Work with Holonomic Sequences

Define new holonomic sequences and sum them and prove identities etc.

In[1]:=

Jacobsthal = 

  DifferenceRoot[

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

     y[1] == 1}]];

In[2]:=

Sum[Jacobsthal[2 i]^2, {i, 0, m - 1}]

Out[2]=

In[3]:=

Sum[Jacobsthal[2 i + 1]^2, {i, 0, m - 1}]

Out[3]=

In[4]:=

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]:=

FullSimplify[Jacobsthal[2 k + 1] == 8 Jacobsthal[2 k - 2] + 3, 

 k \[Element] Integers]

Out[5]=