#384 - Rudin-Shapiro Sequence
Define the sequence \(a(n)\) as the number of adjacent pairs of ones in the binary expansion of \(n\) (possibly overlapping).
E.g.: \(a(5) = a(101_2) = 0\), \(a(6) = a(110_2) = 1\), \(a(7) = a(111_2) = 2\).
Define the sequence \(b(n) = (-1)^{a(n)}\).
This sequence is called the Rudin-Shapiro sequence.
Also consider the summatory sequence of \(b(n)\): \(s(n) = \sum \limits_{i = 0}^n b(i)\).
The first couple of values of these sequences are:
| \(n\) | \(0\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) |
| \(a(n)\) | \(0\) | \(0\) | \(0\) | \(1\) | \(0\) | \(0\) | \(1\) | \(2\) |
| \(b(n)\) | \(1\) | \(1\) | \(1\) | \(-1\) | \(1\) | \(1\) | \(-1\) | \(1\) |
| \(s(n)\) | \(1\) | \(2\) | \(3\) | \(2\) | \(3\) | \(4\) | \(3\) | \(4\) |
The sequence \(s(n)\) has the remarkable property that all elements are positive and every positive integer \(k\) occurs exactly \(k\) times.
Define \(g(t,c)\), with \(1 \le c \le t\), as the index in \(s(n)\) for which \(t\) occurs for the \(c\)'th time in \(s(n)\).
E.g.: \(g(3,3) = 6\), \(g(4,2) = 7\) and \(g(54321,12345) = 1220847710\).
Let \(F(n)\) be the Fibonacci sequence defined by:
\(F(0)=F(1)=1\) and
\(F(n)=F(n-1)+F(n-2)\) for \(n \gt 1\).
Define \(GF(t)=g(F(t),F(t-1))\).
Find \(\sum GF(t)\) for \(2 \le t \le 45\).
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=384. Published Sunday, 13th May 2012, 02:00 am. Solved by 391 members at time of mirroring.
Why this is useful
Numerical Computing. Matrix methods and linear-recurrence acceleration are the machinery behind covariance work, PCA, and lattice/transition models (Phases 4, 5, 8).
We classify relevance honestly - not every Euler problem is a trading application.
Prerequisites
Lessons that prepare you:
1.3 Proof Techniques: Direct, Contrapositive, and Contradiction · 19.14 Computational Complexity, Feasibility Estimation, and Proving Algorithms Correct · 19.1 Divisibility, GCD, and the Euclidean Algorithm · 19.7 Dynamic Programming: Memoization and Tabulation · 19.9 Graph Algorithms: BFS, DFS, Dijkstra, and Minimum Spanning Trees · 19.13 Matrix Exponentiation and Linear Recurrence Acceleration · 19.3 Modular Arithmetic, Inverses, and Fast Exponentiation · 19.6 Recurrence Relations and Generating Functions · 19.2 Primes, Sieves, and Integer Factorization · 2.7 Sequences, Series, Convergence, and Power Series
Recommended stepping-stone problems: #312 · #194 · #364
Concepts: graph-theory number-theory sequences-series brute-force-reduction
Likely techniques: matrix-exponentiation memoization
Learning mode
Pick how much scaffolding you want. Your choice is remembered per problem.
Understand the problem
- What exactly is the input to problem 384? Is it a bound (45), a supplied dataset, or a definition you must generate from?
- What is the required output - restate it precisely: a single sum.
- Write out, in your own words, the definition of rudin-shapiro sequence as the statement gives it. Which integers/objects are excluded by that definition?
- What do the arguments of s(n), a(n) mean, and what is the value's type (count, sum, probability)?
- What are the edge cases: the smallest legal object, zero/one, ties, and the boundary at exactly 45?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 45 and the cost of testing one.
- Which graph-theory fact would, if true, collapse the search - and can you state it as a testable claim before you look for a proof?
Predict & plan (before you code)
- Predict the strategy: in one sentence, what will your solution do? (The classification says graph-theory / matrix-exponentiation - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 45, and the wall-clock time you expect. Write both down now.
- Predict the key data structure: what is stored, keyed by what, and how large will it get at full scale?
- Predict the failure mode: what is most likely to break - an off-by-one on the bound, a definition misread, precision, or memory?
- Predict the output of the small case from rung 3 BEFORE running it (the statement says: "E.g.: a(5) = a(101_2) = 0, a(6) = a(110_2) = 1, a(7) = a(111_2) = 2.") - then run it. A surprise here is worth more than an hour of debugging later.
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Optimization
You have a correct answer. That is the start of the learning, not the end.
- Reduce the time complexity. What is the bottleneck, and what mathematical fact removes it?
- Reduce memory. Can you stream, or keep only the last k states?
- Replace brute force with a closed form, a sieve, a recurrence, or a symmetry argument.
- Prove the optimized version computes the same thing.
- Compare two implementations and time them.
Explain it
Which step of your solution were you least confident about, and what evidence would settle it?
What did you try first, and what specifically made you abandon it - a proof, a timing, or a wrong small-case answer?
Where did the graph-theory structure do the real work? Name the single observation that collapsed the search space.
Could you have reached the matrix-exponentiation idea faster? Which words in the statement were pointing at it, and did you notice them?
What was the bug that cost you the most time, and what CLASS of bug was it (off-by-one, definition misread, precision, state under-specified)?
How would your solution change if the bound 45 were multiplied by 1000? Does it survive, or does it need a different idea?
What is the honest complexity of what you wrote (not what you intended), and where is the remaining slack?
Which problem you have already solved is this most similar to, and what is the shared skeleton - is it really 'matrix-exponentiation' underneath?
State the transferable technique in one sentence, without mentioning this problem's story at all.
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Low confidence schedules this problem for spaced review, even if you solved it.
Mastery check
- Variation: change the bound (or a rule) in the statement. Does your method still work? What breaks first?
- Constraints: if the limit were 10× larger, which step fails, and what would you replace it with?
- Related problem: #312 · #194 · #364
- Transfer: where else does this technique appear? Name a lesson and a real computational setting.
- Spaced re-attempt: come back after the review interval and re-solve it with no hints.