New Scientist Enigma 642 – Fibonacciesque
by Kenneth Armstrong
From Issue #1796, 23rd November 1991 [link]
My children had been practising addition, forming additive sequences of numbers, and when the numbers got too large for them, extending the sequence the other way using subtraction and getting some negative numbers. Eventually, they noticed that one of their sequences had numbers in it divisible by 2, 3, 5 and 7 but none divisible by 11. Part of that sequence was:
… –2 3 1 4 …
the rule going from left to right being, of course, that one term plus the following one gives the next.
The children soon saw too that every sequence they wrote down had terms divisible by 2 and 3, and probably 7 also.
What I want you to find is the two sequences like this, each having no terms divisible by 5, 11 or 13. In each sequence the four consecutive terms we want should have:
just the first term negative;
the third term 3;
and the fourth term less than 100
(In fact, one of the sequences will have terms divisible by 17 and none divisible by 19, while the other will have terms divisible by 19 and none divisible by 17, but you don’t need these facts to find them).
Please send in the four terms of the two sequences.