cp's OEIS Frontend

A316672 Numbers k for which 120*k + 169 is a square.

%I A316672 #34 Sep 08 2022 08:46:22
%S A316672 -1,0,1,3,10,14,17,22,36,43,48,56,77,87,94,105,133,146,155,169,204,
%T A316672 220,231,248,290,309,322,342,391,413,428,451,507,532,549,575,638,666,
%U A316672 685,714,784,815,836,868,945,979,1002,1037,1121,1158,1183,1221,1312,1352,1379,1420
%N A316672 Numbers k for which 120*k + 169 is a square.
%C A316672 All terms of A303305 belong to this sequence.
%H A316672 Colin Barker, <a href="/A316672/b316672.txt">Table of n, a(n) for n = 1..1000</a>
%H A316672 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,2,-2,0,0,-1,1).
%F A316672 O.g.f.: x*(-1 + x + x^2 + 2*x^3 + 9*x^4 + 2*x^5 + x^6 + x^7 - x^8)/((1 + x)^2*(1 - x)^3*(1 + x^2)^2).
%F A316672 a(n) = a(1-n) = a(n-1) + 2*a(n-4) - 2*a(n-5) - a(n-8) + a(n-9).
%F A316672 a(n) = (30*n^2 - 2*(15 + 3*(-1)^n + 10*i^(n*(n+1)))*n + 2*(5 + (-1)^n)*i^(n*(n+1)) + 3*(-1)^n - 79)/64, with i = sqrt(-1). Therefore:
%F A316672 a(4*k+1) = (3*k + 2)*(5*k - 1)/2;
%F A316672 a(4*k+2) = k*(15*k + 13)/2, first bisection of A303305;
%F A316672 a(4*k+3) = (k + 1)*(15*k + 2)/2, second bisection of A303305 (see A051869);
%F A316672 a(4*k+4) = (3*k + 1)*(5*k + 6)/2.
%p A316672 select(k->issqr(120*k+169),[$-1..1500]); # _Muniru A Asiru_, Jul 10 2018
%t A316672 LinearRecurrence[{1, 0, 0, 2, -2, 0, 0, -1, 1}, {-1, 0, 1, 3, 10, 14, 17, 22, 36}, 60]
%o A316672 (Magma) [k: k in [0..1500] | IsSquare(120*k+169)];
%o A316672 (Sage) print([k for k in (0..1500) if is_square(120*k+169)])
%o A316672 (PARI) isok(n) = issquare(120*n+169); \\ _Michel Marcus_, Jul 11 2018
%o A316672 (PARI) Vec(x*(-1 + x + x^2 + 2*x^3 + 9*x^4 + 2*x^5 + x^6 + x^7 - x^8)/((1 + x)^2*(1 - x)^3*(1 + x^2)^2) + O(x^40)) \\ _Colin Barker_, Jul 18 2018
%Y A316672 Cf. A051869, A303305.
%Y A316672 Subsequence of A047283.
%Y A316672 Cf. Numbers k for which 8*(2*h+1)*k + (2*h-1)^2 is a square: A000217 (h=0), A001318 (h=1), A085787 (h=2), A118277 (h=3), A195160 (h=4), A195313 (h=5), A277082 (h=6), this sequence (h=7), A303813 (h=8), A303298 (h=9); A303815 (h=13).
%K A316672 sign,easy
%O A316672 1,4
%A A316672 _Bruno Berselli_, Jul 10 2018