Integers in base i-1

Consider the Gaussian integer i-1. A base i-1 representation of a Gaussian integer a+bi is a finite sequence of digits d_n-1d_n-2...d₁d₀ such that:

• a+bi = d_n-1(i-1)^n-1 + d_n-2(i-1)^n-2 + ... + d₁(i-1) + d₀
• Each d_k is in {0,1}
• There are no leading zeroes, i.e. d_n-1 ≠ 0, unless a+bi is itself 0

Here are base i-1 representations of a few Gaussian integers: 11+24i → 111010110001101 24-11i → 110010110011 8+0i → 111000000 -5+0i → 11001101 0+0i → 0

Remarkably, every Gaussian integer has a unique base i-1 representation!

Define f(a+bi) as the number of 1s in the unique base i-1 representation of a+bi. For example, f(11+24i) = 9 and f(24-11i) = 7.

Define B(L) as the sum of f(a+bi) for all integers a, b such that |a| ≤ L and |b| ≤ L. For example, B(500) = 10795060.

Find B(10¹⁵) mod 1 000 000 007.

Written by gamwe6 who lives and works in San Francisco building useful things. You should follow him on Twitter