Notes on terms A104266(n) where n mod 4 = 2 =========================================== Let k = (n-2)/4, so n = 4k + 2. The first k digits are all 9's: <----- 1st k dgts -----> <------------------------- remaining 3k+2 digits ------------------------> 2 81 6 9 78121 10 99 78811236 14 999 78881115136 18 9999 78918111112681 22 99999 78917811111478921 26 999999 78917921111135242816 30 9999999 78918138111111226346761 34 99999999 78918147611111112515384644 38 999999999 78918148921111111112398242916 42 9999999999 78918148932111111111113333185156 46 99999999999 78918148931611111111111754955536964 50 999999999999 78918148932181111111111114358185598881 54 9999999999999 78918148932188111111111111351491155616836 58 99999999999999 78918148932195811111111111126927193645238441 62 999999999999999 78918148932199221111111111112332219711264421521 66 9999999999999999 78918148932199788111111111111227234725423186811236 70 99999999999999999 78918148932199932211111111111122571472396119842114921 74 999999999999999999 78918148932199978881111111111112252226735572759681115136 78 9999999999999999999 78918148932199989462111111111111225111129483276238227762361 82 99999999999999999999 78918148932199989463611111111111122511111261779538398775393124 86 999999999999999999999 78918148932199989463661111111111112251111119853398519537536151489 90 9999999999999999999999 78918148932199989463736111111111111225111111184229511377353214769424 94 99999999999999999999999 78918148932199989463742811111111111122511111111255121774683317894613796 98 999999999999999999999999 78918148932199989463742741111111111112251111111131836732788671792622256769 (The discussion below applies when k > 0; for the special case k = 0, i.e., n = 2, we get a(2) = 81.) Since a(n) is just below an even power of 10, sqrt(a(n)) will be just below a power of 10; as a result, successively smaller squares in the vicinity of a(n), while becoming smaller overall, will have 2nd-half digit substrings that get larger; e.g., 999800^2 = 999600 040000 999799^2 = 999598 040401 999798^2 = 999596 040804 where the falling square roots 999800, 999799, 999798, ... correspond to squares whose first halves are likewise falling (999600, 999598, 999596, ...), but whose second halves are rising (040000, 040401, 040804, ...). Let Q2 represent the "2nd quarter" of the digits, i.e., more precisely, the number obtained by concatenating "0." and the k+1 digits that follow immediately after the k 9's at the beginning of the number. Q2 cannot exceed 0.7891814893221080445... (which is 1 - 2/sqrt(90)), or else the 2nd half of the number will begin with something between 1111111... and 0000000..., and will thus include at least one 0. However, the digit string 7891814893221080445 itself includes a 0, immediately after the 7891814893221, so a tighter upper bound on Q2 is given by 0.7891814893219999999..., the largest number smaller than 0.7891814893221080445... that does not include a zero. But an n-digit square that begins with k 9's followed immediately by a string beginning with 7891814893219999999... would have 0's in its second half, since the second n/2 digits of the square would begin with 11111111111122500004.... Thus, a still smaller square must be used to raise the value of the digits immediately after the 111111111111225 (recall that lowering the square raises the value of the 2nd half) so that they contain no 0's (i.e., so that they begin with at least 1111111..., and thus the 2nd half will begin with at least 1111111111112251111111...). So instead of having Q2 = 0.7891814893219999999..., we find that Q2 cannot exceed 0.78918148932199989463742936620511272054032... (which is 1 - 2/sqrt(90/1.000000000001026), rather than 1 - 2/sqrt(90); this derives from the fact that 90/1.000000000001026 = 10/0.11111111111122511111111111111111...). From the above, we can obtain the result that an n-digit term a(n) where n = 4k + 2 cannot exceed UpperBound(a(n)) = 10^n * (1 - (10^-k)* 2 / sqrt(90/1.000000000001026)). E.g., we get that a(98) <= 10^98 * (1 - (10^-((98-2)/4))* 2 / sqrt(90/1.000000000001026)) a(98) <= 9999999999999999999999997891814893219998946374293 6620511272054031988917774992848086238673136310993 and, indeed, a(98) = 9999999999999999999999997891814893219998946374274 1111111111112251111111131836732788671792622256769; in terms of square roots, the upper bound gives sqrt(a(98)) <= 9999999999999999999999998945907446609999473187146 and in fact sqrt(a(98)) = 9999999999999999999999998945907446609999473187137 (a difference of just 9). It appears that the 1st half of the digits approaches the concatenation of k 9's and 10^(k+1) times the above upper bound, and that the 2nd half approaches 1111111111112251111111...; see the illustration below. <----------------- 1st n/2 dgts ----------------> <------------ remaining n/2 digits -------------> 2 8 1 6 978 121 10 99788 11236 14 9997888 1115136 18 999978918 111112681 22 99999789178 11111478921 26 9999997891792 1111135242816 30 999999978918138 111111226346761 34 99999999789181476 11111112515384644 38 9999999997891814892 1111111112398242916 42 999999999978918148932 111111111113333185156 46 99999999999789181489316 11111111111754955536964 50 9999999999997891814893218 1111111111114358185598881 54 999999999999978918148932188 111111111111351491155616836 58 99999999999999789181489321958 11111111111126927193645238441 62 9999999999999997891814893219922 1111111111112332219711264421521 66 999999999999999978918148932199788 111111111111227234725423186811236 70 99999999999999999789181489321999322 11111111111122571472396119842114921 74 9999999999999999997891814893219997888 1111111111112252226735572759681115136 78 999999999999999999978918148932199989462 111111111111225111129483276238227762361 82 99999999999999999999789181489321999894636 11111111111122511111261779538398775393124 86 9999999999999999999997891814893219998946366 1111111111112251111119853398519537536151489 90 999999999999999999999978918148932199989463736 111111111111225111111184229511377353214769424 94 99999999999999999999999789181489321999894637428 11111111111122511111111255121774683317894613796 98 9999999999999999999999997891814893219998946374274 1111111111112251111111131836732788671792622256769