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A217441 Numbers k such that 26*k+1 is a square.

%I A217441 #62 Mar 17 2022 07:04:29
%S A217441 0,24,28,100,108,228,240,408,424,640,660,924,948,1260,1288,1648,1680,
%T A217441 2088,2124,2580,2620,3124,3168,3720,3768,4368,4420,5068,5124,5820,
%U A217441 5880,6624,6688,7480,7548,8388,8460,9348,9424,10360,10440,11424,11508,12540,12628
%N A217441 Numbers k such that 26*k+1 is a square.
%C A217441 Equivalently, numbers of the form m*(26*m+2), where m = 0,-1,1,-2,2,-3,3,...
%C A217441 Also, integer values of 2*h*(h+1)/13.
%H A217441 Bruno Berselli, <a href="/A217441/b217441.txt">Table of n, a(n) for n = 1..1000</a>
%H A217441 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A217441 G.f.: 4*x^2*(6 + x + 6*x^2)/((1 + x)^2*(1 - x)^3).
%F A217441 a(n) = a(-n+1) = (26*n*(n-1) + 11*(-1)^n*(2*n - 1) - 3)/4 + 3 = (26*n + 11*(-1)^n - 15)*(26*n + 11*(-1)^n - 11)/104.
%F A217441 26*a(2*n-1)+1 = A175886(4*n-3)^2, 26*a(2*n)+1 = A175886(4*n)^2.
%F A217441 Sum_{n>=2} 1/a(n) = 13/2 - cot(Pi/13)*Pi/2. - _Amiram Eldar_, Mar 17 2022
%p A217441 A217441:=proc(q)
%p A217441 local n;
%p A217441 for n from 1 to q do if type(sqrt(26*n+1), integer) then print(n);
%p A217441 fi; od; end:
%p A217441 A217441(1000); # _Paolo P. Lava_, Feb 19 2013
%t A217441 Select[Range[0, 13000], IntegerQ[Sqrt[26 # + 1]] &]
%t A217441 CoefficientList[Series[4 x (6 + x + 6 x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* _Vincenzo Librandi_, Aug 18 2013 *)
%t A217441 LinearRecurrence[{1,2,-2,-1,1},{0,24,28,100,108},50] (* _Harvey P. Dale_, Nov 03 2019 *)
%o A217441 (Magma) [n: n in [0..13000] | IsSquare(26*n+1)];
%o A217441 (Magma) I:=[0,24,28,100,108]; [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
%o A217441 (PARI) a(n)=is(n)=issquare(26*n+1) \\ _Charles R Greathouse IV_, Oct 16 2015
%Y A217441 Cf. similar sequences listed in A219257.
%Y A217441 Cf. A174768 (the squares A174768^2 belong to the sequence), A175886.
%K A217441 nonn,easy
%O A217441 1,2
%A A217441 _Bruno Berselli_, Nov 14 2012