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%I #25 Nov 02 2024 18:22:02
%S 53,61,67,71,73,73,71,67,61,53,43,31,17,1,-17,-37,-59,-83,-109,-137,
%T -167,-199,-233,-269,-307,-347,-389,-433,-479,-527,-577,-629,-683,
%U -739,-797,-857,-919,-983,-1049,-1117,-1187,-1259,-1333,-1409,-1487,-1567,-1649,-1733,-1819,-1907,-1997,-2089,-2183,-2279
%N a(n) = -n^2 + 9*n + 53.
%C Quadratic equation derived from the four primes 61, 67, 71, 73 using the method of common differences. Many of the initial terms are primes.
%H Michael M. Ross, <a href="http://www.naturalnumbers.org">Natural Numbers</a>.
%H Robert Sacks, <a href="http://www.numberspiral.com/p/common_diff.html">Number Spiral: Method of Common Differences</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F From _Arkadiusz Wesolowski_, Oct 24 2013: (Start)
%F a(n) = -A186950(n+19).
%F G.f.: (53 - 98*x + 43*x^2)/(1 - x)^3. (End)
%F From _Elmo R. Oliveira_, Nov 02 2024: (Start)
%F E.g.f.: (53 + 8*x - x^2)*exp(x).
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
%e For n=8, -1*8^2 + 9*8 + 53 = 61.
%t Table[ - n^2 + 9*n + 53, {n, 0, 46}] (* _Arkadiusz Wesolowski_, Oct 24 2013 *)
%t LinearRecurrence[{3,-3,1},{53,61,67},60] (* _Harvey P. Dale_, Apr 04 2024 *)
%o (PARI) a(n) = -n^2 + 9*n + 53 \\ _Michel Marcus_, Jun 30 2013
%o (Magma) [-n^2+9*n+53 : n in [0..46]]; // _Arkadiusz Wesolowski_, Oct 24 2013
%Y Cf. A186950.
%K sign,easy
%O 0,1
%A _Michael M. Ross_, Mar 13 2007