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Numbers k such that 5*k-1 is a perfect square.
4

%I #37 Feb 27 2020 18:19:18

%S 1,2,10,13,29,34,58,65,97,106,146,157,205,218,274,289,353,370,442,461,

%T 541,562,650,673,769,794,898,925,1037,1066,1186,1217,1345,1378,1514,

%U 1549,1693,1730,1882,1921,2081,2122,2290,2333,2509,2554,2738,2785,2977

%N Numbers k such that 5*k-1 is a perfect square.

%H Harry J. Smith, <a href="/A062317/b062317.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).

%F a(n) = ((2+5*(n-1)/2)^2 + 1)/5 if n is odd; a(n) = ((3+5*(n-2)/2)^2 + 1)/5 if n is even.

%F From _R. J. Mathar_, Jan 30 2010: (Start)

%F a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).

%F G.f.: x*(1+x+6*x^2+x^3+x^4)/((1+x)^2*(1-x)^3). (End)

%F a(n) = (10*n*(n-1) + 5 - (6*n-3)*(-1)^n)/8. - _Eric Simon Jacob_, Jan 20 2020

%t f[n_]:=IntegerQ[Sqrt[5*n-1]]; Select[Range[0,8! ],f[ # ]&] (* _Vladimir Joseph Stephan Orlovsky_, Feb 19 2010 *)

%t LinearRecurrence[{1,2,-2,-1,1},{1,2,10,13,29},50] (* _Harvey P. Dale_, Dec 29 2018 *)

%o (PARI) je=[]; for(n=1,5000, if(issquare(5*n-1),je=concat(je,n))); je

%o (PARI) { n=0; for (m=1, 10^9, if (issquare(5*m - 1), write("b062317.txt", n++, " ", m); if (n==1000, break)) ) } \\ _Harry J. Smith_, Aug 04 2009

%Y Cf. A036666.

%K nonn,easy

%O 1,2

%A _Santi Spadaro_, Jul 12 2001

%E More terms from _Jason Earls_, Jul 14 2001