A168244 a(n) = 1 + 3*n - 2*n^2.

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

Offset: 0

Author

A.K. Devaraj, Nov 21 2009

Keywords

• sign
• easy

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

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A168235, A168240, A165806, A165808, A165809, A045944, A091823.

Programs

Magma

[1+3*n-2*n^2: n in [1..50]]; // Vincenzo Librandi, Jul 10 2012

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 *)

PARI

a(n) = 1 + 3*n - 2*n^2; \\ Altug Alkan, Apr 09 2016

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

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