|
%I #23 Mar 08 2022 19:55:28
%S 0,4,44,440,4360,43164,427284,4229680,41869520,414465524,4102785724,
%T 40613391720,402031131480,3979697923084,39394948099364,
%U 389969783070560,3860302882606240,38213059042991844,378270287547312204,3744489816430130200,37066627876753989800
%N Nonnegative n such that 6*n^2 + 6*n + 1 is a square.
%H Harvey P. Dale, <a href="/A105038/b105038.txt">Table of n, a(n) for n = 0..1000</a>
%H Editors, L'Intermédiaire des Mathématiciens, <a href="/A072256/a072256.pdf">Query 4500: The equation x(x+1)/2 = y*(y+1)/3</a>, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (I).
%H Editors, L'Intermédiaire des Mathématiciens, <a href="/A072256/a072256_1.pdf">Query 4500: The equation x(x+1)/2 = y*(y+1)/3</a>, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (II).
%H Editors, L'Intermédiaire des Mathématiciens, <a href="/A072256/a072256_2.pdf">Query 4500: The equation x(x+1)/2 = y*(y+1)/3</a>, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (III).
%H Editors, L'Intermédiaire des Mathématiciens, <a href="/A072256/a072256_3.pdf">Query 4500: The equation x(x+1)/2 = y*(y+1)/3</a>, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (IV).
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (11,-11,1).
%F G.f.: 4*x/(1-11*x+11*x^2-x^3).
%F a(0)=0, a(1)=4, a(2)=44, a(n)=11*a(n-1)-11*a(n-2)+a(n-3). - _Harvey P. Dale_, Sep 29 2013
%F a(n) = (-6-(5-2*sqrt(6))^n*(-3+sqrt(6))+(3+sqrt(6))*(5+2*sqrt(6))^n)/12. - _Colin Barker_, Mar 05 2016
%F a(n) = (A072256(n+1) - 1)/2.
%t CoefficientList[Series[4x/(1-11x+11x^2-x^3),{x,0,30}],x] (* or *) LinearRecurrence[{11,-11,1},{0,4,44},30] (* _Harvey P. Dale_, Sep 29 2013 *)
%o (PARI) for(n=0,427284,if(issquare(6*n*(n+1)+1),print1(n,",")))
%o (PARI) Vec(4*x/(1-11*x+11*x^2-x^3)+O(x^99)) \\ _Charles R Greathouse IV_, Nov 13 2012
%Y Cf. A001652, A001570, A049629, A105040, A104240, A077288, A105036, A103200, A105037.
%K nonn,easy
%O 0,2
%A _Gerald McGarvey_, Apr 03 2005
%E More terms from _Vladeta Jovovic_, Apr 05 2005
%E Incorrect conjectures deleted by _N. J. A. Sloane_, Nov 24 2010
|
