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Find what comes next in the following sequence: 0,1,4,9,7,7,9,13,10,9,1,? Bonus: What is the rule?

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The $n$th term of the sequence is

the sum of the digits of $n^2$ (in base $10$) so the next term is $1+2+1=4$.

For the bonus (what numbers never appear in the sequence),

if $k$ is not divisible by $9$ and $k-1$ is not divisible by $3$, i.e. $k$ modulo $9$ is in $\{2,3,5,6,8\}$, then $k$ cannot be in the sequence because the sum of digits of $n^2$ is the same modulo $9$ as $n^2$ itself, which must be $0,1,4$ or $7$. If $k$ is divisible by $9$, then note that $33...3$ with $\frac{k}{9}$ $3$'s squared is $11...1088...89$ with $\frac{k}{9}-1$ $1$'s and $8$'s, so its sum of digits is $k$. Otherwise, note that $33...35$ with $\frac{k-7}{3}$ $3$'s squared is $11...122...25$ with $\frac{k-7}{3}$ $1$'s and $\frac{k-4}{3}$ $2$'s; this has sum of digits $k$. Therefore, since $k=1$ and $k=4$ obviously appear in the sequence, the numbers that never appear in the sequence are the positive integers that are $2,3,5,6$ or $8$ modulo $9$.

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  • $\begingroup$ Bonus: what are the numbers that never appear in this sequence? $\endgroup$ – WhatsUp May 15 at 16:43
  • $\begingroup$ +1 for solving my bonus question! ^_^ $\endgroup$ – WhatsUp May 15 at 18:24

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