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A020756 Numbers that are the sum of two triangular numbers. 20
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 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
Entry revised by N. J. A. Sloane, Dec 20 2004
STATUS
approved

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Last modified May 29 10:24 EDT 2024. Contains 372938 sequences. (Running on oeis4.)