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A226389 Triangular numbers representable as triangular(x)*triangular(y)+1.

%I A226389 #9 Jun 06 2013 08:09:02
%S A226389 1,10,91,136,946,1711,1891,5671,8911,10585,11026,11935,13861,19306,
%T A226389 21736,26335,32131,36856,44551,49141,65341,107416,138601,239086,
%U A226389 305371,351541,366796,459361,849556,873181,933661,1100386,1413721,1516411,1904176,2297296,2467531,3837835
%N A226389 Triangular numbers representable as triangular(x)*triangular(y)+1.
%C A226389 Cases with x=y are included.
%C A226389 The associated indices in A000217 are 1, 4, 13, 16, 43, 58, 61, 106, 133, 145, 148, 154,...
%H A226389 Giovanni Resta, <a href="/A226389/b226389.txt">Table of n, a(n) for n = 1..1000</a>
%e A226389 10 = 3 * 3 + 1,
%e A226389 91 = 6 * 15 + 1,
%e A226389 3837835 = 378 * 10153 + 1.
%p A226389 A000217inv:=proc(n) local t1; t1:=floor(sqrt(2*n)); if n = t1*(t1+1)/2 then return t1 ; else return -1; end if; end proc:
%p A226389 A000217 := proc(n)
%p A226389     n*(n+1)/2 ;
%p A226389 end proc:
%p A226389 isA226389 := proc(n)
%p A226389     local Tx, Ty;
%p A226389     if n = 1 then
%p A226389         return true;
%p A226389     elif A000217inv(n) >= 0 then
%p A226389         for x from 0 do
%p A226389             Tx := A000217(x) ;
%p A226389             if Tx+1 > n then
%p A226389                 return false;
%p A226389             end if;
%p A226389             for y from 0 to x do
%p A226389                 Ty := A000217(y) ;
%p A226389                 if Tx*Ty+1 >n then
%p A226389                     break;
%p A226389                 elif Tx*Ty+1 = n then
%p A226389                     return true;
%p A226389                 end if;
%p A226389             end do:
%p A226389         end do:
%p A226389     else
%p A226389         false;
%p A226389     end if;
%p A226389 end proc:
%p A226389 for n from 0 do
%p A226389     Tn := A000217(n) ;
%p A226389     if isA226389(Tn) then
%p A226389         printf("%d,\n",Tn) ;
%p A226389     end if;
%p A226389 end do: # _R. J. Mathar_, Jun 06 2013
%Y A226389 Cf. A000217, A188630, A225390.
%K A226389 nonn
%O A226389 1,2
%A A226389 _Alex Ratushnyak_, Jun 06 2013