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%I #51 Dec 11 2021 11:49:11
%S 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,
%T 34,36,37,38,39,42,43,45,46,48,49,51,55,56,57,58,60,61,64,65,66,67,69,
%U 70,72,73,76,78,79,81,83,84,87,88,90,91,92,93,94,97,99,100,101,102,105,106,108
%N Numbers that are the sum of two triangular numbers.
%C 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
%C A052343(a(n)) > 0; union of A118139 and A119345. - _Reinhard Zumkeller_, May 15 2006
%C Also union of A051533 and A000217. - _Ant King_, Nov 29 2010
%H T. D. Noe, <a href="/A020756/b020756.txt">Table of n, a(n) for n = 1..10000</a>
%H John A. Ewell, <a href="http://www.jstor.org/stable/2324243">On Sums of Triangular Numbers and Sums of Squares</a>, The American Mathematical Monthly, 99:8 (October 1992), pp. 752-757.
%H T. Khovanova, K. Knop, and A. Radul, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Khovanova/khova4.html">Baron Munchhausen's Sequence</a>, J. Int. Seq. 13 (2010) # 10.8.7.
%H L. K. Mork, Keith Sullivan, Trenton Vogt, and Darin J. Ulness, <a href="https://ajc.maths.uq.edu.au/pdf/80/ajc_v80_p305.pdf">A group theoretical approach to the partitioning of integers: Application to triangular numbers, squares, and centered polygonal numbers</a>, Australasian J. Comb. (2021) Vol. 80, No. 3, 305-321.
%F 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)
%F 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
%F 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
%F Closed under the operation f(x, y) = 4*x*y + x + y.
%t 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 *)
%t Take[Union[Total/@Tuples[Accumulate[Range[0,20]],2]],80] (* _Harvey P. Dale_, May 02 2012 *)
%o (PARI) v=vector(200); vc=0; for (x=0,10, for (y=0,10,v[vc++ ]=x^2+y*(y+1))); v=vecsort(v); v
%o (PARI) 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
%o (Haskell)
%o a020756 n = a020756_list !! (n-1)
%o a020756_list = filter ((> 0) . a052343) [0..]
%o -- _Reinhard Zumkeller_, Jul 25 2014
%Y Complement of A020757.
%Y Cf. A051533 (sums of two positive triangular numbers, A001481 (sums of two squares), A002378, A000217.
%Y Cf. A052343.
%K nonn,nice
%O 1,3
%A _David W. Wilson_
%E Entry revised by _N. J. A. Sloane_, Dec 20 2004
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