|
| |
|
|
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
|
| |
|
|
