A119345 -id:A119345 - cp's OEIS frontend

A052343 Number of ways to write n as the unordered sum of two triangular numbers (zero allowed).

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

Offset: 0

Christian G. Bower, Jan 23 2000

Number of ways of writing n as a sum of a square and twice a triangular number (zeros allowed). - Michael Somos, Aug 18 2003
a(A020757(n))=0; a(A020756(n))>0; a(A119345(n))=1; a(A118139(n))>1. - Reinhard Zumkeller, May 15 2006
Also, number of ways to write 4n+1 as the unordered sum of two squares of nonnegative integers. - Vladimir Shevelev, Jan 21 2009
The average value of a(n) for n <= x is Pi/4 + O(1/sqrt(x)). - Vladimir Shevelev, Feb 06 2009

			G.f. = 1 + x + x^2 + x^3 + x^4 + 2*x^6 + x^7 + x^9 + x^10 + x^11 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Vladimir Shevelev, Binary additive problems: recursions for numbers of representations, arXiv:0901.3102 [math.NT], 2009-2013.
Vladimir Shevelev, Binary additive problems: theorems of Landau and Hardy-Littlewood type, arXiv:0902.1046 [math.NT], 2009-2012.

Cf. A000217, A052344, A052345 (greedy inverse), A052346, A052347, A052348, A053587, A056303, A056304.
Cf. A053692, A093518, A121444, A259285, A259287, A260415, A260516.
Cf. A005369, A010052.

a052343 = (flip div 2) . (+ 1) . a008441
-- Reinhard Zumkeller, Jul 25 2014

A052343 := proc(n)
    local a,t1idx,t2idx,t1,t2;
    a := 0 ;
    for t1idx from 0 do
        t1 := A000217(t1idx) ;
        if t1 > n then
            break;
        end if;
        for t2idx from t1idx do
            t2 := A000217(t2idx) ;
            if t1+t2 > n then
                break;
            elif t1+t2 = n then
                a := a+1 ;
            end if;
        end do:
    end do:
    a ;
end proc: # R. J. Mathar, Apr 28 2020

Length[PowersRepresentations[4 # + 1, 2, 2]] & /@ Range[0, 101] (* Ant King, Dec 01 2010 *)
d1[k_]:=Length[Select[Divisors[k],Mod[#,4]==1&]];d3[k_]:=Length[Select[Divisors[k],Mod[#,4]==3&]];f[k_]:=d1[k]-d3[k];g[k_]:=If[IntegerQ[Sqrt[4k+1]],1/2 (f[4k+1]+1),1/2 f[4k+1]];g[#]&/@Range[0,101] (* Ant King, Dec 01 2010 *)
a[ n_] := Length @ Select[ Table[ Sqrt[n - i - i^2], {i, 0, Quotient[ Sqrt[4 n + 1] - 1, 2]}], IntegerQ]; (* Michael Somos, Jul 28 2015 *)
a[ n_] := Length @ FindInstance[ {j >= 0, k >= 0, j^2 + k^2 + k == n}, {k, j}, Integers, 10^9]; (* Michael Somos, Jul 28 2015 *)

{a(n) = if( n<0, 0, sum(i=0, (sqrtint(4*n + 1) - 1)\2, issquare(n - i - i^2)))}; /* Michael Somos, Aug 18 2003 */

a(n) = ceiling(A008441(n)/2). - Reinhard Zumkeller, Nov 03 2009
G.f.: (Sum_{k>=0} x^(k^2 + k)) * (Sum_{k>=0} x^(k^2)). - Michael Somos, Aug 18 2003
Recurrence: a(n) = Sum_{k=1..r(n)} r(2n-k^2+k) - C(r(n),2) - a(n-1) - a(n-2) - ... - a(0), n>=1,a (0)=1, where r(n)=A000194(n+1) is the nearest integer to square root of n+1. For example, since r(6)=3, a(6) = r(12) + r(10) + r(6) - C(3,2) - a(5) - ... - a(0) = 4 + 3 + 3 - 3 - 0 - 1 - 1 - 1 - 1 - 1 = 2. - Vladimir Shevelev, Feb 06 2009
a(n) = A025426(8n+2). - Max Alekseyev, Mar 09 2009
a(n) = (A002654(4n+1) + A010052(4n+1)) / 2. - Ant King, Dec 01 2010
a(2*n + 1) = A053692(n). a(4*n + 1) = A259287(n). a(4*n + 3) = A259285(n). a(6*n + 1) = A260415(n). a(6*n + 4) = A260516(n). - Michael Somos, Jul 28 2015
a(3*n) = A093518(n). a(3*n + 1) = A121444(n). a(9*n + 2) = a(n). a(9*n + 5) = a(9*n + 8) = 0. - Michael Somos, Jul 28 2015
Convolution of A005369 and A010052. - Michael Somos, Jul 28 2015

