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%I #14 Jun 28 2020 14:23:15
%S 2,4,8,8,22,-6,72,-92,266,-426,838,-1172,1432,-398,-3614,15140,-41274,
%T 95126,-195698,370876,-652384,1063442,-1570116,1961852,-1560168,
%U -1401888,11023226,-36000318,93408538,-214275608,450374202,-879254356,1599245876,-2695464868,4138070460,-5539280974
%N Extrapolation for (n + 1)-st prime made by fitting least-degree polynomial to first n primes.
%C Construct the least-degree polynomial p(x) which fits the first n primes (p has degree n-1 or less). Then predict the next prime by evaluating p(n+1).
%C Can anything be said about the pattern of positive and negative values?
%C Row sums of triangle A095195. - _Reinhard Zumkeller_, Oct 10 2013
%H Jonathan Wellons, <a href="/A140119/b140119.txt">Table of n, a(n) for n = 1..1500</a>
%F a(n) = Sum_{i=1..n} prime(i) * (-1)^(n-i) * C(n,i-1).
%e The lowest-order polynomial having points (1,2), (2,3), (3,5) and (4,7) is f(x) = 1/6 (-x^3 +9x^2 -14x +18). When evaluated at x = 5, f(5) = 8.
%o (Haskell)
%o a140119 = sum . a095195_row -- _Reinhard Zumkeller_, Oct 10 2013
%o (PARI) a(n) = sum(i=1, n, prime(i)*(-1)^(n-i)*binomial(n, i-1)); \\ _Michel Marcus_, Jun 28 2020
%Y Cf. A082594, A140118.
%K sign
%O 1,1
%A Jonathan Wellons (wellons(AT)gmail.com), May 08, 2008