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A089982
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Triangular numbers that can be expressed as the sum of 2 positive triangular numbers.
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7
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6, 21, 36, 55, 66, 91, 120, 136, 171, 210, 231, 276, 351, 378, 406, 496, 561, 666, 703, 741, 820, 861, 946, 990, 1035, 1081, 1176, 1225, 1326, 1378, 1431, 1485, 1540, 1596, 1653, 1711, 1770, 1891, 1953, 2016, 2080, 2211, 2278, 2346, 2556, 2701, 2775, 2850
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OFFSET
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1,1
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COMMENTS
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LINKS
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FORMULA
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EXAMPLE
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Generally, A000217(A000217(n)) = A000217(A000217(n)-1) + A000217(n) and so is automatically included. These are 6=T(3), 21=T(6), 55=T(10), etc. Other solutions occur when a partial sum from x to y is triangular, e.g., 15 + 16 + 17 + 18 = 66 = T(11), so T(14) + T(11) = T(18). This particular example arises since 10+4k is triangular (at k=14, 10 + 4k = 66), and we therefore have a solution.
All other solutions occur when 3+2k, 6+3k, 10+4k, etc. -- in general, T(j) + j*k -- is triangular.
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MATHEMATICA
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trn[i_]:=Module[{trnos=Accumulate[Range[i]], t2s}, t2s=Union[Total/@ Tuples[ trnos, 2]]; Intersection[trnos, t2s]] (* Harvey P. Dale, Nov 08 2011 *)
Select[Range[75], ! PrimeQ[#^2 + (# + 1)^2] &] /. Integer_ -> (Integer^2 + Integer)/2 (* Arkadiusz Wesolowski, Dec 03 2015 *)
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PROG
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(PARI) t(i) = i*(i+1)/2;
{ v=vector(100, i, t(i)); y=vector(100); c=0; for (i=1, 30, for (j=i, 30, x=t(i)+t(j); f=0; for (k=1, 100, if (x==v[k], f=1; break)); if (f==1, y[c++ ]=x))); select(x->(x>0), vecsort(y, , 8)) } \\ slightly edited by Michel Marcus, Apr 15 2021
(PARI) lista(nn) = {for (n=1, nn, my(t = n*(n+1)/2); for (k=1, n-1, if (ispolygonal(t - k*(k+1)/2, 3), print1(t, ", "); break; )); ); } \\ Michel Marcus, Apr 15 2021
(Python)
from itertools import count, takewhile
def aupto(lim):
t = list(takewhile(lambda x: x<=lim, (i*(i+1)//2 for i in count(1))))
s = set(a+b for i, a in enumerate(t) for b in t[i:])
return sorted(s & set(t))
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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More terms from Lambert Klasen (Lambert.Klasen(AT)gmx.net) and David Wasserman, Sep 23 2005
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STATUS
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approved
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