

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
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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(n1)  3*a(n2) + a(n3).
G.f.: 1 + x*(27*x+x^2)/(1x)^3.
a(n) = a(n2) + (2)*sqrt((8)*a(n1) + 17), n > 1.  Klaus Purath, Jul 08 2021


MATHEMATICA

LinearRecurrence[{3, 3, 1}, {2, 1, 8}, 60] (* Harvey P. Dale, Jun 06 2015 *)


PROG

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


CROSSREFS



KEYWORD

sign,easy


AUTHOR



EXTENSIONS

Edited, definition simplified, sequence extended beyond a(5) by R. J. Mathar, Nov 23 2009


STATUS

approved