A020756 Numbers that are the sum of two triangular numbers.

Original entry on oeis.org

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

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

A125018 Numbers == 1 (mod 4) with a unique partition as a sum of 2 squares x^2 + y^2.

Original entry on oeis.org

1, 5, 9, 13, 17, 29, 37, 41, 45, 49, 53, 61, 73, 81, 89, 97, 101, 109, 113, 117, 121, 137, 149, 153, 157, 173, 181, 193, 197, 229, 233, 241, 245, 257, 261, 269, 277, 281, 293, 313, 317, 333, 337, 349, 353, 361, 369, 373, 389, 397, 401, 405, 409, 421, 433

Offset: 1

Artur Jasinski, Nov 16 2006

nonn

			5 = 1^2 + 2^2, 9 = 0^2 + 3^2, 13 = 2^2 + 3^2, 17 = 1^2 + 4^2, 29 = 2^2 + 5^2, ... - _Michael Somos_, Jul 25 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002145, A119345, A124982, A124983.

Select[4 * Range[0, 100] + 1, Length @ PowersRepresentations[#, 2, 2] == 1 &] (* Amiram Eldar, Mar 12 2020 *)

isok(n)= {if (n % 4 != 1, return(0)); A000161(n) == 1;} \\ Michel Marcus, Nov 02 2013

More terms from Michel Marcus, Nov 02 2013

A274794 Numbers n such that n^3 is the sum of two triangular numbers in exactly one way.

Original entry on oeis.org

0, 1, 3, 4, 7, 9, 10, 19, 24, 25, 34, 37, 39, 42, 49, 54, 55, 72, 73, 78, 85, 87, 93, 94, 102, 108, 109, 118, 138, 142, 147, 157, 160, 165, 168, 175, 192, 195, 202, 210, 214, 220, 228, 232, 243, 247, 249, 250, 252, 253, 258, 267, 273, 274, 279, 289, 297, 312, 333

Offset: 1

Altug Alkan, Jul 07 2016

nonn

A115104 is a subsequence. Terms such that 4*n^3 + 1 is not prime are 24, 337, 457, 750, 840, 1015, ...

			3 is a term because 3^3 = 27 = 6 + 21.

Cf. A000217, A000578, A020756, A052343, A115104, A119345.

Select[Range@ 333, Length[PowersRepresentations[4 #^3 + 1, 2, 2]] == 1 &] (* after Ant King at A052343, or *)
nn = 20; t = (#^2 + #)/2 & /@ Range[0, nn^3]; Select[Range[0, nn], Function[n, Count[Transpose@ {#, n^3 - #} &@ Range[0, Floor[n^3/2]], k_ /; Times @@ Boole@ Map[MemberQ[t, #] &, k] == 1] == 1]] (* Michael De Vlieger, Jul 07 2016 *)

a052343(n) = sum(i=0, (sqrtint(4*n + 1) - 1)\2, issquare(n - i - i^2));
lista(nn) = for(n=0, nn, if(a052343(n^3) == 1, print1(n, ", ")));

cp's OEIS Frontend

A052343 Number of ways to write n as the unordered sum of two triangular numbers (zero allowed).

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A020756 Numbers that are the 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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A125018 Numbers == 1 (mod 4) with a unique partition as a sum of 2 squares x^2 + y^2.

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A274794 Numbers n such that n^3 is the sum of two triangular numbers in exactly one way.

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