A085780 -id:A085780 - cp's OEIS frontend

A188630 Triangular numbers that are the product of two triangular numbers greater than 1.

36, 45, 210, 630, 780, 990, 1540, 2850, 3570, 4095, 4851, 8778, 11781, 15400, 17955, 19110, 21528, 25200, 26565, 26796, 33930, 37128, 40755, 43956, 61425, 61776, 70125, 79800, 105570, 113050, 122265, 145530, 176715, 189420, 192510, 246753, 270480, 303810, 349866, 437580, 500500, 526851

Offset: 1

T. D. Noe, Apr 06 2011

nonn

For squares, it is a simple matter to find squares that are the product of squares greater than 1. Is there a simple procedure for triangular numbers? That is, given n, is it easy to determine whether T(n) is the product of T(i) * T(j) for some i,j > 1?
Breiteig mentions this problem, but does not solve it. The problem can be extended to any polygonal number; for example, when is a pentagonal number the product of two pentagonal numbers? See A188660 and A188663 for the oblong and pentagonal cases.
Sequence A001571 gives the indices of triangular numbers that are 3 times another triangular number. For example, A001571(4) is 132; T(132) is 8778, which equals 3*T(76). Note that A061278 is the companion sequence, whose 4th term is 76. As with the oblong numbers covered by Breiteig, the triangular numbers in this sequence appear to satisfy linear recursions.

			210 = T(20) = 10 * 21 = T(4) * T(6).

Donovan Johnson, Table of n, a(n) for n = 1..4082
Trygve Breiteig, When is the product of two oblong numbers another oblong?, Math. Mag. 73 (2000), 120-129.

Cf. A000217 (triangular numbers), A085780 (products of two triangular numbers), A140089 (products of two triangular numbers > 1).
Subsequence of A068143 (more than 2 factors allowed).
Cf. A001571, A061278, A188660, A188663.
See also A379609.

A188630 := proc(limit) local t,E,n,k,c,b,ist; E:=NULL;
t := proc(n) option remember; iquo(n*(n+1), 2) end;
ist := proc(n) option remember; n = t(floor(sqrt(2*n))) end;
for n from 2 do
    c := t(n); if c > limit then break fi;
    for k from 2 do
        b := c*t(k); if b > limit then break fi;
        if ist(b) then E := E, b fi;
od od; sort({E}) end:
A188630(200000); # Peter Luschny, Dec 21 2017

TriangularQ[n_] := IntegerQ[Sqrt[1 + 8 n]]; TriIndex[n_] := Floor[(-1 + Sqrt[1 + 8*n])/2]; lim = 10^6; nMax = TriIndex[lim/3]; tri = Table[n (n + 1)/2, {n, 2, nMax}]; Union[Reap[Do[num = tri[[i]]*tri[[j]]; If[TriangularQ[num], Sow[num]], {i, TriIndex[Sqrt[lim]]}, {j, i, TriIndex[lim/tri[[i]]] - 1}]][[2, 1]]]
Module[{upto=530000,maxr},maxr=Ceiling[(Sqrt[1+8*Ceiling[upto/3]]-1)/2]; Union[Select[Times@@@Tuples[Rest[Accumulate[Range[maxr]]],2], IntegerQ[ Sqrt[1+8#]]&&#<=upto&]]] (* Harvey P. Dale, Jun 12 2012 *)

A072389 Numbers of the form x(x+1) y*(y+1) ("bipronics") with x and y nonnegative integers.

Original entry on oeis.org

0, 4, 12, 24, 36, 40, 60, 72, 84, 112, 120, 144, 180, 220, 240, 252, 264, 312, 336, 360, 364, 400, 420, 432, 480, 504, 540, 544, 600, 612, 660, 672, 684, 760, 792, 840, 864, 900, 924, 936, 1012, 1080, 1092, 1104, 1120, 1200, 1260, 1300, 1320

Offset: 1

Stuart M. Ellerstein (ellerstein(AT)aol.com), Jul 20 2002

Nonnegative numbers k = a*b = c*d, where a+b = c+d+1. - Yifan Xie, Jun 28 2024

			a(3) = 1*2*2*3 = 12.

John Drake, Table of n, a(n) for n = 1..10000

Cf. A053990 (sequence of bipronics with x and y distinct).

