Letâs fix a position $ i $, and count the number of sequences where the repeated pair is at $ i, i+1 $, and no other consecutive digits are equal.
["Title: How to Fix a Fix Position $ i $ and Count Valid Sequences with a Repeated Pair at $ i, i+1 $", "---", "Introduction", "In many string processing and combinatorics problems, identifying and fixing a specific position — such as $ i $ — in a sequence is crucial for analyzing structural patterns. This article explores a classic pattern matching problem: fix position $ i $ where a repeated pair occurs at $ (i, i+1) $, and exactly one such pair exists with all other consecutive digits being unequal. We’ll also provide a systematic method to count all valid sequences satisfying these constraints, which is valuable in algorithm design, data validation, and sequence analysis.", "---", "### Understanding the Problem", "Suppose you are given a digit sequence (string) composed of digits from $ 0 $ to $ 9 $. The goal is to:", "- Fix position $ i $ such that digits at $ i $ and $ i+1 $ are equal, forming a repeated adjacent pair.\n- Ensure that this repeated pair is isolated: no other consecutive equal digits appear elsewhere in the sequence.\n- Count how many such valid sequences exist, under these rules.", "This problem arises in testing digit pattern constraints, validating pseudo-random sequences, and designing efficient neuromorphic algorithms for sequence verification — especially in brain-inspired computational models used in bioengineering.", "---", "### Key Definitions", "- Repeated Pair at $ i, i+1 $: $ s[i] = s[i+1] $, and $ s[i] $ is the only occurrence in the sequence where any digit is repeated consecutively.\n- Isolation Condition: For all $ j <br/>\neq i $, $ s[j] <br/>\neq s[j+1] $, i.e., no other equal adjacent digits.", "---", "### Step-by-Step Approach", "#### Step 1: Fix the Repeated Pair at Position $ i $, $ s[i] = s[i+1] = d $, $ d \in {0,1,...,9} $", "Each digit $ d $ from 0 to 9 can be inserted at position $ i $, but must respect:\n- Isolation: $ s[i-1] <br/>\neq d $ and $ s[i+2] <br/>\neq d $ unless at boundary.\n- No other consecutive duplicates.", "#### Step 2: Segment the Sequence Around $ i $", "The sequence $ s $ is split into three regions:", "- Left segment: $ s[0..i-1] $\n- Middle segment: just $ s[i] = s[i+1] = d $\n- Right segment: $ s[i+2..n-1] $", "Each region must adhere to the no-consecutive-repeats rule.", "#### Step 3: Enforce Isolation Using Valid Digits and Transitions", "- To the left: if $ i > 0 $, $ s[i-1] <br/>\neq d $.\n So valid first digits $ s[i-1] $ (if exists) are in $ {0..9} \setminus {d} $.", "- To the right: if $ i+2 < n-1 $, $ s[i+2] <br/>\neq d $.\n Valid next digits $ s[i+2] $ are also in $ {0..9} \setminus {d} $.", "#### Step 4: Count Valid Assignments for Left and Right Segments", "Each valid prefix/suffix must form a sequence with no repeating consecutive digits. This is equivalent to counting non-repetitive strings of length $ \ell $, where each character is a digit, and no two identical digits are adjacent.", "Let $ f(\ell, c) $ be the number of strings of length $ \ell $ over 10 digits (0–9) with no consecutive equal digits, starting (or continuing) with digit $ c $.", "We’ll derive a recurrence to compute total valid combinations.", "---", "### Recursive Count: Counting Non-Repeating Sequences", "Let $ A(\ell, d) $ be the number of valid sequences of length $ \ell $, ending with digit $ d $, with no two equal adjacent digits.", "Then:", "- $ A(0, d) = 1 $ for any digit $ d $ (empty string).\n- For $ \ell \geq 1 $:\n $$\n A(\ell, d) = \sum_{\substack{e=0 \ e <br/>\neq d}}^{9} A(\ell-1, e)\n $$", "This recurrence arises because the previous digit must differ from $ d $.", "Alternatively, define $ T(\ell) $ as the total number of sequences of length $ \ell $ with no consecutive equal digits:", "- $ T(1) = 10 $\n- $ T(\ell) = 10 \ imes 9^{\ell-1} $ for $ \ell \geq 1 $", "But due to isolation and local constraints, we refine this.", "---", "### Key Insight: Isolated Pair at $ i $", "Because only $ s[i]=s[i+1]=d $ is repeated, and elsewhere no two adjacent digits match, the left and right segments must themselves be non-repeating, and:", "- $ s[i-1] <br/>\neq d $\n- $ s[i+2] <br/>\neq d $\n- Both segments can be any valid non-repeating subsequence, as long as no match with $ d $ across boundaries.", "Let $ L = i $ be the length of the left segment (from 0 to $ i-1 $).