A165892 -id:A165892 - cp's OEIS frontend

A214838 Triangular numbers of the form k^2 + 2.

3, 6, 66, 171, 2211, 5778, 75078, 196251, 2550411, 6666726, 86638866, 226472403, 2943171003, 7693394946, 99981175206, 261348955731, 3396416785971, 8878171099878, 115378189547778, 301596468440091, 3919462027838451, 10245401755863186, 133146330756959526, 348042063230908203

Offset: 1

Alex Ratushnyak, Mar 07 2013

Corresponding k values are in A077241.

Except 3, all terms are in A089982: in fact, a(2) = 3+3 and a(n) = (k-2)*(k-1)/2+(k+1)*(k+2)/2, where k = sqrt(a(n)-2) > 2 for n > 2. [Bruno Berselli, Mar 08 2013]

			2211 is in the sequence because 2211 = 47^2 + 2.

Index entries for linear recurrences with constant coefficients, signature (1,34,-34,-1,1).

Cf. A000217, A001110, A001219, A006454, A029549, A069017, A077241, A089982, A164055, A165892.

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(-3*(x^4+x^3-14*x^2+x+1)/((x-1)*(x^2-6*x+1)*(x^2+6*x+1)))); // Bruno Berselli, Mar 08 2013

LinearRecurrence[{1, 34, -34, -1, 1}, {3, 6, 66, 171, 2211}, 25] (* Bruno Berselli, Mar 08 2013 *)

t[n]:=((5-2*sqrt(2))*(1+(-1)^n*sqrt(2))^(2*floor(n/2))+(5+2*sqrt(2))*(1-(-1)^n*sqrt(2))^(2*floor(n/2))-2)/4$
makelist(expand(t[n]*(t[n]+1)/2), n, 1, 25); /* Bruno Berselli, Mar 08 2013 */

for(n=1, 10^9, t=n*(n+1)/2; if(issquare(t-2), print1(t,", "))); \\ Joerg Arndt, Mar 08 2013

import math
for i in range(2, 1<<32):
      t = i*(i+1)//2 - 2
      sr = int(math.sqrt(t))
      if sr*sr == t:
          print(f'{sr:10} {i:10} {t+2}')

G.f.: -3*x*(x^4+x^3-14*x^2+x+1)/((x-1)*(x^2-6*x+1)*(x^2+6*x+1)). - Joerg Arndt, Mar 08 2013

a(n) = A000217(t), where t = ((5-2*sqrt(2))*(1+(-1)^n*sqrt(2))^(2*floor(n/2))+(5+2*sqrt(2))*(1-(-1)^n*sqrt(2))^(2*floor(n/2))-2)/4. - Bruno Berselli, Mar 08 2013

A165893 Numbers n with property that n(n+2)(n+4) is a triangular number.

Original entry on oeis.org

0, 1, 3, 11, 30, 40, 122, 130, 1972

Offset: 1

Zak Seidov, Sep 29 2009

Abscissas of integral points on the elliptic curve: m^2 = 8*n^3 + 48*n^2 + 64*n + 1.

Cf. A001219 Triangular numbers of form a(a+1)(a+2), A165892 Triangular numbers of form a(a+2)(a+4).

TNQ[n_]:=IntegerQ[Sqrt[1+8n]];Select[Range[7500],TNQ[ #(#+2)(#+4)]&]

fini, full keywords from Max Alekseyev, Oct 01 2009

Initial 0 added by Zak Seidov, Oct 04 2009 at the suggestion of Alexander R. Povolotsky.

A226500 Triangular numbers representable as 3 * x^2.

Original entry on oeis.org

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

Offset: 1

Alex Ratushnyak, Jun 09 2013

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

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

Cf. A000217, A001219, A165892, A069017.

#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;
}

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 *)

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

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

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A214838 Triangular numbers of the form k^2 + 2.

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A165893 Numbers n with property that n(n+2)(n+4) is a triangular number.

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A226500 Triangular numbers representable as 3 * x^2.

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