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%I #23 Mar 31 2023 17:26:12
%S 21,171,703,990,3741,4186,6786,8778,30628,38781,77028,188191,203203,
%T 219453,318801,359128,416328,678030,763230,928203,1023165,1342341,
%U 1505980,1983036,2114596,2185095,2349028,2795430,3219453,3744216,4928230,6049981,7036876,7478778
%N Larger member of a pair of triangular numbers whose sum and difference are triangular.
%D Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966, p. 197, nr. 8.
%F a(n) = A000217(A185223(n)). - _R. J. Mathar_, Feb 11 2018
%e a(2) = 171, since the pair of triangular numbers 171 = 18*(18+1)/2 and 105 = 14*(14+1)/2 produce the sum 276 = 23*(23+1)/2 and the difference 66 = 11*(11+1)/2 which are both triangular numbers.
%t Module[{trs=Accumulate[Range[3900]]},Union[Select[Sort/@Subsets[trs,{2}],AllTrue[{Sqrt[ 8Total[#]+ 1],Sqrt[8Abs[#[[1]]-#[[2]]]+1]},OddQ]&]]][[All,2]]//Sort (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale_, Dec 02 2018 *)
%o (PARI) lista(nn) = {v = vector(nn, n, n*(n+1)/2); for (n=2, nn, for (k=1, n-1, if (ispolygonal(v[n]+v[k], 3) && ispolygonal(v[n]-v[k], 3), print1(v[n], ", "));););} \\ _Michel Marcus_, Jan 08 2015
%Y Cf. A000217, A185129, A185223, A185233, A185243, A185253, A185257, A185258.
%K nonn
%O 1,1
%A _Martin Renner_, Jan 20 2012
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