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

1,3

Comments

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

Links

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.

Formula

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.

Mathematica

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 *)

Prog

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

Crossrefs

Complement of A020757.

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

Cf. A052343.

Sequence in context: A039148 A065904 A039108 * A051382 A026514 A285974

Adjacent sequences: A020753 A020754 A020755 * A020757 A020758 A020759

Keyword

nonn,nice

Author

David W. Wilson

Extensions

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

Status

approved