Cf. A085780 (one quarter of this).

A003056(n)=(sqrtint(8*n+1)-1)\2
list(lim)=my(v=List([0]), t); for(a=1, A003056(lim\4), t=a*(a+1); for(b=a, A003056(lim\t\2), listput(v, b*(b+1)*t))); Set(v) \\ Charles R Greathouse IV, Jul 11 2024

More terms from James Sellers, Jul 23 2002

A085782 Numbers that can be written as a product of triangular numbers.

Original entry on oeis.org

0, 1, 3, 6, 9, 10, 15, 18, 21, 27, 28, 30, 36, 45, 54, 55, 60, 63, 66, 78, 81, 84, 90, 91, 100, 105, 108, 120, 126, 135, 136, 150, 153, 162, 165, 168, 171, 180, 189, 190, 198, 210, 216, 225, 231, 234, 243, 252, 253, 270, 273, 276, 280, 300, 315, 324, 325, 330, 351

Offset: 1

Jon Perry, Jul 23 2003

nonn

These are the numbers that appear at least once in A007425 (the number of ordered factorizations of n as n = r*s*t). - Matthew Vandermast, Jul 26 2003
Contains A085780 as a subsequence (exactly 2 factors); 27,54,81,... are the first elements in the complement. - M. F. Hasler, Apr 03 2008
The number of r-orthotopes of any size contained in a n1*n2*...*nr r-orthotope, where a(n)=t(n1)*t(n2)*...*t(nr) and t(k)=A000217(k). - Dimitri Boscainos, Aug 27 2015

			54 = 6*3*3 = t(3)*t(2)*t(2).

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000217, A007425, A085780, A334115.

More terms from Max Alekseyev and Jon E. Schoenfield, Sep 04 2009

A104473 a(n) = binomial(n+2,2)*binomial(n+6,2).

Original entry on oeis.org

15, 63, 168, 360, 675, 1155, 1848, 2808, 4095, 5775, 7920, 10608, 13923, 17955, 22800, 28560, 35343, 43263, 52440, 63000, 75075, 88803, 104328, 121800, 141375, 163215, 187488, 214368, 244035, 276675, 312480, 351648, 394383, 440895, 491400, 546120, 605283, 669123

Offset: 0

Zerinvary Lajos, Apr 18 2005

			a(0) = C(0+2,2)*C(0+6,2) = C(2,2)*C(6,2) = 1*15 = 155.
a(6) = 1*3*5 + 2*4*6 + 3*5*7 + 4*6*8 + 5*7*9 + 6*8*10 + 7*9*11 = 1848.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A000579, A033275, A034856, A062264.

Subsequence of A085780.

[Binomial(n+2, 2)*Binomial(n+6, 2): n in [0..50]]; // Vincenzo Librandi, Apr 28 2014

f[n_] := Binomial[n + 2, 2] Binomial[n + 6, 2]; Table[f[n], {n,0,40}] (* Robert G. Wilson v, Apr 20 2005 *)
CoefficientList[Series[3 (5-4*x+x^2)/(1-x)^5, {x,0,40}], x] (* Vincenzo Librandi, Apr 28 2014 *)

a(n)=binomial(n+2,2)*binomial(n+6,2) \\ Charles R Greathouse IV, Jun 07 2013

def A104473(n): return binomial(n+2,2)*binomial(n+6,2)
print([A104473(n) for n in range(51)]) # G. C. Greubel, Mar 05 2025

a(n) = (1/4)*(n+1)*(n+2)*(n+5)*(n+6).
a(n) = A034856(n+2)^2 - 1. - J. M. Bergot, Dec 14 2010
G.f.: 3*(5-4*x+x^2)/(1-x)^5. - Colin Barker, Sep 21 2012
a(n) = Sum_{i=1..n+1} i*(i+2)*(i+4). - Bruno Berselli, Apr 28 2014
a(n) = A000217(n)*A000217(n+4) = 3*A033275(n+4). - R. J. Mathar, Nov 29 2015
From Amiram Eldar, Aug 30 2022: (Start)
Sum_{n>=0} 1/a(n) = 43/450.
Sum_{n>=0} (-1)^n/a(n) = 16*log(2)/15 - 154/225. (End)
From G. C. Greubel, Mar 05 2025: (Start)
a(n) = 90*A000579(n+6)/A000279(n+3).
E.g.f.: (1/4)*(60 + 192*x + 114*x^2 + 20*x^3 + x^4)*exp(x). (End)

A104676 a(n) = binomial(n+2,2) * binomial(n+7,2).

