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A220082 Numbers k such that 10*k-1 is a square.

%I A220082 #36 Nov 19 2023 18:44:02
%S A220082 1,5,17,29,53,73,109,137,185,221,281,325,397,449,533,593,689,757,865,
%T A220082 941,1061,1145,1277,1369,1513,1613,1769,1877,2045,2161,2341,2465,2657,
%U A220082 2789,2993,3133,3349,3497,3725,3881,4121,4285,4537,4709,4973,5153,5429,5617,5905
%N A220082 Numbers k such that 10*k-1 is a square.
%C A220082 Equivalently, numbers of the form m*(10*m+6)+1, where m=0,-1,1,-2,2,-3,3,...
%H A220082 Bruno Berselli, <a href="/A220082/b220082.txt">Table of n, a(n) for n = 1..1000</a>
%H A220082 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A220082 G.f.: x*(1+4*x+10*x^2+4*x^3+x^4)/((1+x)^2*(1-x)^3).
%F A220082 a(n) = a(-n+1) = (10*n*(n-1)-(2*n-1)*(-1)^n+3)/4.
%F A220082 For the definition: 10*a(n)-1 = ((10*n-(-1)^n-5)/2)^2.
%F A220082 a(n) = A212570(n)-A212570(n-1) = 4*A085787(n-1)+1 = A132356(n-1)-(2*n-1)*(-1)^n.
%p A220082 A220082:=proc(q)
%p A220082 local n;
%p A220082 for n from 1 to q do if type(sqrt(10*n-1), integer) then print(n);
%p A220082 fi; od; end:
%p A220082 A220082(1000); # _Paolo P. Lava_, Feb 19 2013
%t A220082 Select[Range[0, 6000], IntegerQ[Sqrt[10 # - 1]] &]
%t A220082 CoefficientList[Series[(1 + 4 x + 10 x^2 + 4 x^3 + x^4) / ((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* _Vincenzo Librandi_, Aug 18 2013 *)
%t A220082 LinearRecurrence[{1,2,-2,-1,1},{1,5,17,29,53},50] (* _Harvey P. Dale_, Nov 19 2023 *)
%o A220082 (Magma) [n: n in [1..6000] | IsSquare(10*n-1)]; /* or (see the first comment): */ [1] cat [m*(10*m+6)+1: m in [-n,n], n in [1..24]];
%o A220082 (Magma) I:=[1,5,17,29,53]; [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..60]]; // _Vincenzo Librandi_, Aug 18 2013
%Y A220082 Cf. A085787, A132356 (numbers n such that 10*n+1 is a square).
%Y A220082 Cf. numbers n such that k*n-1 is a square: A002522 (k=1), A001844 (k=2), A062317 (k=5).
%K A220082 nonn,easy
%O A220082 1,2
%A A220082 _Bruno Berselli_, Dec 05 2012