A217758 Triangular numbers of the form k^2 + k - 1.

1, 55, 1891, 64261, 2183005, 74157931, 2519186671, 85578188905, 2907139236121, 98757155839231, 3354836159297755, 113965672260284461, 3871478020690373941, 131516287031212429555, 4467682281040532230951, 151769681268346883422801

Offset: 1

Author

Alex Ratushnyak, Mar 23 2013

Keywords

• nonn
• easy

Comments

Triangular numbers belonging to A028387. - Bruno Berselli, Apr 18 2018

Since this sequence lists triangular numbers t such that 4*t + 5 = u^2 and 8*t + 1 = v^2 by definition of triangular numbers, 2*u^2 - 9 = 2*(4*t + 5) - 9 = 8*t + 1 = v^2, that is, sqrt(4*a(n) + 5) is in A075841. - Altug Alkan, Apr 18 2018

Examples

			Triangular(10) = 55 = 7^2 + 7 - 1, so 55 is in the sequence.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Crossrefs

Cf. A000217, A028387, A069017, A075841.

Programs

C

#include 
typedef unsigned long long U64;
U64 rootPronic(U64 a) {
    U64 sr = 1L<<32, s, b;
    while (a < sr*(sr-1))  sr>>=1;
    for (b = sr>>1; b; b>>=1) {
        s = sr+b;
        if (a >= s*(s-1))  sr = s;
    }
    return sr;
}
int main() {
  U64 a, i, t;
  for (i=0; i < 1L<<32; ++i) {
      a = i*(i+1)/2 + 1;
      t = rootPronic(a);
      if (a == t*(t-1))  printf("%llu %llu %llu\n", i, t, a-1);
  }
  return 0;
}

Mathematica

a[1]=1; a[2]=55; a[3]=1891; a[n_] := a[n] = 35*a[n-1] - 35*a[n-2] + a[n-3]; Array[a,20] (* Giovanni Resta, Mar 24 2013 *)
Table[Floor@(9 (17 + Sqrt@ 288)^n*(3 - Sqrt@ 8)/32), {n, 0, 16}] (* or *)
CoefficientList[Series[-x (1 + 20 x + x^2)/((x - 1) (x^2 - 34 x + 1)), {x, 0, 16}], x] (* Michael De Vlieger, Oct 08 2016 *)

PARI

Vec(x*(1+20*x+x^2)/(1-35*x+35*x^2-x^3)+O(x^66)) \\ Joerg Arndt, Mar 25 2013

Formula

From Giovanni Resta, Mar 24 2013: (Start)

G.f.: -x*(1 + 20*x + x^2) / ( (x - 1)*(x^2 - 34*x + 1) ).

a(n) = floor(9 * (17 + sqrt(288))^n * (3 - sqrt(8))/32). (End)

a(n) = (A075841(n)^2 - 5)/4. - Altug Alkan, Apr 18 2018