I'm trying to solve some problems. Can anyone provide answers? Or post more succinct code? Or provide other questions of a similar nature?

First, what is the last integer that results from fib(n+1)/n, going up to 35?

Last[Select[Table[Fibonacci[n + 1]/n, {n, 1, 35}], IntegerQ]]

How many squares are in an approximation of a circle of radius 26? I believe it might be 4*Sum[Floor[Sqrt[26^2 - k^2]], {k, 1, 26}]. Here's a picture to help:

Graphics[{Line[{First[#], Last[#]}], Line[Reverse /@ {First[#], Last[#]}]} & /@ SplitBy[Select[Tuples[Range[-25, 25], {2}], Norm[#] <= 26 &], First]]

How many circles are in this triangle?

Graphics[Circle[#, 1/2] & /@ Flatten[Table[a {1, 0} + b {-1/2, Sqrt[3]/2}, {a, 1, 63}, {b, 1, a - 1}], 1]]

For that matter, what is the Topological Index of this triangular grid?

What is the magic sum in the following magic square of consecutive primes?

What is the missing coefficient in the following "Swiss knife polynomial"? It seems to give the same values as cos(x Pi/2).

How many nonsingular 2x2 matrices are there, mod 7? I believe the code might be

Length[Select[Partition[#, 2] & /@ Tuples[Range[0, 6], {4}], Det[#, Modulus -> 7] != 0 &]]

These matrices can make a multiplication table. Here's a part of it.

g = SortBy[Select[Partition[#, 2] & /@ Tuples[Range[0, 6], {4}], Det[#, Modulus -> 7] != 0 &], Det[#, Modulus -> 7] &]; ArrayPlot[Table[Det[Mod[g[[a]] g[[b]], 7], Modulus -> 7], {a, 1, 366}, {b, 1, 366}], ImageSize -> 750]

Staying on 7, how many permutations of {1,2,3,4,5,6,7} have a maximum increasing run of length 2? I believe the code might be

Length[Select[Drop[Permutations[Range[7]], -1], Not[MemberQ[Partition[Sign[Differences[#]], 2, 1], {1, 1}]] &]]

I know that 31067664 == EulerPhi[IntegerReverse[31067664]]. Are there smaller examples where phi(rev(n))=n?

Last[Select[Range[10000], # == EulerPhi[IntegerReverse[#]] &]]

How many spanning trees does this graph have?

If anyone can provide answers for any of these, I'd be very appreciative.