\nLet $ R = n - (i+2) $ be the length of the right segment.", "Now:", "- Number of valid left sequences of length $ L $ ending in any digit $ <br/>\neq d $:\n For each valid start digit $ s[i-1] \in D_L = {0..9} \setminus {d} $, count how many sequences exist. But since transitions depend only on different digits, and we allow any start or end, we use total valid transitions.", "A known result:\nThe number of sequences of length $ L $ with no consecutive repeats, using 10 digits, starting (or just existing) with any digit ≠ $ d $, is:", "$$\nL_{\ ext{left}} \cdot 9^{L-1}\n$$\nIf $ L > 0 $, because first digit has 9 choices (≠ $ d $), each next has 9 (≠ previous). If $ L = 0 $, it’s 1 (empty).", "Similarly:", "$$\nR_{\ ext{right}} = R \cdot 9^{R-1}, \quad R > 0\n$$", "But ensure:", "- $ s[i] = s[i+1] = d $\n- $ s[i-1] <br/>\neq d $ ⇒ left sequence ends in ≠ $ d $\n- $ s[i+2] <br/>\neq d $ ⇒ right sequence starts in ≠ $ d $", "Therefore, total valid sequences for fixed $ d $ and $ i $:", "$$\n\ ext{Total}(d,i) = \left(\n \begin{cases}\n1 & i = 0 \\n\left(9 \cdot 9^{i-1}\right) & \ ext{if } i > 0 \\n\left(9 \cdot 9^{i-1}\right) & \ ext{if } i < n-2 \\n\ ext{exclude cross match risk} & \ ext{otherwise}\n\end{cases}\n\right.)\n$$", "Actually, independence: left and right are independent given isolation.", "Thus:", "$$\n\ ext{Total}(d,i) = \n\begin{cases}\n10 & \ ext{if } i = 0 \ ext{ and } i+2 = n-1, \ ext{ but risky} \\n9 \cdot 9^{i-1} \cdot (9 \cdot 9^{R-1}) & \ ext{if } i > 0 \ ext{ and } i+2 < n-1 \\n9 \cdot 9^{i-1} \cdot (9 \cdot 9^{R-1}) & \ ext{if } i = 0 \ ext{ but } i+2 = n-1? \ ext{ check index}\n\end{cases}\n$$", "Simplify:", "- Left: $ i $ digits $ s[0..i-1] $, $ s[i] = d $: requires $ s[i-1] <br/>\neq d $.\n If $ i = 0 $: no $ s[-1] $, so only constraint is $ d <br/>\ne d $ trivial → 1 option. Then right: $ R = n - i - 2 = n - 2 $, so right count is $ 9 \cdot 9^{R-1} = 9^{R} $ if $ R \geq 1 $, or 1 if $ R = 0 $.", "- If $ i > 0 $, $ s[i-1] <br/>\neq d $, left count: $ 9 \cdot 9^{i-1} = 9^i $\n- Right: $ R = n - i - 2 $, if $ R \geq 1 $, count is $ 9^R $; if $ R = 0 $, 1 (empty)", "- If $ i = 0 $, $ i+2 = 2 $. Then $ R = n - 2 $.\n Constraint: $ s[2] <br/>\neq d $. Since $ s[0] = s[1] = d $, $ s[2] <br/>\ne d $ ensures isolation.", "Left: $ i = 0 $, so no digit before, just $ s[0] = d $.\n But right: $ s[2] \in {0..9} \setminus {d} $, so 9 choices. Each such choice defines a full sequence only if $ s[2] <br/>\ne d $, which is enforced.", "Also no adjacent repeats: $ s[0]=s[1]=d $, no repeat → ok.\n $ s[1]=d, s[2] <br/>\ne d $ → no repeat.\n $ s[2] $ to $ s[n-1] $? No, right segment is $ s[2..n-1] $, length $ R = n - 2 $. But our earlier $ R $ was $ n - i - 2 = n - 2 $, correct.", "But since only two blocks, and no overlap, and all transitions valid under isolation, number of valid sequences is:", "$$\n (\ ext{# valid left} = 1) \ imes (\ ext{# valid right} = 9^{R}) \quad \ ext{where } R = n - 2\n $$", "But wait: left sequence is just $ s[0]=d $. No other restriction, since $ i=0 $, $ s[i-1] $ doesn’t exist. So only require $ s[2] <br/>\ne d $, which gives 9 options for $ s[2] $, then the rest can follow? Wait — no: the right segment must be a valid non-repeating sequence of length $ R = n - 2 $, starting with $ s[2] \in {0..9}\setminus{d} $, and each next digit ≠ previous.", "So number of valid right sequences of length $ R \geq 1 $ is $ 9^R $, and $ R = 0 $ only if $ n = 2 $, $ i=0 $. But $ n=2 $, $ i=0 $, then $ s[0]=d, s[1]=d $, but then other adjacent pair exists — violates “only at $ i,i+1 $”.", "Contradiction!", "---", "### Critical Correction: No Other Equal Pairs Absolutely", "If $ i = 0 $, $ s[0] = s[1] = d $, but $ s[2], \dots $ may repeat ⇒ unless $ n = 2 $, but then $ s[1] = d $ and $ s[2] $? No, $ i+2 = 2 $, so $ s[2] $ not checked directly, but in a length-2 sequence, $ s[1] $ and $ s[2] $ don’t exist. But in a sequence of length 3, $ s[1] $ and $ s[2] $ are adjacent.", "So for $ i = 0 $, we require:", "- $ s[0] = s[1] = d $\n- $ s[2] <br/>\ne d $ (if exists, i.e., $ n \geq 3 $), but if $ n = 2 $, then $ i+2 = 2 $, but indices out, so only check $ i = 0 $, no contradiction.", "But in a sequence of length 2: $ s[0], s[1] = d,d $. Only one adjacent pair — this one is repeated. So it can be valid if no other positions.", "But for $ i=0 $, $ i+2 = 2 $, which in $ n=2 $ refers to index 2, invalid → so no constraint from right.", "But if $ n \geq 3 $, then $ s[2] <br/>\ne d $ to prevent further repetition starting at $ i=1 $.", "Hence:", "- For $ i = 0 $: isolation only requires $ s[2] <br/>\ne d $ if $ n \geq 3 $; no constraint from right segment.\n- Right segment must still be valid, length $ R = n - 2 $, starting with $ s[2] \in {0..9}\setminus{d} $, and no consecutive repeats.", "So number of valid sequences for $ i=0 $:\n- Left: fixed, 1 choice\n- Right: $ R = n - 2 $: if $ R = 0 $ (n=2), then 1 sequence (just"]
