A001219 -id:A001219 - cp's OEIS frontend

A097571 Numbers n such that triangular number t(n) (see A000217) = n(n+1)/2 is a product of three consecutive integers.

0, 3, 15, 20, 44, 608, 22736

Offset: 1

N. J. A. Sloane, Aug 29 2004

Replacing "three" by "two" we get A001652.
Replacing "three" by "N" we get: {15} for N = 4 and 5, {2079} for N = 6 and no solutions for N >= 7. - J. B. M. Melissen.
t(a(n)) equals x*(x+1)*(x+2) for x = [0, 1, 4, 5, 9, 56, 636]n. - _Zak Seidov, Jun 21 2013

S. P. Mohanty, Which triangular numbers are products of three consecutive integers?, Acta Mathematica Hungarica 1991, Volume 58, Issue 1-2, pp 31-36.

Cf. A000217, A001219, A001652.

(Sqrt[8#+1]-1)/2&/@Select[Table[n(n+1)(n+2),{n,0,23000}],OddQ[Sqrt[8#+1]]&] (* Harvey P. Dale, Jan 12 2023 *)

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

Original entry on oeis.org

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

A165892 Triangular numbers of form n(n+2)(n+4).

Original entry on oeis.org

0, 15, 105, 2145, 32640, 73920, 1906128, 2299440, 7692030528

Offset: 1

Zak Seidov, Sep 29 2009

Values of (m^2-1)/8 corresponding to ordinates of integral points on the elliptic curve: m^2 = 8*n^3 + 48*n^2 + 64*n + 1.

Corresponding values of n are given in A165893.

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

TNQ[m_]:=IntegerQ[Sqrt[1+8*m]];Do[If[TNQ[m=n*(n+2)*(n+4)],Print[m]],{n,2*10^3}]
Select[Table[n(n+2)(n+4),{n,0,2000}],OddQ[Sqrt[8#+1]]&] (* Harvey P. Dale, Feb 07 2015 *)

"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.

A227027 Triangular numbers representable as x!/y! with y < x-1.

Original entry on oeis.org

6, 120, 210, 990, 7140, 185136, 242556, 2162160, 8239770, 258474216, 279909630, 9508687656, 323015470680, 10973017315470, 372759573255306, 12662852473364940, 430164224521152660, 14612920781245825506, 496409142337836914550

Offset: 1

Alex Ratushnyak, Jun 27 2013

nonn

Triangular numbers in A045619, except A045619(1)=0. The sequence is infinite because A029549 is a subsequence. According to Melissen's comment in A097571,  y > x-7.
The sequence of x's producing a(n): A227026.
a(2) and a(3) have two representations:
a(2) = 120 = 5*4*3*2 = 6*5*4.
a(3) = 210 = 7*6*5 = 15*14.

			990 is in the sequence since 990 = 11!/8! = 11*10*9 is a ratio of factorials and 990 = (44)(44 + 1)/2 is a triangular number.

Cf. A000217, A001219, A029549, A045619, A097571, A227026.

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.

A165519 Integers k for which k(k+1)(k+2) is a triangular number.

Original entry on oeis.org

-2, -1, 0, 1, 4, 5, 9, 56, 636

Offset: 1

Ant King, Sep 28 2009

This sequence is complete; there are no other integers k for which k(k+1)(k+2) is a triangular number.

Integers k such that 8*k*(k+1)*(k+2)+1 is a square. - Robert Israel, Nov 07 2014

			The third triangular number which is a product of three consecutive integers is 4*5*6=120=T(15), but 4 is the fifth integer k for which k(k+1)(k+2) is a triangular number, so a(5)=4.

R. K. Guy, "Figurate Numbers", D3 in Unsolved Problems in Number Theory, 2nd ed., New York, Springer-Verlag, 1994, p. 148.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 19.

S. P. Mohanty, Which triangular numbers are products of three consecutive integers?, Acta Math Hung 58, 31-36 (1991).

Cf. A000217, A001219.

[-2,-1] cat [n: n in [0..1000] | IsSquare(8*n^3+24*n^2 +16*n+1)]; // Vincenzo Librandi, Nov 10 2014

select(x -> issqr(8*x^3 + 24*x^2 + 16*x+1), [$-2..1000]); # Robert Israel, Nov 07 2014

TriangularNumberQ[k_]:=If[IntegerQ[1/2 (Sqrt[1+8k]-1)],True,False]; Select[Range[750],TriangularNumberQ[ # (#+1)(#+2)] &]
With[{nos=Partition[Range[0,1000],3,1]},Transpose[Select[nos, IntegerQ[ (Sqrt[1+8Times@@#]-1)/2]&]][[1]]] (* Harvey P. Dale, Dec 25 2011 *)

isok(k) = ispolygonal(k*(k+1)*(k+2), 3); \\ Michel Marcus, Oct 31 2014

Initial 0 added by Alexander R. Povolotsky, Sep 29 2009

Initial -2 and -1 added by Alex Ratushnyak, Nov 07 2014

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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A097571 Numbers n such that triangular number t(n) (see A000217) = n(n+1)/2 is a product of three consecutive integers.

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

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A165892 Triangular numbers of form n(n+2)(n+4).

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A227027 Triangular numbers representable as x!/y! with y < x-1.

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

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A165519 Integers k for which k(k+1)(k+2) is a triangular number.

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A226500 Triangular numbers representable as 3 * x^2.

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