A305290 - OEIS

%I #35 Jun 04 2018 11:15:30

%S 0,-7,31,-86,182,-333,549,-844,1228,-1715,2315,-3042,3906,-4921,6097,

%T -7448,8984,-10719,12663,-14830,17230,-19877,22781,-25956,29412,

%U -33163,37219,-41594,46298,-51345,56745,-62512,68656,-75191,82127,-89478,97254,-105469,114133,-123260,132860

%N Numbers k such that 4*k + 1 is a perfect cube, sorted by absolute values.

%H Colin Barker, <a href="/A305290/b305290.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (-3,-2,2,3,1).

%F G.f.: x^2*(-7 + 10*x - 7*x^2)/((1 - x)*(1 + x)^4).

%F a(n) = -3*a(n-1) - 2*a(n-2) + 2*a(n-3) + 3*a(n-4) + a(n-5).

%F a(n) = (-1 - A016755(n-1)*(-1)^n)/4.

%F a(n) + a(-n) = (-1)^n*2^((1-(-1)^n)/2).

%F (n - 2)*(4*n^2 - 16*n + 19)*a(n) + (12*n^2 - 36*n + 31)*a(n-1) - (n - 1)*(4*n^2 - 8*n + 7)*a(n-2) = 0.

%F From _Colin Barker_, May 30 2018: (Start)

%F a(n) = n*(4*n^2 + 6*n + 3)/2 for n even.

%F a(n) = -(n + 1)*(4*n^2 + 2*n + 1)/2 for n odd.

%F (End)

%p seq(coeff(series(x^2*(-7+10*x-7*x^2)/((1-x)*(1+x)^4), x,50),x,n),n=1..45); # _Muniru A Asiru_, May 31 2018

%t LinearRecurrence[{-3, -2, 2, 3, 1}, {0, -7, 31, -86, 182}, 45] (* _Jean-François Alcover_, Jun 04 2018 *)

%o (PARI) concat(0, Vec(-x^2*(7 - 10*x + 7*x^2) / ((1 - x)*(1 + x)^4) + O(x^40))) \\ _Colin Barker_, Jun 04 2018

%Y Cf. A016755.

%Y Cf. A000290: k such that 4*k is a square.

%Y Cf. A002378: k such that 4*k+1 is a square.

%Y Cf. A033431: k such that 4*k is a nonnegative cube.

%Y Cf. A305291: k such that 4*k+3 is a cube.

%Y Cf. A141046: k such that 4*k is a fourth power.

%Y Cf. 4*A219086: k such that 4*k+1 is a fourth power.

%K sign,easy

%O 1,2

%A _Bruno Berselli_, May 29 2018