Oresme numbers

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{ …, −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 14^th century, the scholar and cleric Nicole Oresme found the value of the infinite series of Oresme numbers 

O(n) := n 2 n , n   ≥   1,

to be (Horadam)

${\begin{array}{l}\displaystyle {\sum _{i=1}^{\infty }{\frac {i}{2^{i}}}=2.}\end{array}}$

The partial sums of the above series are

${\begin{array}{l}\displaystyle {\sum _{i=1}^{n}{\frac {i}{2^{i}}}=2^{-n}(2^{n+1}-n-2)=2-\left({\frac {n+2}{2^{n}}}\right).}\end{array}}$

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

${\begin{array}{l}\displaystyle {G_{\{2^{n+1}-n-2\}}(x)={\frac {x}{(1-2x)(1-x)^{2}}},\quad n\geq 1.}\end{array}}$

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, ...}

External links

• A.F. Horadam, Oresme Numbers.