# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a220186 Showing 1-1 of 1 %I A220186 #42 Nov 01 2022 17:03:34 %S A220186 0,8,800,78408,7683200,752875208,73774087200,7229107670408, %T A220186 708378777612800,69413891098384008,6801852948864020000, %U A220186 666512175097575576008,65311391306613542428800,6399849835873029582446408,627119972524250285537319200 %N A220186 Numbers n >= 0 such that n^2 + n*(n+1)/2 is a square. %C A220186 Equivalently, numbers n such that triangular(2*n) - triangular(n) is a square. %H A220186 Index entries for linear recurrences with constant coefficients, signature (99,-99,1). %F A220186 a(n) = A098308(2*n-2). %F A220186 a(1) = 0, a(2) = 8, a(3) = 800 and a(n) = 99*a(n-1)-99*a(n-2)+a(n-3) for n>3. - _Giovanni Resta_, Apr 12 2013 %F A220186 G.f.: -8*x^2*(x+1) / ((x-1)*(x^2-98*x+1)). - _Colin Barker_, May 31 2013 %F A220186 a(n) = (49+20*sqrt(6))^(-n)*(49+20*sqrt(6)-2*(49+20*sqrt(6))^n+(49-20*sqrt(6))*(49+20*sqrt(6))^(2*n))/12. - _Colin Barker_, Mar 05 2016 %F A220186 a(n) = 8*A108741(n). - _R. J. Mathar_, Feb 19 2017 %t A220186 a[n_]:=Floor[(1/12)*(49 + 20*Sqrt[6])^n]; Table[a[n],{n,0,10}] (* _Giovanni Resta_, Apr 12 2013 *) %t A220186 LinearRecurrence[{99,-99,1},{0,8,800},20] (* _Harvey P. Dale_, Nov 01 2022 *) %o A220186 (C) %o A220186 #include %o A220186 #include %o A220186 int main() { %o A220186 unsigned long long a, i, t; %o A220186 for (i=0; i < (1L<<32); ++i) { %o A220186 a = (i*i) + ((i+1)*i/2); %o A220186 t = sqrt(a); %o A220186 if (a == t*t) printf("%llu\n", i); %o A220186 } %o A220186 return 0; %o A220186 } %o A220186 (PARI) lista(nn) = for(n=0, nn, if(issquare(n^2 + n*(n+1)/2), print1(n, ", "))); \\ _Altug Alkan_, Mar 05 2016 %Y A220186 Cf. A005449 (n^2 + n(n+1)/2). %Y A220186 Cf. A011916 (numbers n such that n^2 + n(n+1)/2 is a triangular number). %Y A220186 Cf. A014105 (n^2 + n(n+1)). %Y A220186 Cf. A084703 (numbers n such that n^2 + n(n+1) is a square). %Y A220186 Cf. A220185 (numbers n such that n^2 + n(n+1) is an oblong number). %K A220186 nonn,easy %O A220186 1,2 %A A220186 _Alex Ratushnyak_, Apr 12 2013 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE