A052343 -id:A052343 - cp's OEIS frontend

A020756 Numbers that are the sum of two triangular numbers.

0, 1, 2, 3, 4, 6, 7, 9, 10, 11, 12, 13, 15, 16, 18, 20, 21, 22, 24, 25, 27, 28, 29, 30, 31, 34, 36, 37, 38, 39, 42, 43, 45, 46, 48, 49, 51, 55, 56, 57, 58, 60, 61, 64, 65, 66, 67, 69, 70, 72, 73, 76, 78, 79, 81, 83, 84, 87, 88, 90, 91, 92, 93, 94, 97, 99, 100, 101, 102, 105, 106, 108

Offset: 1

David W. Wilson

The possible sums of a square and a promic, i.e., x^2+n(n+1), e.g., 3^2 + 2*3 = 9 + 6 = 15 is present. - Jon Perry, May 28 2003
A052343(a(n)) > 0; union of A118139 and A119345. - Reinhard Zumkeller, May 15 2006
Also union of A051533 and A000217. - Ant King, Nov 29 2010

T. D. Noe, Table of n, a(n) for n = 1..10000
John A. Ewell, On Sums of Triangular Numbers and Sums of Squares, The American Mathematical Monthly, 99:8 (October 1992), pp. 752-757.
T. Khovanova, K. Knop, and A. Radul, Baron Munchhausen's Sequence, J. Int. Seq. 13 (2010) # 10.8.7.
L. K. Mork, Keith Sullivan, Trenton Vogt, and Darin J. Ulness, A group theoretical approach to the partitioning of integers: Application to triangular numbers, squares, and centered polygonal numbers, Australasian J. Comb. (2021) Vol. 80, No. 3, 305-321.

Complement of A020757.
Cf. A051533 (sums of two positive triangular numbers), A001481 (sums of two squares), A002378, A000217.
Cf. A052343.

a020756 n = a020756_list !! (n-1)
a020756_list = filter ((> 0) . a052343) [0..]
-- Reinhard Zumkeller, Jul 25 2014

q[k_] := If[! Head[Reduce[m (m + 1) + n (n + 1) == 2 k && 0 <= m && 0 <= n, {m, n}, Integers]] === Symbol, k, {}]; DeleteCases[Table[q[i], {i, 0, 108}], {}] (* Ant King, Nov 29 2010 *)
Take[Union[Total/@Tuples[Accumulate[Range[0,20]],2]],80] (* Harvey P. Dale, May 02 2012 *)

v=vector(200); vc=0; for (x=0,10, for (y=0,10,v[vc++ ]=x^2+y*(y+1))); v=vecsort(v); v

is(n)=my(f=factor(4*n+1));for(i=1,#f~,if(f[i,1]%4==3 && f[i,2]%2, return(0))); 1 \\ Charles R Greathouse IV, Jul 05 2013

Numbers n such that 4n+1 is the sum of two squares, i.e. such that 4n+1 is in A001481. Hence n is a member if and only if 4n+1 = odd square * product of distinct primes of form 4k+1. (Fred Helenius and others, Dec 18 2004)
Equivalently, we may say that a positive integer n can be partitioned into a sum of two triangular numbers if and only if every 4 k + 3 prime factor in the canonical form of 4 n + 1 occurs with an even exponent. - Ant King, Nov 29 2010
Also, the values of n for which 8n+2 can be partitioned into a sum of two squares of natural numbers. - Ant King, Nov 29 2010
Closed under the operation f(x, y) = 4*x*y + x + y.

Entry revised by N. J. A. Sloane, Dec 20 2004

A053603 Number of ways to write n as an ordered sum of two nonzero triangular numbers.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 1, 2, 0, 2, 0, 2, 1, 2, 0, 0, 4, 0, 2, 0, 1, 2, 2, 0, 2, 2, 0, 2, 0, 2, 1, 4, 0, 0, 2, 0, 2, 2, 2, 2, 0, 0, 3, 2, 0, 0, 4, 0, 2, 2, 0, 4, 0, 0, 0, 2, 3, 2, 2, 0, 2, 2, 0, 0, 2, 2, 2, 2, 0, 2, 2, 0, 3, 2, 0, 0, 4, 0, 0, 2, 0, 6, 0, 2, 2, 0, 0, 2, 2, 0, 1, 2, 2

Offset: 0

N. J. A. Sloane, Jan 20 2000

nonn

a(A051611(n)) = 0; A051533(a(n)) > 0. - Reinhard Zumkeller, Jun 27 2013

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A000217, A007294, A051611, A051533, A052343-A052348, A053604.

