cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

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

Original entry on oeis.org

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

Views

Author

A.K. Devaraj, Nov 21 2009

Keywords

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

Crossrefs

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

Programs

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

A168235 1+5*n+7*n^2.

Original entry on oeis.org

13, 39, 79, 133, 201, 283, 379, 489, 613, 751, 903, 1069, 1249, 1443, 1651, 1873, 2109, 2359, 2623, 2901, 3193, 3499, 3819, 4153, 4501, 4863, 5239, 5629, 6033, 6451, 6883, 7329, 7789, 8263, 8751, 9253, 9769, 10299, 10843, 11401, 11973, 12559, 13159, 13773
Offset: 1

Views

Author

A.K. Devaraj, Nov 21 2009

Keywords

Comments

Consider the quadratic cyclotomic polynomial f(x) = x^2+x+1 and the quotients defined by f(x + n*f(x))/f(x). a(n) is the quotient at x=2.
See A168240 for x=3 or A168244 for x= 1+sqrt(-5).

Examples

			When x = 2, f(x) = 7. Hence at n=1, f( x + f(x))/f(x) = 13 = a(1).

Links

Crossrefs

Cf. A165806, A165808, A165809.

Programs

Formula

a(1)=13, a(2)=39, a(3)=79, a(n)=3*a(n-1)-3*a(n-2)+a(n-3). - Harvey P. Dale, Feb 07 2015
From G. C. Greubel, Apr 09 2016: (Start)
G.f.: (1 + 10*x + 3*x^2)/(1-x)^3.
E.g.f.: (1 + 12*x + 7*x^2)*exp(x). (End)

Extensions

Edited, definition simplified, sequence extended beyond a(8) by R. J. Mathar, Nov 23 2009
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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