# What is the next number in the sequence 0,0,16,28,77,144,264?

0,0,16,28,77,132,264

What is the next number in the sequence?

• I want to say it's rot13(… oh, wait, that's not gonna work here. – msh210 Feb 23 at 12:48
• rot13(ner lbh fher gurer fubhyqa'g or gjryir yrff va 144?) – Michał Wójcik Feb 23 at 20:09

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" $$\rightarrow \text{len}(a_7) = 22$$

$$a_8 = 484 = 22 \times 22 \rightarrow$$ 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 $$m \geq 5$$

• Great answer and explanation. Well done! – UnidentifiedX Feb 24 at 11:25
• However, the answer is literally 484, not the length of 484 :D – UnidentifiedX Feb 25 at 11:21
• Yep, I just added it to show what happens with the sequence in next steps :D – Michał Wójcik Feb 25 at 14:21