Cf. A010054.

a053603 n = sum $ map (a010054 . (n -)) $
                  takeWhile (< n) $ tail a000217_list
-- Reinhard Zumkeller, Jun 27 2013

nmax = 100; m0 = 10; A053603 := Table[a[n], {n, 0, nmax}]; Clear[counts]; counts[m_] := counts[m] = (Clear[a]; a[A053603);%20counts%5Bm%20=%20m0%5D;%20counts%5Bm%20=%202*m%5D;%20While%5B%20counts%5Bm%5D%20!=%20counts%5Bm/2%5D,%20m%20=%202*m%5D;%20A053603%20(*%20_Jean-Fran%C3%A7ois%20Alcover">] = 0; Do[k = i*(i+1)/2 + j*(j+1)/2; a[k] = a[k]+1, {i, 1, m}, {j, 1, m}]; A053603); counts[m = m0]; counts[m = 2*m]; While[ counts[m] != counts[m/2], m = 2*m]; A053603 (* _Jean-François Alcover, Sep 05 2013 *)

istriang(n)={n>0 && issquare(8*n+1);}
a(n) = { my(t=1, ct=0, j=1); while (tJoerg Arndt, Sep 05 2013

G.f.: ( Sum_{k>=1} x^(k*(k+1)/2) )^2. - Ilya Gutkovskiy, Dec 24 2016

a(n) = Sum_{k=1..n-1} c(k) * c(n-k), where c(n) = A010054(n). - Wesley Ivan Hurt, Jan 06 2024

A230121 Number of ways to write n = x + y + z (0 < x <= y <= z) such that x(x+1)/2 + y(y+1)/2 + z*(z+1)/2 is a triangular number.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 1, 2, 1, 1, 0, 2, 1, 2, 1, 2, 3, 2, 2, 6, 1, 3, 5, 1, 2, 3, 5, 2, 1, 3, 3, 3, 4, 3, 8, 2, 5, 11, 2, 5, 8, 4, 6, 4, 9, 4, 6, 5, 4, 6, 3, 8, 8, 5, 8, 10, 7, 7, 11, 8, 6, 7, 8, 5, 9, 7, 6, 8, 7, 7, 8, 13, 9, 11, 10, 7, 22, 9, 10, 13, 3, 6, 10, 8, 17, 12, 7, 9, 10, 16, 6, 18, 18, 10, 15, 9, 12, 20, 5

Offset: 1

Zhi-Wei Sun, Oct 10 2013

nonn

Conjecture: (i) a(n) > 0 except for n = 1, 2, 4, 5, 7, 12. Moreover, for each n = 20, 21, ... there are three distinct positive integers x, y and z with x + y + z = n such that x*(x+1)/2 + y*(y+1)/2 + z*(z+1)/2 is a triangular number.
(ii) A positive integer n cannot be written as x + y + z (x, y, z > 0) with x^2 + y^2 + z^2 a square if and only if n has the form 2^r*3^s or the form 2^r*7, where r and s are nonnegative integers.
(iii) Any integer n > 14 can be written as a + b + c + d, where a, b, c, d are positive integers with a^2 + b^2 + c^2 + d^2 a square. If n > 20 is not among 22, 28, 30, 38, 44, 60, then we may require additionally that a, b, c, d are pairwise distinct.
(iv) For each integer n > 50 not equal to 71, there are positive integers a, b, c, d with a + b + c + d = n such that both a^2 + b^2 and c^2 + d^2 are squares.
Part (ii) and the first assertion in part (iii) were confirmed by Chao Huang and Zhi-Wei Sun in 2021. - Zhi-Wei Sun, May 09 2021

			a(16) = 1 since 16 = 3 + 6 + 7 and 3*4/2 + 6*7/2 + 7*8/2 = 55 = 10*11/2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..3000
Chao Huang and Zhi-Wei Sun, On partitions of integers with restrictions involving squares, arXiv:2105.03416 [math.NT], 2021.
Zhi-Wei Sun, Diophantine problems involving triangular numbers and squares, a message to Number Theory List, Oct. 11, 2013.

Cf. A000217, A000290.

Cf. A008443, A053604, A053603, A052343, A052344, A008441.

SQ[n_]:=IntegerQ[Sqrt[n]]
T[n_]:=n(n+1)/2
a[n_]:=Sum[If[SQ[8(T[i]+T[j]+T[n-i-j])+1],1,0],{i,1,n/3},{j,i,(n-i)/2}]
Table[a[n],{n,1,100}]

a(n)=my(t=(n+1)*n/2,s);sum(x=1,n\3,s=t-n--*x;sum(y=x,n\2,is_A000217(s-(n-y)*y))) \\ - M. F. Hasler, Oct 11 2013

A052344 Number of ways to write n as the unordered sum of two nonzero triangular numbers.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 2, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 2, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 2, 1, 0, 0, 2, 0, 1, 1, 0, 2, 0, 0, 0, 1, 2, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 2, 1, 0, 0, 2, 0, 0, 1, 0, 3, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 2, 0, 0, 1, 0, 1, 1, 1

Offset: 0

Christian G. Bower, Jan 23 2000

nonn

Number of ways to write 8*n+2 as the unordered sum of two odd squares > 1. - Robert Israel, Feb 24 2016

Number of partitions of 2n into two promic numbers > 1. - Wesley Ivan Hurt, Jun 09 2021

T. D. Noe, Table of n, a(n) for n = 0..10000

Cf. A000217, A052343-A052348, A053587, A056303, A056304.
Column k=2 of A319797.
Cf. A005369, A010054.

G:= (1/8)*(JacobiTheta2(0, sqrt(q))^2-4*JacobiTheta2(0, sqrt(q))*q^(1/8)+2*JacobiTheta2(0, q))/q^(1/4):
S:= series(G,q,1001):
seq(coeff(S,q,j),j=0..1000); # Robert Israel, Feb 24 2016

nn=150; tri=Accumulate[Range[nn]]; t=Table[0, {tri[[-1]]}]; Do[n=tri[[i]]+tri[[j]]; If[n <= tri[[-1]], t[[n]]++], {i,nn}, {j,i}]; t=Prepend[t,0]

G.f.: (Theta_2(sqrt(x))^2 - 4*x^(1/8)*Theta_2(sqrt(x)) + 2*Theta_2(x))/(8*x^(1/4)) where Theta_2 is a Jacobi theta function. - Robert Israel, Feb 24 2016
a(n) = Sum_{k=1..n} c(k) * c(2*n-k), where c(n) is the characteristic function of promic numbers (A005369). - Wesley Ivan Hurt, Jun 09 2021
a(n) = Sum_{k=1..floor(n/2)} c(k) * c(n-k), where c = A010054. - Wesley Ivan Hurt, Jan 06 2024

A053587 Indices of A052344 (ways to write n as sum of two nonzero triangular numbers) where record values are reached.

Original entry on oeis.org

2, 16, 81, 471, 1056, 1381, 6906, 17956, 34531, 40056, 200281, 520731, 1001406, 1482081, 7410406, 19267056, 37052031, 60765331, 303826656, 789949306, 1519133281, 3220562556, 13429138206, 16102812781, 41867313231, 80514063906, 196454315931, 711744324931

Offset: 1

Jeremy Rouse, Jan 19 2000

The subsequence of primes begins: 2, 1381, 1519133281 [Jonathan Vos Post, Feb 01 2011].

			The order of the terms is ignored when deciding in how many ways the sum can be expressed. For example, a(2) does not equal 9, although 9 = 3 + 6 = 6 + 3.
a(2) = 16 because 16 = 1 + 15 = 6 + 10. a(3) = 81 because 81 = 3 + 78 = 15 + 66 = 36 + 55.

Kenneth Korbin, Problem 665, The College Mathematics Journal Vol. 30 No. 5, Nov. 1999. Asks for the 48th number in this sequence.

Probably differs from A052348 only at n=1, 2, 4.

Cf. A000217, A052343, A052344, A052345, A052346, A052347, A052348.

More terms from Christian G. Bower, Jan 23 2000
a(25)-a(26) from Donovan Johnson, Jun 26 2010
a(27)-a(28) from Donovan Johnson, Mar 20 2013

A052346 Smallest number which is the sum of two positive triangular numbers in exactly n different ways.

