TL_h(n). The number of new triangles produced at level h in iteration n. rows h, columns n. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 1 3 6 12 24 48 96 192 384 768 1536 3072 6144 12288 24576 49152 98304 196608 393216 786432 2 0 0 3 9 18 36 72 144 288 576 1152 2304 4608 9216 18432 36864 73728 147456 294912 589824 3 0 0 0 9 27 54 108 216 432 864 1728 3456 6912 13824 27648 55296 110592 221184 442368 884736 4 0 0 0 0 27 81 162 324 648 1296 2592 5184 10368 20736 41472 82944 165888 331776 663552 1327104 5 0 0 0 0 0 90 270 540 1080 2160 4320 8640 17280 34560 69120 138240 276480 552960 1105920 2211840 6 0 0 0 0 0 0 288 864 1728 3456 6912 13824 27648 55296 110592 221184 442368 884736 1769472 3538944 7 0 0 0 0 0 0 0 927 2781 5562 11124 22248 44496 88992 177984 355968 711936 1423872 2847744 5695488 8 0 0 0 0 0 0 0 0 2988 8964 17928 35856 71712 143424 286848 573696 1147392 2294784 4589568 9179136 9 0 0 0 0 0 0 0 0 0 9621 28863 57726 115452 230904 461808 923616 1847232 3694464 7388928 14777856 10 0 0 0 0 0 0 0 0 0 0 30987 92961 185922 371844 743688 1487376 2974752 5949504 11899008 23798016 11 0 0 0 0 0 0 0 0 0 0 0 99801 299403 598806 1197612 2395224 4790448 9580896 19161792 38323584 12 0 0 0 0 0 0 0 0 0 0 0 0 321426 964278 1928556 3857112 7714224 15428448 30856896 61713792 13 0 0 0 0 0 0 0 0 0 0 0 0 0 1035216 3105648 6211296 12422592 24845184 49690368 99380736 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3334113 10002339 20004678 40009356 80018712 160037424 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10738152 32214456 64428912 128857824 257715648 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 34584291 103752873 207505746 415011492 17 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 111385377 334156131 668312262 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 358738083 1076214249 19 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1155385170 Then at n = 100 ... Sum area of colored triangles = 4.705882351163920272455109651041179608629950136035195299338101344101657508229627724263363347905389927457052091274846401829305280793833901492738173426714216158149584057213132837205193936824798583984375 by the way ........... 80/17 ~= 4.7058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235 The total area of the ninjastar = 9.99999999999999999999999999999053366913734785833505925721660656524392152277869172394275665283203125 And total colored area / total area = 0.4705882351163920272455109651045634352563633440239950554254489897595301467963190698977119009378870045885291395457987796541511420102123393105894128169678345628494728141878008288525895398939087928967352241854456172440800315097138195252177950437450190469024420663540639636668718432292798612473922870920110965427706472621758815437999631281158269645269197884915526434285377473918857400162550997456257919489101631269769318851506454462613548454317157445809908476239768653997804202613079468213420587885565247352169258901674567151474393918604343500997994418645875643550025614522267918440393906684521290955389078859247609298945866848131256904706254323586343930140258963384511951470497843000992083297539332563671973842608336344067800147314168471158150391588548487345232539557532379633164424828662971409954834883790046253666330499341102903693281487915206924239549103334042817837492856620521958365346310695880256119473025552328682827078136409819810282375579451898024486019282708617206475520149193529700102222759439 by the way .................. 8/17 ~= 0.4705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058824 Total number of new triangles produced in each iteration n: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 1 3 9 30 96 309 996 3207 10329 33267 107142 345072 1111371 3579384 11528097 37128459 119579361 385128390 1240380240 3994883733 Total number of triangles present in each iteration n: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 1 4 13 43 139 448 1444 4651 14980 48247 155389 500461 1611832 5191216 16719313 53847772 173427133 558555523 1798935763 5793819496 Area of each colored triangle produced in iteration n: 1, 0.25, 0.0625, 0.015625, 0.00390625, 0.0009765625, 0.000244140625, 0.00006103515625, 0.0000152587890625, 0.000003814697265625, 9.5367431640625E-7, 2.384185791015625E-7, 5.9604644775390625E-8, 1.490116119384765625E-8, 3.7252902984619140625E-9, 9.31322574615478515625E-10, 2.3283064365386962890625E-10, 5.82076609134674072265625E-11, 1.4551915228366851806640625E-11, 3.63797880709171295166015625E-12 Area of whole ninja star at iteration n: 4, 7, 8.5, 9.25, 9.625, 9.8125, 9.90625, 9.953125, 9.9765625, 9.98828125, 9.994140625, 9.9970703125, 9.99853515625, 9.999267578125, 9.9996337890625, 9.99981689453125, 9.999908447265625, 9.9999542236328125, 9.99997711181640625, 9.999988555908203125 Total area of colored triangles contained within the ninja star at iteration n: 1, 1.75, 2.3125, 2.781250, 3.15625000, 3.4580078125, 3.701171875000, 3.89691162109375, 4.0545196533203125, 4.181423187255859375, 4.28360176086425781250, 4.3658733367919921875000, 4.432116210460662841796875, 4.48545318841934204101562500, 4.5283986963331699371337890625, 4.562977268360555171966552734375, 4.59081900794990360736846923828125, 4.6132364306831732392311096191406250, 4.631286338786594569683074951171875000, 4.64581964114404399879276752471923828125 The fraction of area in the ninja star that is colored at iteration n: 0.25, 0.25000, 0.27206, 0.30068, 0.32792, 0.35241, 0.37362, 0.39153, 0.40640, 0.41863, 0.42861, 0.43672, 0.44328, 0.44858, 0.45286, 0.45631, 0.45909, 0.46133, 0.46313, 0.46458 Total number of new triangles produced in each iteration n: Full Comma Separated List 1, 3, 9, 30, 96, 309, 996, 3207, 10329, 33267, 107142, 345072, 1111371, 3579384, 11528097, 37128459, 119579361, 385128390, 1240380240, 3994883733 Total number of triangles present in each iteration n: Full Comma Separated List 1, 4, 13, 43, 139, 448, 1444, 4651, 14980, 48247, 155389, 500461, 1611832, 5191216, 16719313, 53847772, 173427133, 558555523, 1798935763, 5793819496