Original entry on oeis.org

21, 84, 216, 450, 825, 1386, 2184, 3276, 4725, 6600, 8976, 11934, 15561, 19950, 25200, 31416, 38709, 47196, 57000, 68250, 81081, 95634, 112056, 130500, 151125, 174096, 199584, 227766, 258825, 292950, 330336, 371184, 415701, 464100, 516600, 573426, 634809, 700986

Offset: 0

Zerinvary Lajos, Apr 22 2005

			If n=0 then C(0+2,0+0)*C(0+7,2) = C(2,0)*C(7,2) = 1*21 = 21.
If n=8 then C(8+2,8+0)*C(8+7,2) = C(10,8)*C(15,2) = 45*105 = 4725.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A062190.

Subsequence of A085780.

A104676:= func< n | Binomial(n+2,2)*Binomial(n+7,2) >;
[A104676(n): n in [0..50]]; // G. C. Greubel, Mar 01 2025

A104676:=n->binomial(n+2,2)*binomial(n+7,2): seq(A104676(n), n=0..50); # Wesley Ivan Hurt, Mar 30 2017

Table[Binomial[n + 2, 2] Binomial[n + 7, 2], {n, 0, 37}] (* Michael De Vlieger, Nov 29 2015 *)

a(n) = binomial(n+2,2)*binomial(n+7,2); \\ Michel Marcus, Nov 29 2015

def A104676(n): return binomial(n+2,2)*binomial(n+7,2)
print([A104676(n) for n in range(51)]) # G. C. Greubel, Mar 01 2025

From R. J. Mathar, Nov 29 2015: (Start)
a(n) = A000217(n+1) * A000217(n+6).
G.f.: 3*(7 - 7*x + 2*x^2)/(1-x)^5. (End)
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5). - Wesley Ivan Hurt, Jan 25 2022
From Amiram Eldar, Aug 30 2022: (Start)
Sum_{n>=0} 1/a(n) = 7/100.
Sum_{n>=0} (-1)^n/a(n) = 7/180. (End)
E.g.f.: (1/4)*(84 + 252*x + 138*x^2 + 22*x^3 + x^4)*exp(x). - G. C. Greubel, Mar 01 2025

A105938 a(n) = binomial(n+2,2)*binomial(n+5,2).

Original entry on oeis.org

10, 45, 126, 280, 540, 945, 1540, 2376, 3510, 5005, 6930, 9360, 12376, 16065, 20520, 25840, 32130, 39501, 48070, 57960, 69300, 82225, 96876, 113400, 131950, 152685, 175770, 201376, 229680, 260865, 295120, 332640, 373626, 418285, 466830, 519480, 576460

Offset: 0

Zerinvary Lajos, Apr 27 2005

			If n=0 then C(0+2,0)*C(0+5,2) = C(2,0)*C(5,2) = 1*10 = 10.
If n=9 then C(9+2,9)*C(9+5,2) = C(11,9)*C(14,2) = 55*91 = 5005.

Nathaniel Johnston, Table of n, a(n) for n = 0..5000
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 9.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Subsequence of A085780.

Cf. A000096, A000217, A000389, A062145.

A105938:= func< n | 30*Binomial(n+5,5)/(n+3) >;
[A105938(n): n in [0..40]]; // G. C. Greubel, Mar 11 2025

a:= n-> binomial(n+2,n)*binomial(n+5,2):
seq(a(n), n=0..40);  # Alois P. Heinz, Oct 16 2008

Table[n(n+1)(n+3)(n+4)/4, {n, 100}] (* Vladimir Joseph Stephan Orlovsky, Jun 26 2011 *)
Table[Binomial[n + 2, n] Binomial[n + 5, 2], {n, 0, 40}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {10, 45, 126, 280, 540}, 40] (* Harvey P. Dale, Sep 05 2013 *)

def A105938(n): return 30*binomial(n+5,5)//(n+3)
print([A105938(n) for n in range(41)]) # G. C. Greubel, Mar 11 2025

