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 A055129 Repunits in different bases: table by antidiagonals of numbers written in base k as a string of n 1's. 13
 1, 1, 2, 1, 3, 3, 1, 4, 7, 4, 1, 5, 13, 15, 5, 1, 6, 21, 40, 31, 6, 1, 7, 31, 85, 121, 63, 7, 1, 8, 43, 156, 341, 364, 127, 8, 1, 9, 57, 259, 781, 1365, 1093, 255, 9, 1, 10, 73, 400, 1555, 3906, 5461, 3280, 511, 10, 1, 11, 91, 585, 2801, 9331, 19531, 21845, 9841, 1023, 11 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 LINKS Michael De Vlieger, Table of n, a(n) for n = 1..11325 (1 <= n <= 150). FORMULA T(n, k) = (k^n-1)/(k-1) [with T(n, 1) = n] = T(n-1, k)+k^(n-1) = (k+1)*T(n-1, k)-k*T(n-2, k) [with T(0, k) = 0 and T(1, k) = 1]. From Werner Schulte, Aug 29 2021 and Sep 18 2021: (Start) T(n,k) = 1 + k * T(n-1,k) for k > 0 and n > 1. Sum_{m=2..n} T(m-1,k)/Product_{i=2..m} T(i,k) = (1 - 1/Product_{i=2..n} T(i,k))/k for k > 0 and n > 1. Sum_{n > 1} T(n-1,k)/Product_{i=2..n} T(i,k) = 1/k for k > 0. Sum_{i=1..n} k^(i-1) / (T(i,k) * T(i+1,k)) = T(n,k) / T(n+1,k) for k > 0 and n > 0. (End) EXAMPLE T(3,5)=31 because 111 base 5 represents 25+5+1=31.       1       1       1       1       1       1       1       2       3       4       5       6       7       8       3       7      13      21      31      43      57       4      15      40      85     156     259     400       5      31     121     341     781    1555    2801       6      63     364    1365    3906    9331   19608       7     127    1093    5461   19531   55987  137257 Starting with the second column, the q-th column list the numbers that are written as 11...1 in base q. - John Keith, Apr 12 2021 MAPLE A055129 := proc(n, k)     add(k^j, j=0..n-1) ; end proc: # R. J. Mathar, Dec 09 2015 MATHEMATICA Table[FromDigits[ConstantArray[1, #], k] &[n - k + 1], {n, 11}, {k, n, 1, -1}] // Flatten (* or *) Table[If[k == 1, n, (k^# - 1)/(k - 1) &[n - k + 1]], {n, 11}, {k, n, 1, -1}] // Flatten (* Michael De Vlieger, Dec 11 2016 *) CROSSREFS Rows include A000012, A000027, A002061, A053698, A053699, A053700. Columns (see recurrence) include A000027, A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002275, A016123, A016125. Diagonals include A023037, A031973. Numbers in the table (apart from the first column and first two rows) are ordered in A053696. Sequence in context: A098546 A126277 A253273 * A133804 A185943 A352001 Adjacent sequences:  A055126 A055127 A055128 * A055130 A055131 A055132 KEYWORD base,easy,nonn,tabl AUTHOR Henry Bottomley, Jun 14 2000 STATUS approved

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Last modified July 5 23:22 EDT 2022. Contains 355105 sequences. (Running on oeis4.)