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Find the next number in the sequence
9, 73, 241, 561, 1081, 1849,____

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closed as off-topic by Jaap Scherphuis, Penguino, boboquack, Quintec, El-Guest Sep 11 '18 at 1:58

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is off-topic as it appears to be a mathematics problem, as opposed to a mathematical puzzle. For more info, see "Are math-textbook-style problems on topic?" on meta." – Jaap Scherphuis, Penguino, boboquack, Quintec, El-Guest
If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ This isn't really a puzzle, and the hint makes its 'solution' no more than a mechanical process. $\endgroup$ – Penguino Sep 10 '18 at 21:17
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I think the answer is

$2913 = 14^3 + 13^2$

Reasoning

The $n$th term in the sequence is given by the formula $$ a_n = 8n^3 + 4n^2 - 4n + 1 = (2n)^3 + (2n-1)^2 $$ i.e, the cube of the $n$th even number plus the square of the $n$th odd number.

First differences

64, 168, 320, 520, 768,...

Second differences

104, 152, 200, 248,...

Third differences

48, 48, 48,...

Consequence

Since the third differences appear to be constant (at least the first three given), this means that the sequence will satisfy a cubic polynomial in the positive integers. In this case, it turns out the coefficients are integers so this is probably the right answer.

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  • $\begingroup$ So, this just happened to be a simple interpolation polynomial (which is the easiest way to interpolate). $\endgroup$ – rus9384 Sep 10 '18 at 16:14

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