login
a(n) = -n^2 + 9*n + 23.
7

%I #27 Nov 03 2024 09:32:35

%S 23,31,37,41,43,43,41,37,31,23,13,1,-13,-29,-47,-67,-89,-113,-139,

%T -167,-197,-229,-263,-299,-337,-377,-419,-463,-509,-557,-607,-659,

%U -713,-769,-827,-887,-949,-1013,-1079,-1147,-1217,-1289,-1363,-1439,-1517,-1597,-1679,-1763,-1849,-1937,-2027,-2119,-2213

%N a(n) = -n^2 + 9*n + 23.

%C Derivation is similar to that of A126665, which see for further information.

%H Harvey P. Dale, <a href="/A126719/b126719.txt">Table of n, a(n) for n = 0..1000</a>

%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 _Harvey P. Dale_, Oct 19 2011: (Start)

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), a(0)=23, a(1)=31, a(2)=37.

%F G.f.: ((38 - 13*x)*x - 23)/(x-1)^3. (End)

%F E.g.f.: exp(x)*(23 + 8*x - x^2). - _Elmo R. Oliveira_, Nov 02 2024

%e For n=8, -1*8^2 + 9*8 + 23 = 31.

%p A126719:=n->-n^2+9*n+23: seq(A126719(n), n=0..100); # _Wesley Ivan Hurt_, Jan 20 2017

%t Table[-n^2+9n+23,{n,0,60}] (* or *) LinearRecurrence[{3,-3,1},{23,31,37},60] (* _Harvey P. Dale_, Oct 19 2011 *)

%o (PARI) a(n)=-n^2+9*n+23 \\ _Charles R Greathouse IV_, Oct 07 2015

%Y Cf. A126665.

%K sign,easy

%O 0,1

%A _Michael M. Ross_, Mar 13 2007