the sum of the previous $$n$$ integers in the sequence. So the subsequent integers in the Keith sequence of 197 are \[ \begin{eqnarray*} 1 + 9 + 7 &=& 17 \\ 9 + 7 + 17 &=& 33 \\ 7 + 17 + 33 &=& 57 \\ 17 + 33 + 57 &=& 107 \\ 33 + 57 + 107 &=& 197 \\ &\vdots & \end{eqnarray*} \] As a result, the Keith sequence of 197 starts as

1, 9, 7, 17, 33, 57, 107, 197, 361, 665, 1223, …

Any Keith sequence is strictly increasing, except for its first $$n$$ integers.

A Keith number is an $$n$$-digit ($$n > 1$$) integer $$k$$ ($$k > 9$$) that occurs in its own Keith sequence. The integer 197 is a Keith number because 197 occurs in the Keith sequence of 197 (indicated in blue above).

The reversal $$\bar{k}$$ of an $$n$$-digit ($$n > 0$$) integer $$k = c_1c_2\ldots c_{n-1}c_n$$ ($$k \geq 0$$) is the integer obtained by inverting the order of its digits: $$\bar{k} = c_nc_{n-1}\ldots c_2c_1$$.

A reverse Keith number is an $$n$$-digit ($$n > 1$$) integer $$k$$ ($$k > 9$$) whose reversal occurs in the Keith sequence of $$k$$. The integer 197 is no reverse Keith number because 791 does not occur in the Keith sequence of 197. The integer 341, on the other hand, is a reverse Keith number because 143 occurs in the Keith sequence of 341

3, 4, 1, 8, 13, 22, 43, 78, 143, 264, 485, 892, 1641, …

Assignment

• Write a function keith_step that takes a list (list) of $$n$$ integers (int). The function must modify the given list by removing the first integer and adding the sum of the $$n$$ given integers at the end. Because one integer is removed and one integer is added, the list still consists of $$n$$ integers. The function must also return a reference to the list.

• Write a function keith_sequence that takes an $$n$$-digit ($$n > 1$$) integer $$k$$ (int; $$k > 9$$). The function also has an optional second parameter target that may take an integer $$t$$ (int). If no value is explicitly passed to the parameter target, then $$t \equiv k$$. The function must return a list (list) containing the first $$n$$ consecutive integers in the Keith sequence of $$k$$, where the last of those $$n$$ integers is greater than or equal to $$t$$.

• Write a function iskeith that takes an integer $$k$$ (int). The function also has an optional second parameter reverse that may take a Boolean value (bool; default value: False). The function must return a Boolean value (bool) that indicates whether $$k$$ is a Keith number (reverse=False) or a reverse Keith number (reverse=True).

Example

>>> numbers = [1, 9, 7]
>>> keith_step(numbers)
[9, 7, 17]
>>> numbers
[9, 7, 17]
>>> keith_step(numbers)
[7, 17, 33]
>>> keith_step(numbers)
[17, 33, 57]
>>> keith_step(numbers)
[33, 57, 107]
>>> keith_step(numbers)
[57, 107, 197]
>>> numbers
[57, 107, 197]

>>> keith_sequence(3)
[3]
>>> keith_sequence(11)
[8, 13]
>>> keith_sequence(34)
[29, 47]
>>> keith_sequence(197)
[57, 107, 197]
>>> keith_sequence(1104, target=7000)
[1104, 2128, 4102, 7907]
>>> keith_sequence(3684, target=10000)
[1910, 3684, 7100, 13685]

>>> iskeith(3)
False
>>> iskeith(34, reverse=False)
False
>>> iskeith(197)
True
>>> iskeith(11)
False
>>> iskeith(2580, False)
True
>>> iskeith(86935)
True
>>> iskeith(174680)
True
>>> iskeith(5752090994058710841670361653731519, reverse=False)
True
>>> iskeith(9, True)
False
>>> iskeith(11, reverse=True)
False
>>> iskeith(12, True)
True
>>> iskeith(341, reverse=True)
True
>>> iskeith(5532, True)
True
>>> iskeith(5426705064, reverse=True)
True