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# Oresme numbers

{ , −2048, −896, −384, −160, −64, −24, −8, −2, 0,
 1 2
,
 2 4
,
 3 8
,
 4 16
,
 5 32
,
 6 64
,
 7 128
,
 8 256
, }
In the middle of the 14th century, the scholar and cleric Nicole Oresme found the value of the infinite series of Oresme numbers
O(n) :=
 n 2 n
, n   ≥   1,
${\displaystyle {\begin{array}{l}\displaystyle {\sum _{i=1}^{\infty }{\frac {i}{2^{i}}}=2.}\end{array}}}$

The partial sums of the above series are

${\displaystyle {\begin{array}{l}\displaystyle {\sum _{i=1}^{n}{\frac {i}{2^{i}}}=2^{-n}(2^{n+1}-n-2)=2-\left({\frac {n+2}{2^{n}}}\right).}\end{array}}}$
A125128
 2 n + 1  −  n  −  2, n   ≥   1,
or partial sums of main diagonal of array A125127 of
 k
-step Lucas numbers.
 {1, 4, 11, 26, 57, 120, 247, 502, 1013, 2036, 4083, 8178, 16369, 32752, 65519, 131054, 262125, 524268, 1048555, 2097130, 4194281, 8388584, 16777191, 33554406, ...}

The generating function is

${\displaystyle {\begin{array}{l}\displaystyle {G_{\{2^{n+1}-n-2\}}(x)={\frac {x}{(1-2x)(1-x)^{2}}},\quad n\geq 1.}\end{array}}}$
For nonpositive integers, the Oresme numbers are
 O( − n) =  −  n  2 n =  −  A036289( − n), n   ≥   0.

A036289
 n  2 n, n   ≥   0.
 {0, 2, 8, 24, 64, 160, 384, 896, 2048, 4608, 10240, 22528, 49152, 106496, 229376, 491520, 1048576, 2228224, 4718592, 9961472, 20971520, 44040192, 92274688, 192937984, ...}