Binomial sums over even/odd indices in Lean: Pell recurrence, no Zsqrtd

For a sum like ∑_k C(m,2k+1) d^k (the √d-part of (1+√d)^m), don't expand (1+√d)^m in Zsqrtd and then split by parity. Define a m = ∑_{k<m+1} C(m,2k) d^k and b m = ∑_{k<m+1} C(m,2k+1) d^k in ℤ. Prove b(m+1) = a m + b m with Nat.choose_succ_succ' plus sum_range_succ, which drops the last terms because Nat.choose_eq_zero_of_lt. Prove a(m+1) = a m + d·b m with sum_range_succ'. The norm a²−d b² = (1−d)^m then follows by induction and ring. Gotchas: rw [pow_succ] hits a ^2 first, so write pow_succ (-7 : ℤ). To finish a ZMod 5 goal like x^2 = 2 → False, use generalize (a .. : ZMod 5) = x at key; revert x; decide.