New in Wolfram Mathematica 7: New Number Theory Capabilities  previous | next 
Work with Holonomic Sequences
Define new holonomic sequences and sum them and prove identities etc.
In[1]:=

Click for copyable input
Jacobsthal = 

  DifferenceRoot[

   Function[{y, n}, {y[n] == y[n - 1] + 2 y[n - 2], y[0] == 0, 

     y[1] == 1}]];
In[2]:=

Click for copyable input
Sum[Jacobsthal[2 i]^2, {i, 0, m - 1}]
Out[2]=

In[3]:=

Click for copyable input
Sum[Jacobsthal[2 i + 1]^2, {i, 0, m - 1}]
Out[3]=

In[4]:=

Click for copyable input
FullSimplify[

 Jacobsthal[2 k + 1]^2 + Jacobsthal[2 k + 3]^2 == 

  10 + 8 Jacobsthal[2 k] (34 Jacobsthal[2 k - 2] + 15), 

 k \[Element] Integers]
Out[4]=

In[5]:=

Click for copyable input
FullSimplify[Jacobsthal[2 k + 1] == 8 Jacobsthal[2 k - 2] + 3, 

 k \[Element] Integers]
Out[5]=