Bob's New House

Bob moved to a new house on another long street and noticed that now the sum of the house numbers up to his own house, but excluding it, equals the sum of the numbers from his house to the end of the road, again excluding his house. If the houses are numbered consecutively, starting from 1, what is the number of Bob's house?

Find the number of houses on the road, and Bob's house number if his has 1, 2, 3, or 4 digits.

Source: Similar to a previous problem (3/26/97) from The Penguin Book of Curious and Interesting Puzzles, David Wells, 1992.

Solutions were received from: Carlos Rivera, Nick Baxter, Bill Chapp, Kirk Bresniker, Sorin Ionescu, Alexander Doskey, Denis Borris, Israel Eisenberg, Michael Schooneveldt.

The first few solutions for Bob's house are:

                                Bob's          Last
                               house         house

                                     6              8
                                    35             49
                                   204            288
                                  1189           1681
                                  6930           9800
                                 40391          57121
                                235416         332928
                               1372105        1940449
                               7997214       11309768
Nick Baxter showed that each successive answer can be found from the previous ones with the following equations:
B(i)=6*B(i-1) - B(i-2)
L(i)=6*L(i-1) - L(i-2) + 2
Israel Eisenberg found two other equations only dependent upon one past answer (they can be derived from those above):
B(n+1) = 3*B(n) + 2*L(n) + 1
L(n+1) = 4*B(n) + 3*L(n) + 1
Most solvers used an exhaustive computer search after finding a representative equation between the two variables. Nick Baxter provides this analytical solution:
C(B,2) = C(N+1,2) - C(B+1,2)
or 2B^2 = N(N+1)
Solving for N, N = (-1+sqrt(8n^2+1))/2
For N to be integral, 8n^2+1 must be a square.
Let x^2 = 8b^2+1, a Pell equation.
Sources for Pell Equations: Slide Rule's Games, Eric's Treasure Trove of Mathematics.
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