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A219257 Numbers k such that 11*k+1 is a square.

%I A219257 #55 Feb 16 2024 10:07:35
%S A219257 0,9,13,40,48,93,105,168,184,265,285,384,408,525,553,688,720,873,909,
%T A219257 1080,1120,1309,1353,1560,1608,1833,1885,2128,2184,2445,2505,2784,
%U A219257 2848,3145,3213,3528,3600,3933,4009,4360,4440,4809,4893,5280,5368,5773,5865
%N A219257 Numbers k such that 11*k+1 is a square.
%C A219257 Equivalently, numbers of the form m*(11*m+2), where m = 0,-1,1,-2,2,-3,3,...
%C A219257 Also, integer values of h*(h+2)/11.
%H A219257 Bruno Berselli, <a href="/A219257/b219257.txt">Table of n, a(n) for n = 1..1000</a>
%H A219257 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A219257 G.f.: x^2*(9+4*x+9*x^2)/((1+x)^2*(1-x)^3).
%F A219257 a(n) = a(-n+1) = (22*n*(n-1)+7*(-1)^n*(2*n-1)-1)/8 + 1 = (1/176)*(22*n+7*(-1)^n-15)*(22*n+7*(-1)^n-7).
%F A219257 Sum_{n>=2} 1/a(n) = 11/4 - cot(2*Pi/11)*Pi/2. - _Amiram Eldar_, Mar 15 2022
%t A219257 Select[Range[0, 7000], IntegerQ[Sqrt[11 # + 1]] &]
%t A219257 CoefficientList[Series[x (9 + 4 x + 9 x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* _Vincenzo Librandi_, Aug 18 2013 *)
%o A219257 (Magma) [n: n in [0..7000] | IsSquare(11*n+1)];
%o A219257 (Magma) I:=[0,9,13,40,48]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..50]]; // _Vincenzo Librandi_, Aug 18 2013
%Y A219257 Cf. numbers k such that h*k+1 is a square: A005563 (h=1), A046092 (h=2), A001082 (h=3), A002378 (h=4), A036666 (h=5), A062717 (h=6), A132354 (h=7), A000217 (h=8), A132355 (h=9), A132356 (h=10), A152749 (h=12), A219389 (h=13), A219390 (h=14), A204221 (h=15), A074378 (h=16), A219394 (h=17), A219395 (h=18), A219396 (h=19), A219190 (h=20), A219391 (h=21), A219392 (h=22), A219393 (h=23), A001318 (h=24), A219259 (h=25), A217441 (h=26), A219258 (h=27), A219191 (h=28).
%Y A219257 Cf. A175885 (square roots of 11*a(n)+1).
%K A219257 nonn,easy
%O A219257 1,2
%A A219257 _Bruno Berselli_, Nov 16 2012