G.f.: (10 - 5*x + x^2)/(1-x)^5. - Alois P. Heinz, Oct 16 2008
a(0)=10, a(1)=45, a(2)=126, a(3)=280, a(4)=540; for n>4, a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Harvey P. Dale, Sep 05 2013
a(n) = A000217(n+1)*A000217(n+4). - R. J. Mathar, Nov 29 2015
a(n) = A000096(n+1)*A000096(n+2). - Bruno Berselli, Sep 21 2016
From Amiram Eldar, Jan 06 2021: (Start)
Sum_{n>=0} 1/a(n) = 5/36.
Sum_{n>=0} (-1)^n/a(n) = 1/12. (End)
From G. C. Greubel, Mar 11 2025: (Start)
a(n) = 30*A000389(n+5)/(n+3).
E.g.f.: (1/4)*(40 + 140*x + 92*x^2 + 18*x^3 + x^4)*exp(x). (End)

A110904 Integers that can be expressed as a product of triangular numbers in 3 different ways.

Original entry on oeis.org

630, 3780, 14850, 16380, 21420, 114660, 131670, 159390, 178200, 234360, 401940, 478800, 523260, 556920, 582120, 712530, 749700, 835380, 1455300, 1753290, 1936935, 2086920, 2162160, 2633400, 2841300, 3828825, 4791150, 5821200, 6056820, 6380010, 6396390, 6486480

Offset: 1

Luke T Pebody (ltp1000(AT)cam.ac.uk), Sep 21 2005

nonn

			835380 = 91*9180 = 153*5460 = 630*1326.

Chai Wah Wu, Table of n, a(n) for n = 1..1007

Subsequence of A264961 and of A085780.

from _future_ import division
mmax = 10**3
tmax, A110904_dict = mmax*(mmax+1)//2, {}
ti = 0
for i in range(1,mmax+1):
    ti += i
    p = ti*i*(i-1)//2
    for j in range(i,mmax+1):
        p += ti*j
        if p <= tmax:
            A110904_dict[p] = A110904_dict[p]+1 if p in A110904_dict else 1
        else:
            break
A110904_list = sorted([i for i in A110904_dict if A110904_dict[i] == 3]) # Chai Wah Wu, Nov 29 2015

a(19)-a(20) from R. J. Mathar, Nov 29 2015

A140089 Products of two triangular numbers each > 1.

Original entry on oeis.org

9, 18, 30, 36, 45, 60, 63, 84, 90, 100, 108, 126, 135, 150, 165, 168, 198, 210, 216, 225, 234, 270, 273, 280, 315, 330, 360, 396, 408, 420, 441, 450, 459, 468, 513, 540, 546, 550, 570, 588, 630, 660, 675, 693, 720, 756, 759, 780, 784, 816, 825, 828, 900, 910

Offset: 1

N. J. A. Sloane, Jun 27 2008

nonn

Cf. A000217. Subsequence of A085780.

isA140089 := proc(n)
    for d in numtheory[divisors](n) minus {1} do
        if d^2 > n then
            return false;
        end if;
        if isA000217(d) then
            if isA000217(n/d) then
                return true;
            end if;
        end if;
    end do:
    return false;
end proc:
for n from 1 to 1000 do
    if isA140089(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jul 31 2024

More terms from Rick L. Shepherd and John W. Layman, Jul 03 2008

A225440 Triangular numbers that are the product of three distinct triangular numbers greater than 1.

Original entry on oeis.org

378, 630, 990, 3240, 4095, 4950, 5460, 9180, 15400, 19110, 25200, 31878, 37128, 37950, 39060, 52650, 61425, 79800, 97020, 103740, 105570, 122265, 145530, 157080, 161028, 176715, 192510, 221445, 265356, 288420, 304590, 306936, 346528, 437580, 500500, 545490, 583740

Offset: 1

Alex Ratushnyak, May 08 2013

nonn

Triangular numbers of the form triangular(x) * triangular(y) * triangular(z), x > y > z > 1.

			378 = 3 * 6 * 21.
630 = 3 * 10 * 21.
990 = 3 * 6 * 55.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A000217, A085780, A140089, A188630, A225390.

