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A168244 a(n) = 1 + 3*n - 2*n^2. 9
1, 2, -1, -8, -19, -34, -53, -76, -103, -134, -169, -208, -251, -298, -349, -404, -463, -526, -593, -664, -739, -818, -901, -988, -1079, -1174, -1273, -1376, -1483, -1594, -1709, -1828, -1951, -2078, -2209, -2344, -2483, -2626, -2773, -2924, -3079, -3238, -3401, -3568, -3739, -3914, -4093, -4276, -4463, -4654, -4849 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Consider the quadratic cyclotomic polynomial f(x) = x^2+x+1 and the quotients f(x + n*f(x))/f(x), as in A168235 and A168240. a(n) is the real part of the quotient at x = 1+sqrt(-5).
The imaginary part of the quotient is sqrt(5)*A045944(n).
As stated in short description of A168244 the quotient is in two parts: rational integers (cf. A168244) and rational integer multiples of sqrt(-5). It so happens that the sequence of rational integer coefficients of sqrt(-5) is A045944. - A.K. Devaraj, Nov 22 2009
This sequence contains half of all integers m such that -8*m +17 is an odd square. The other half are found in A091823 multiplied by -1. The squares resulting from A168244 are (4*n - 3)^2, those from A091823 are (4*n + 3)^2. - Klaus Purath, Jul 11 2021
LINKS
FORMULA
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: 1 + x*(2-7*x+x^2)/(1-x)^3.
a(-n) = -A091823(n), a(0) = 1. - Michael Somos, May 11 2014
E.g.f.: (1 + x - 2*x^2)*exp(x). - G. C. Greubel, Apr 09 2016
a(n) = a(n-2) + (-2)*sqrt((-8)*a(n-1) + 17), n > 1. - Klaus Purath, Jul 08 2021
MATHEMATICA
Table[-(n + (n + 1)^2 - 3)/2, {n, -1, 200, 2}] (* Vladimir Joseph Stephan Orlovsky, Jun 26 2011 *)
LinearRecurrence[{3, -3, 1}, {2, -1, -8}, 60] (* Harvey P. Dale, Jun 06 2015 *)
PROG
(Magma) [1+3*n-2*n^2: n in [1..50]]; // Vincenzo Librandi, Jul 10 2012
(PARI) a(n) = 1 + 3*n - 2*n^2; \\ Altug Alkan, Apr 09 2016
CROSSREFS
Sequence in context: A012961 A013124 A012966 * A009828 A084222 A160602
KEYWORD
sign,easy
AUTHOR
A.K. Devaraj, Nov 21 2009
EXTENSIONS
Edited, definition simplified, sequence extended beyond a(5) by R. J. Mathar, Nov 23 2009
a(0)=1 added by N. J. A. Sloane, Apr 09 2016
STATUS
approved

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Last modified June 9 00:06 EDT 2023. Contains 363165 sequences. (Running on oeis4.)