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MATHEMATICA
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initTriangle[ ] := {{1, 0, 1, 0, 1}, {1, 0, 1, 1, 0, 1}, {1, 0, 1, 0, 1, 0, 1}, {1, 0, 1, 1, 1, 1, 0, 1}, {1, 0, 1, 0, 0, 0, 1, 0, 1}, {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}};
nextRow[k_] := Join[{1, 0}, Table[0, k], {0, 1}]
neighborPosList[{i_, j_}] := {{i - 1, j - 1}, {i - 1, j}, {i, j - 1}, {i, j + 1}, {i + 1, j}, {i + 1, j + 1}}
neighborEmptyList[{i_, j_}] := Select[neighborPosList[{i, j}], nTriangle[[Apply[Sequence, #]]] == 0 &]
getCellValue[{i_, j_}] := nTriangle[[i, j]]
a169781[n_] := Module[{newGen = { }, lastGen = {{5, 3}, {5, 7}}, newCount = 0, totalCount = 11, k, p, emptyNs, q, emptyCs, r}, nTriangle = initTriangle[ ]; For[k = 6, k <= n, k++, AppendTo[nTriangle, nextRow[k + 1]]; For[p = 1, p <= Length[lastGen], p++, emptyNs = Select[neighborPosList[lastGen[[p]]], getCellValue[#] == 0 &]; For[q = 1, q <= Length[emptyNs], q++, emptyCs = Select[neighborPosList[emptyNs[[q]]], getCellValue[#] == 1 &]; If[Length[emptyCs] == 1, AppendTo[newGen, emptyNs[[q]]]]]]; For[r = 1, r <= Length[newGen], r++, nTriangle[[Apply[Sequence, newGen[[r]]]]] = 1]; lastGen = newGen; totalCount += Length[newGen]; newGen = { }]; totalCount]/; (n>=2^3 && IntegerQ[Log[2, n]])
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