#include 
typedef unsigned long long U64;
U64 isTriangular(U64 a) {  // ! Must be a < (1<<63)
    U64 s = sqrt(a*2);
    if (a>=(1ULL<<63)) exit(1);
    return (s*(s+1)/2 == a);
}
int compare64(const void *p1, const void *p2) {
  if (*(U64*)p1 == *(U64*)p2) return 0;
  if (*(U64*)p1 < *(U64*)p2) return -1;
  return 1;
}
#define TOP (1<<21)
U64 d[TOP];
int main() {
  U64 c, x, tx, y, ty, z, tz, p = 0;
    for (x = tx = 3; tx <= TOP; tx+=x, ++x) {
    for (y = ty = 3; ty < tx;   ty+=y, ++y) {
    for (z = tz = 3; tz < ty;   tz+=z, ++z) {
    c = tx*ty*tz;
    if (c <= TOP*18 && isTriangular(c))  d[p++] = c;
  }}}
  qsort(d, p, 8, compare64);
  for (x=c=0; cx) printf("%llu, ", y), x=y;
  return 0;
}

A264961 Numbers that are products of two triangular numbers in more than one way.

Original entry on oeis.org

36, 45, 210, 315, 360, 630, 780, 990, 1260, 1386, 1540, 1800, 2850, 2970, 3510, 3570, 3780, 4095, 4788, 4851, 6300, 7920, 8415, 8550, 8778, 9450, 11700, 11781, 14850, 15400, 15561, 16380, 17640, 17955, 18018, 18648, 19110, 20790, 21420, 21450, 21528, 25116, 25200, 26565, 26775, 26796, 27720, 28980

Offset: 1

R. J. Mathar, Nov 29 2015

nonn

One of the factors in the product may be 1 = A000217(1). We count the ways of writing n = A000217(i)*A000217(j) with i <= j, unordered factorizations.

			36 = 1*36 = 6*6. 45 = 1*45 = 3*15. 210 = 1*210 = 10*21. 315 = 3*105 = 15*21. 360 = 3*120 = 10*36. 630 = 1*630 = 3*210 = 6*105. 3780= 6*360 = 10 * 378 = 36*105.

Chai Wah Wu, Table of n, a(n) for n = 1..10602

Subsequence of A085780. A188630 and A110904 are subsequences of this.

A264961ct := proc(n)
    local ct,d ;
    ct := 0 ;
    for d in numtheory[divisors](n) do
        if d^2 > n then
            return ct;
        end if;
        if isA000217(d) then
            if isA000217(n/d) then
                ct := ct+1 ;
            end if;
        end if;
    end do:
    return ct;
end proc:
for n from 1 to 30000 do
    if A264961ct(n) > 1 then
        printf("%d,",n) ;
    end if;
end do:

lim = 10000; t = Accumulate[Range@lim]; f[n_] := Select[{#, n/#} & /@ Select[Divisors@ n, # <= Sqrt@ n && MemberQ[t, #] &], MemberQ[t, Last@ #] &]; Select[Range@ lim, Length@ f@ # == 2 &] (* Michael De Vlieger, Nov 29 2015 *)

from _future_ import division
mmax = 10**3
tmax, A264961_dict = mmax*(mmax+1)//2, {}
ti = 0
for i in range(1,mmax+1):
    ti += i
    p = ti*i*(i-1)//2
    for j in range(i,mmax+1):
        p += ti*j
        if p <= tmax:
            A264961_dict[p] = 2 if p in A264961_dict else 1
        else:
            break
A264961_list = sorted([i for i in A264961_dict if A264961_dict[i] > 1]) # Chai Wah Wu, Nov 29 2015

cp's OEIS Frontend

A188630 Triangular numbers that are the product of two triangular numbers greater than 1.

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A072389 Numbers of the form x*(x+1) * y*(y+1) ("bipronics") with x and y nonnegative integers.

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A085782 Numbers that can be written as a product of triangular numbers.

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A104473 a(n) = binomial(n+2,2)*binomial(n+6,2).

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A104676 a(n) = binomial(n+2,2) * binomial(n+7,2).

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A105938 a(n) = binomial(n+2,2)*binomial(n+5,2).

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A110904 Integers that can be expressed as a product of triangular numbers in 3 different ways.

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A072389 Numbers of the form x(x+1) y*(y+1) ("bipronics") with x and y nonnegative integers.