Find next number in series 3 14 39 84 155?

Answer 1

Any number can be the next number since given any number whatsoever, it is possible to find a polynomial of order 5 that will fit the above sequence and the additional "next" number.

The simplest solution for the 5 numbers in the given sequence is that the sequence is generated by the rule:

Un = n3 + n2 + n for n = 1, 2, 3, ...

In that case the next number is 258.

But, for example, if I want the next number to be 2013 (the current year), all I need to do is to define the rule as:

Un = (117n5 - 1755n4 + 9953n3 - 26317n2 + 32066n - 14040)/8 for n = 1, 2, 3, ...

Answer 2

5, 6, 7, 8, 10, 11, 14, 15, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51 OR 5, 6, 7, 8, 10, 11, 14, 15, 18, 20, 23, 26, 29, 32, 36, 39, 44, 47, 53, 57, 63, 68, 74, 80, 87, 93, 101, 107, 116, 123, 133, 141, 151, 160, 171, 181, 193, 203, 216, 227, 241, 253, 268, 281, 297, 311 or 5, 6, 7, 8, 10, 11, 14, 15, 18, 20, 25, 27, 33, 35, 43, 47, 56, 62, 71, 80, 92, 103, 118, 132, 149, 168, 190, 212, 240, 267, 300, 336, 375, 419, 468, 521, 580, 646, 717, 798, 885, 981, 1087, 1204, 1332, 1473, 1626, 1797, 1980, 2187, 2405 The last one is Expansion of Product_{ n >= 2, n not of the form 2^k-1 } (1+x^n). The second one is Expansion of 1/((1-x)(1-x^2)(1-x^5)(1-x^12)). And the first one is nonsquares mod 52. This is a great question for showing that a sequence is not unique. This is a sequence not a series since there is not indication we are adding any terms.

Answer 3

15-30 is next