Return to Answer

There seems to be an error in the sequence. However, the result is the same even after this mistake:

$\text{len}(a_{n})$ is the length of $a_{n}$ in English (only letters, no spaces)

$a_1 = a_2 = 0$
$a_n = \text{len}(a_{n-1}) \times \text{len}(a_{n-2})$ for $n \geq 3$

$a_1 = 0 \rightarrow$ length of "zero" $\rightarrow \text{len}(a_1) = 4$

$a_2 = 0 \rightarrow$ length of "zero" $\rightarrow \text{len}(a_2) = 4$

$a_3 = 16 = 4 \times 4 \rightarrow$ length of "sixteen" $\rightarrow \text{len}(a_3) = 7$

$a_4 = 28 = 7 \times 4 \rightarrow$ length of "twenty eight" $\rightarrow \text{len}(a_4) = 11$

$a_5 = 77 = 11 \times 7 \rightarrow$ length of "seventy seven" $\rightarrow \text{len}(a_5) = 12$

Here we have a mistake (or some additional rule that requires you to change from 11 to 12):
$a_6 = 144 = 12 \times 12 \rightarrow$ length of "one hundred and forty four" $\rightarrow \text{len}(a_6) = 22$

But for the correct value we get the same result:
$a_6 = 132 = 12 \times 11 \rightarrow$ length of "one hundred and thirty two" $\rightarrow \text{len}(a_6) = 22$

$a_7 = 264 = 22 \times 12 \rightarrow$ length of "two hundred and sixty four" $\rightarrow \text{len}(a_7) = 22$

$a_8 = 484 = 22 \times 22 \rightarrow$ that's the final answer

There seems to be an error in the sequence. However, the result is the same even after this mistake:

$\text{len}(a_{n})$ is the length of $a_{n}$ in English (only letters, no spaces)

$a_1 = a_2 = 0$
$a_n = \text{len}(a_{n-1}) \times \text{len}(a_{n-2})$ for $n \geq 3$

$a_1 = 0 \rightarrow$ length of "zero" $\rightarrow \text{len}(a_1) = 4$

$a_2 = 0 \rightarrow$ length of "zero" $\rightarrow \text{len}(a_2) = 4$

$a_3 = 16 = 4 \times 4 \rightarrow$ length of "sixteen" $\rightarrow \text{len}(a_3) = 7$

$a_4 = 28 = 7 \times 4 \rightarrow$ length of "twenty eight" $\rightarrow \text{len}(a_4) = 11$

$a_5 = 77 = 11 \times 7 \rightarrow$ length of "seventy seven" $\rightarrow \text{len}(a_5) = 12$

Here we have a mistake (or some additional rule that requires you to change from 11 to 12):
$a_6 = 144 = 12 \times 12 \rightarrow$ length of "one hundred and forty four" $\rightarrow \text{len}(a_6) = 22$

But for the correct value we get the same result:
$a_6 = 132 = 12 \times 11 \rightarrow$ length of "one hundred and thirty two" $\rightarrow \text{len}(a_6) = 22$

$a_7 = 264 = 22 \times 12 \rightarrow$ length of "two hundred and sixty four" $\rightarrow \text{len}(a_7) = 22$

Final answer:
$a_8 = 484 = 22 \times 22 \rightarrow$ length of "four hundred and eighty four" $\rightarrow \text{len}(a_8) = 24$

Bonus:

$a_{9} = 528 = 24 \times 22 \rightarrow$ length of "five hundred and twenty eight" $\rightarrow \text{len}(a_{9}) = 25$

$a_{10} = 600 = 25 \times 24 \rightarrow$ length of "six hundred" $\rightarrow \text{len}(a_{10}) = 10$

$a_{11} = 250 = 10 \times 25 \rightarrow$ length of "two hundred and fifty" $\rightarrow \text{len}(a_{11}) = 18$

$a_{12} = 180 = 18 \times 10 \rightarrow$ length of "one hundred and eighty" $\rightarrow \text{len}(a_{12}) = 19$

$a_{13} = 342 = 19 \times 18 \rightarrow$ length of "three hundred and forty two" $\rightarrow \text{len}(a_{13}) = 23$

$a_{14} = 437 = 23 \times 19 \rightarrow$ length of "four hundred and thirty seven" $\rightarrow \text{len}(a_{14}) = 25$

$a_{15} = 575 = 25 \times 23 \rightarrow$ length of "five hundred and seventy five" $\rightarrow \text{len}(a_{15}) = 25$

$a_{16} = 625 = 25 \times 25 \rightarrow$ length of "six hundred and twenty five" $\rightarrow \text{len}(a_{16}) = 23$

$a_{17} = 575 = 23 \times 25 \rightarrow$ length of "five hundred and seventy five" $\rightarrow \text{len}(a_{17}) = 25$

$a_{18} = 575 = 25 \times 23 \rightarrow$ length of "five hundred and seventy five" $\rightarrow \text{len}(a_{18}) = 25$

$a_{19} = 625 = 25 \times 25 \rightarrow$ length of "six hundred and twenty five" $\rightarrow \text{len}(a_{19}) = 23$

$\ldots$

So, starting from $a_{15}$:

