A226500 Triangular numbers representable as 3 * x^2.

0, 3, 300, 29403, 2881200, 282328203, 27665282700, 2710915376403, 265642041604800, 26030209161894003, 2550694855824007500, 249942065661590841003, 24491771739980078410800, 2399943688452386093417403, 235169989696593857076494700, 23044259046577745607403063203

Offset: 1

Author

Alex Ratushnyak, Jun 09 2013

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 1..503
• Index entries for linear recurrences with constant coefficients, signature (99,-99,1).

Crossrefs

Cf. A029549 (triangular numbers representable as x^2 + x).

Cf. A000217, A001219, A165892, A069017.

Programs

C

#include 
#include 
typedef unsigned long long U64;
U64 isTriangular(U64 a) {   // input must be < 1ULL<<63
    U64 r = sqrt(a*2);
    return (r*(r+1) == a*2);
}
int main() {
  for (U64 j, i = 0; (j=i*i*3) < (1ULL<<63); i++)
      if (isTriangular(j)) printf("%llu, ", j);
  return 0;
}

Mathematica

a[1]=0; a[2]=3; a[3]=300; a[n_] := a[n] = 99*(a[n-1] - a[n-2]) + a[n-3]; Array[a, 10] (* Giovanni Resta, Jun 09 2013 *)
Rest@ CoefficientList[Series[3 x^2 (1 + x)/((1 - x) (1 - 98 x + x^2)), {x, 0, 16}], x] (* or *)
3 LinearRecurrence[{99, -99, 1}, {0, 1, 100}, 16] (* Michael De Vlieger, Mar 03 2016, latter after Vincenzo Librandi at A108741 *)

Formula

a(n) = 99*a(n-1) - 99*a(n-2) + a(n-3), for n > 3. a(n) = floor((49 + 20*sqrt(6))^(n-1)/32). - Giovanni Resta, Jun 09 2013

G.f.: 3*x^2*(1+x)/((1-x)*(1-98*x+x^2)); a(n)=3*A108741(n-1). - Joerg Arndt, Jun 10 2013

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))/32. - Colin Barker, Mar 03 2016

Extensions

a(12)-a(15) from Giovanni Resta, Jun 09 2013