Original entry on oeis.org

1, 2, 16, 81, 471, 1056, 1381, 11781, 6906, 17956, 34531, 123256, 40056, 4462656, 305256, 448906, 200281, 1957231, 520731, 10563906, 1001406, 11222656, 539550781, 3454506, 1482081, 75865156, 7172606106, 8852431, 25035156, 334020781, 13018281, 38531031, 7410406, 7014160156

Offset: 0

Christian G. Bower, Jan 23 2000

nonn

From Chai Wah Wu, Oct 20 2023: (Start)
Other terms:
a(35) =  42980356
a(36) =  19267056
a(38) =  1289707656
a(39) =  2782318906
a(40) =  37052031
a(41) =  256720506
a(42) =  325457031
a(45) =  221310781
a(47) =  550240551
a(48) =  60765331
a(50) =  2200089531
a(54) =  327539956
a(56) =  926300781
a(59) =  7629645156
a(60) =  481676406
a(63) =  4598740656
a(64) =  303826656
a(68) =  6418012656
a(71) =  4579579956
a(72) =  789949306
a(80) =  1519133281
a(81) =  9498658731
a(84) =  12041910156
a(90) =  8188498906
a(96) =  3220562556
a(108) =  13429138206
(End)

			a(4) = 471 because 471 is the sum of two positive triangular numbers in exactly 4 different ways (as 300+171, 351+120, 435+36, and 465 + 6), and there is no smaller number that has this property.

Cf. A000217, A052343, A052344, A052345, A052347, A052348, A053587, A064816.

a(27), a(28) = 8852431, 25035156; a(26) not yet found
a(26) from Donovan Johnson, Nov 17 2008
Name edited (added the qualifier "positive"), example edited, and a(29)-a(32) added by Jon E. Schoenfield, Jul 16 2017
a(33) from Chai Wah Wu, Oct 20 2023

A119345 Numbers having exactly one representation as sum of two triangular numbers.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 9, 10, 11, 12, 13, 15, 18, 20, 22, 24, 25, 27, 28, 29, 30, 34, 37, 38, 39, 43, 45, 48, 49, 57, 58, 60, 61, 64, 65, 67, 69, 70, 73, 78, 79, 83, 84, 87, 88, 90, 92, 93, 97, 99, 100, 101, 102, 105, 108, 110, 112, 114, 115, 119, 127, 130, 132, 135, 137, 139, 142

Offset: 1

Reinhard Zumkeller, May 15 2006

nonn

A052343(a(n)) = 1; gives A020756 together with A118139.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A020757, A000217.

Cf. A020756, A052343, A118139.

a119345 n = a119345_list !! (n-1)
a119345_list = filter ((== 1) . a052343) [0..]
-- Reinhard Zumkeller, Jul 25 2014

trn=SortBy[{First[#],Last[#],Total[#]}& /@ (Union[Sort/@Tuples[Accumulate[Range[0,70]],{2}]]),Last]; Take[With[{x=Transpose[trn][[3]]}, Complement[Union[x], Union[Flatten[Select[Split[x], Length[#]>1&]]]]],70]  (* Harvey P. Dale, Feb 14 2011 *)
nn=100; tri=Table[n(n+1)/2,{n,0,nn}]; sums=Select[Flatten[Table[tri[[i]]+tri[[j]], {i,nn}, {j,i}]], #

A118139 Numbers expressible as the sum of two triangular numbers in at least two different ways.

Original entry on oeis.org

6, 16, 21, 31, 36, 42, 46, 51, 55, 56, 66, 72, 76, 81, 91, 94, 106, 111, 120, 121, 123, 126, 133, 136, 141, 146, 156, 157, 171, 172, 174, 181, 186, 191, 196, 198, 210, 211, 216, 225, 226, 231, 237, 241, 246, 256, 259, 268, 276, 281, 286, 289, 291, 297, 301, 306

Offset: 1

Greg Huber, May 13 2006

A052343(a(n)) > 1; gives A020756 together with A119345. - Reinhard Zumkeller, May 15 2006

			a(1) = 6 = 0 + 6 = 3 +3.
a(2) = 16 = 1 + 15 = 6 + 10.
a(3) = 21 = 0 + 21 = 6 + 15.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A020757, A000217.

Cf. A052343.

a118139 n = a118139_list !! (n-1)
a118139_list = filter ((> 1) . a052343) [0..]
-- Reinhard Zumkeller, Jul 25 2014

Sort[Transpose[Select[Tally[Total/@(Union[Sort/@Tuples[Accumulate[ Range[ 0,30]],2]])],#[[2]]>1&]][[1]]] (* Harvey P. Dale, Jul 21 2015 *)

More terms from Reinhard Zumkeller, May 15 2006

A224928 Numbers of pairs {x, y} such that x <= y and triangular(x) + triangular(y) = 2^n.