$a_{3m} = 575$ for $m \geq 5$

$a_{3m+1} = 625$ for $m \geq 5$

$a_{3m+2} = 575$ for $m \geq 5$

There seems to be an error in the sequence. However, the result is the same even after this mistake:

$\text{len}(a_{n})$ is the length of $a_{n}$ in English (only letters)

$a_1 = a_2 = 0$
$a_n = \text{len}(a_{n-1}) \times \text{len}(a_{n-2})$ for $n \geq 3$

$a_1 = 0 \rightarrow$ length of "zero" is 4

$a_2 = 0 \rightarrow$ length of "zero" is 4

$a_3 = 16 = 4 \times 4 \rightarrow$ length of "sixteen" is 7

$a_4 = 28 = 7 \times 4 \rightarrow$ length of "twenty eight" is 11

$a_5 = 77 = 11 \times 7 \rightarrow$ length of "seventy seven" is 12

Here we have a mistake (or some additional rule that requires you to change from 11 to 12):
$a_6 = 144 = 12 \times 12 \rightarrow$ length of "one hundred and forty four" is 22

But for the correct value we get the same result:
$a_6 = 132 = 12 \times 11 \rightarrow$ length of "one hundred and thirty two" is 22

$a_7 = 264 = 22 \times 12 \rightarrow$ length of "two hundred and sixty four" is 22

$a_8 = 484 = 22 \times 22 \rightarrow$ that's the final answer

There seems to be an error in the sequence. However, the result is the same even after this mistake:

$\text{len}(a_{n})$ is the length of $a_{n}$ in English (only letters, no spaces)

$a_1 = a_2 = 0$
$a_n = \text{len}(a_{n-1}) \times \text{len}(a_{n-2})$ for $n \geq 3$

$a_1 = 0 \rightarrow$ length of "zero" $\rightarrow \text{len}(a_1) = 4$

$a_2 = 0 \rightarrow$ length of "zero" $\rightarrow \text{len}(a_2) = 4$

$a_3 = 16 = 4 \times 4 \rightarrow$ length of "sixteen" $\rightarrow \text{len}(a_3) = 7$

$a_4 = 28 = 7 \times 4 \rightarrow$ length of "twenty eight" $\rightarrow \text{len}(a_4) = 11$

$a_5 = 77 = 11 \times 7 \rightarrow$ length of "seventy seven" $\rightarrow \text{len}(a_5) = 12$

Here we have a mistake (or some additional rule that requires you to change from 11 to 12):
$a_6 = 144 = 12 \times 12 \rightarrow$ length of "one hundred and forty four" $\rightarrow \text{len}(a_6) = 22$

But for the correct value we get the same result:
$a_6 = 132 = 12 \times 11 \rightarrow$ length of "one hundred and thirty two" $\rightarrow \text{len}(a_6) = 22$

$a_7 = 264 = 22 \times 12 \rightarrow$ length of "two hundred and sixty four" $\rightarrow \text{len}(a_7) = 22$

$a_8 = 484 = 22 \times 22 \rightarrow$ that's the final answer

There seems to be an error in the sequence. However, the result is the same in both cases:

$\text{len}(a_{n})$ is the length of $a_{n}$ in English (only letters)

$a_n = \text{len}(a_{n-1}) \times \text{len}(a_{n-2})$

0 = 0 * 0 $\rightarrow$ length of "zero" is 4
0 = 4 * 0 $\rightarrow$ length of "zero" is 4
16 = 4 * 4 $\rightarrow$ length of "sixteen" is 7
28 = 7 * 4 $\rightarrow$ length of "twenty eight" is 11
77 = 11 * 7 $\rightarrow$ length of "seventy seven" is 12

Here we have a mistake (or some additional rule that requires you to change from 11 to 12):
144 = 12 * 12 $\rightarrow$ length of "one hundred and forty four" is 22

But for the correct value we get the same result:
132 = 12 * 11 $\rightarrow$ length of "one hundred and thirty two" is 22

264 = 22 * 12 $\rightarrow$ length of "two hundred and sixty four" is 22
484 = 22 * 22 $\rightarrow$ that's the final answer

There seems to be an error in the sequence. However, the result is the same even after this mistake:

$\text{len}(a_{n})$ is the length of $a_{n}$ in English (only letters)

$a_1 = a_2 = 0$
$a_n = \text{len}(a_{n-1}) \times \text{len}(a_{n-2})$ for $n \geq 3$

$a_1 = 0 \rightarrow$ length of "zero" is 4 

$a_2 = 0 \rightarrow$ length of "zero" is 4 

$a_3 = 16 = 4 \times 4 \rightarrow$ length of "sixteen" is 7 

$a_4 = 28 = 7 \times 4 \rightarrow$ length of "twenty eight" is 11 

$a_5 = 77 = 11 \times 7 \rightarrow$ length of "seventy seven" is 12

Here we have a mistake (or some additional rule that requires you to change from 11 to 12):
$a_6 = 144 = 12 \times 12 \rightarrow$ length of "one hundred and forty four" is 22

But for the correct value we get the same result:
$a_6 = 132 = 12 \times 11 \rightarrow$ length of "one hundred and thirty two" is 22

$a_7 = 264 = 22 \times 12 \rightarrow$ length of "two hundred and sixty four" is 22 

$a_8 = 484 = 22 \times 22 \rightarrow$ that's the final answer