cp's OEIS Frontend

A222390 Nonnegative integers m such that 10*m*(m+1)+1 is a square.

%I A222390 #30 Sep 08 2022 08:46:04
%S A222390 0,3,15,132,588,5031,22347,191064,848616,7255419,32225079,275514876,
%T A222390 1223704404,10462309887,46468542291,397292260848,1764580902672,
%U A222390 15086643602355,67007605759263,572895164628660,2544524437949340,21754929612286743,96624921036315675
%N A222390 Nonnegative integers m such that 10*m*(m+1)+1 is a square.
%C A222390 a(n+1)/a(n) tends alternately to (7+2*sqrt(10))/3 and (13+4*sqrt(10))/3; a(n+2)/a(n) tends to A176398^2.
%C A222390 Subsequence of A014601.
%H A222390 Bruno Berselli, <a href="/A222390/b222390.txt">Table of n, a(n) for n = 1..500</a>
%H A222390 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,38,-38,-1,1).
%F A222390 G.f.: 3*x*(1+4*x+x^2)/((1-x)*(1-6*x-x^2)*(1+6*x-x^2)).
%F A222390 a(n) = a(-n+1) = a(n-1)+38*a(n-2)-38*a(n-3)-a(n-4)+a(n-5).
%F A222390 a(n) = -1/2+((5+t*(-1)^n)*(3-t)^(2*floor(n/2))+(5-t*(-1)^n)*(3+t)^(2*floor(n/2)))/20, where t=sqrt(10).
%F A222390 2*a(n)+1 = A221875(n).
%t A222390 LinearRecurrence[{1, 38, -38, -1, 1}, {0, 3, 15, 132, 588}, 23]
%t A222390 CoefficientList[Series[3 x (1 + 4 x + x^2)/((1 - x) (1 - 6 x - x^2) (1 + 6 x - x^2)), {x, 0, 25}], x] (* _Vincenzo Librandi_, Aug 18 2013 *)
%o A222390 (Magma) m:=22; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(3*(1+4*x+x^2)/((1-x)*(1-6*x-x^2)*(1+6*x-x^2))));
%o A222390 (Magma) I:=[0,3,15,132,588]; [n le 5 select I[n] else Self(n-1) +38*Self(n-2)-38*Self(n-3)-Self(n-4)+Self(n-5): n in [1..25]]; // _Vincenzo Librandi_, Aug 18 2013
%o A222390 (Maxima) makelist(expand(-1/2+((5+(-1)^n*sqrt(10))*(3-sqrt(10))^(2*floor(n/2))+(5-(-1)^n*sqrt(10))*(3+sqrt(10))^(2*floor(n/2)))/20), n, 1, 23);
%o A222390 (PARI) x='x+O('x^30); concat([0], Vec(3*x*(1+4*x+x^2)/((1-x)*(1-6*x-x^2)*(1+6*x-x^2)))) \\ _G. C. Greubel_, Jul 15 2018
%Y A222390 Cf. nonnegative integers m such that k*m*(m+1)+1 is a square: A001652 (k=2), A001921 (k=3), A001477 (k=4), A053606 (k=5), A105038 (k=6), A105040 (k=7), A053141 (k=8), this sequence (k=10), A105838 (k=11), A061278 (k=12), A104240 (k=13); A105063 (k=17), A222393 (k=18), A101180 (k=19), A077259 (k=20) [incomplete list].
%Y A222390 Cf. A221875.
%K A222390 nonn,easy
%O A222390 1,2
%A A222390 _Bruno Berselli_, Feb 18 2013