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A220185 Numbers n such that n^2 + n(n+1) is an oblong number (A002378).

%I A220185 #38 Sep 08 2022 08:46:04
%S A220185 0,10,348,11830,401880,13652098,463769460,15754509550,535189555248,
%T A220185 18180690368890,617608282987020,20980500931189798,712719423377466120,
%U A220185 24211479893902658290,822477596969312915748,27940026817062736477150,949138434183163727307360
%N A220185 Numbers n such that n^2 + n(n+1) is an oblong number (A002378).
%C A220185 Also numbers n such that the sum of the hexagonal numbers H(n) and H(n+1) is equal to m^2 + (m+1)^2 for some m. - _Colin Barker_, Dec 10 2014
%C A220185 Also nonnegative integers x in the solutions to 4*x^2-2*y^2+2*x-2*y = 0, the corresponding values of y being A251867. - _Colin Barker_, Dec 10 2014
%H A220185 Colin Barker, <a href="/A220185/b220185.txt">Table of n, a(n) for n = 1..654</a>
%H A220185 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (35,-35,1).
%F A220185 For n>1, a(n) = A089928(n*4-5).
%F A220185 From _Bruno Berselli_, Apr 12 2013: (Start)
%F A220185 G.f.: 2*x^2*(5-x)/((1-x)*(1-34*x+x^2)).
%F A220185 a(n) = ((1+sqrt(2))^(4n-3)+(1-sqrt(2))^(4n-3)-2)/8.
%F A220185 a(n+2) = 10*A029546(n)-2*A029546(n-1). (End)
%F A220185 a(n) = 35*a(n-1)-35*a(n-2)+a(n-3). - _Colin Barker_, Dec 10 2014
%F A220185 a(n) = A251867(n) - A001542(n-1)^2. - _Alexander Samokrutov_, Sep 05 2015
%e A220185 a(3) = A089928(7) = 348.
%p A220185 f:= gfun:-rectoproc({a(n)=35*(a(n-1)-a(n-2))+a(n-3),a(1)=0,a(2)=10,a(3)=348},a(n),remember):
%p A220185 map(f, [$1..50]); # _Robert Israel_, Sep 06 2015
%t A220185 LinearRecurrence[{35, - 35, 1}, {0, 10, 348}, 20] (* _Vincenzo Librandi_, Sep 06 2015 *)
%o A220185 (C)
%o A220185 #include <stdio.h>
%o A220185 #include <math.h>
%o A220185 typedef unsigned long long U64;
%o A220185 U64 rootPronic(U64 a) {
%o A220185     U64 sr = 1L<<31, s, b;
%o A220185     if (a < sr*(sr+1)) {
%o A220185           sr>>=1;
%o A220185           while (a < sr*(sr+1))  sr>>=1;
%o A220185     }
%o A220185     for (b = sr>>1; b; b>>=1) {
%o A220185             s = sr+b;
%o A220185             if (a >= s*(s+1))  sr = s;
%o A220185     }
%o A220185     return sr;
%o A220185 }
%o A220185 int main() {
%o A220185   U64 a, n, r, t;
%o A220185   for (n=0; n < (1L<<31); n++) {
%o A220185     a = (n*(n+1)) + n*n;
%o A220185     t = rootPronic(a);
%o A220185     if (a == t*(t+1))  printf("%llu\n", n);
%o A220185   }
%o A220185 }
%o A220185 (PARI) concat(0, Vec(2*x^2*(5-x)/((1-x)*(1-34*x+x^2))+O(x^100))) \\ _Colin Barker_, Dec 10 2014
%o A220185 (Magma) [Floor(((1+Sqrt(2))^(4*n-3)+(1-Sqrt(2))^(4*n-3)-2)/8): n in [1..20]]; // _Vincenzo Librandi_, Sep 08 2015
%Y A220185 Cf. A002378, A014105 (n^2 + n(n+1)), A029546, A084703 (numbers n such that n^2 + n(n+1) is a square).
%Y A220185 Cf. A000290, A000384, A251867.
%K A220185 nonn,easy
%O A220185 1,2
%A A220185 _Alex Ratushnyak_, Apr 12 2013
%E A220185 More terms from _Bruno Berselli_, Apr 12 2013