Original entry on oeis.org

1, 1, 1, 0, 2, 0, 1, 0, 3, 0, 2, 0, 4, 0, 1, 0, 8, 0, 2, 0, 4, 0, 4, 0, 8, 0, 2, 0, 24, 0, 2, 0, 8, 0, 8, 0, 8, 0, 2, 0, 32, 0, 4, 0, 16, 0, 4, 0, 32, 0, 4, 0, 32, 0, 4, 0, 4, 0, 8, 0, 16, 0, 2, 0, 32, 0, 6, 0, 48, 0, 16, 0, 16, 0, 8, 0, 384, 0, 4, 0, 16, 0, 16, 0, 16, 0, 8, 0, 768, 0, 2, 0, 8, 0, 4, 0, 32, 0, 32, 0, 256

Offset: 0

Alex Ratushnyak, May 08 2013

nonn

Conjectures:
1. a(n) = 0 for odd n > 1.
2. a(n) is even for even n > 14.

			2^1 = 1 + 1, the only representation of 2 as a sum of two triangular numbers, so a(1)=1.
2^4 = 16 = 1+15 = 6+10, two representations, so a(4) = 2.
2^8 = 256 = 3+253 = 66+190 = 120+136, so a(8) = 3.
2^12 = 4096 = 1+4095 = 91+4005 = 1540+2556 = 2016+2080, so a(12) = 4.

Antti Karttunen, Table of n, a(n) for n = 0..329

Cf. A000217, A052343, A006307, A225437.

#include 
#include 
typedef unsigned long long U64;
U64 isTriangular(U64 a) {      // ! Must be a <= (1<<63)
    U64 s = sqrt(a*2);
    if (a>=(1ULL<<63)) {
        if (a==(1ULL<<63)) return 0;
        printf("Error: a = %llu\n", a), exit(1);
    }
    return (s*(s+1)/2 == a);
}
int main() {
  U64 c, n, x, tx;
  for (n = 1; n; n+=n) {
    for (c = x = tx = 0; tx*2 <= n; ++x, tx+=x)
      if (isTriangular(n - tx))
        ++c;//, printf("(%llu+%llu) ", tx, n-tx);
    printf("%llu, ", c);
  }
  return 0;
}

A008441(n) = if(!n,n,sumdiv(4*n + 1, d, (d%4==1) - (d%4==3)));
A052343(n) = if(!n,1,my(u=A008441(n)); ((u\2)+(u%2)));
A224928(n) = A052343(2^n); \\ Antti Karttunen, May 24 2021

a(n) = A052343(2^n).

More terms from Antti Karttunen, May 24 2021

A347627 Number of partitions of n into at most 4 triangular numbers.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 2, 2, 3, 3, 2, 4, 4, 2, 4, 4, 3, 5, 4, 3, 5, 6, 4, 6, 4, 4, 6, 6, 4, 6, 8, 5, 7, 6, 4, 8, 8, 6, 6, 8, 6, 8, 8, 6, 9, 9, 6, 10, 9, 6, 10, 10, 6, 8, 10, 7, 11, 13, 8, 9, 10, 10, 10, 10, 7, 13, 14, 9, 10, 10, 10, 13, 14, 8, 10

Offset: 0

Ilya Gutkovskiy, Sep 09 2021

nonn

Cf. A000217, A002636, A007294, A052343, A110534, A319814, A347628.

cp's OEIS Frontend

A020756 Numbers that are the sum of two triangular numbers.

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A053603 Number of ways to write n as an ordered sum of two nonzero triangular numbers.

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A230121 Number of ways to write n = x + y + z (0 < x <= y <= z) such that x*(x+1)/2 + y*(y+1)/2 + z*(z+1)/2 is a triangular number.

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A052344 Number of ways to write n as the unordered sum of two nonzero triangular numbers.

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A053587 Indices of A052344 (ways to write n as sum of two nonzero triangular numbers) where record values are reached.

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A052346 Smallest number which is the sum of two positive triangular numbers in exactly n different ways.

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A119345 Numbers having exactly one representation as sum of two triangular numbers.

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A118139 Numbers expressible as the sum of two triangular numbers in at least two different ways.

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A230121 Number of ways to write n = x + y + z (0 < x <= y <= z) such that x(x+1)/2 + y(y+1)/2 + z*(z+1)/2 is a triangular number.