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Joseph Likar, <a href="/A371283/b371283_1.txt">Table of n, a(n) for n = 1..10000</a>

a(51) onwards by Joseph Likar, Aug 25 2024

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Joseph Likar, <a href="/A371288/b371288_1.txt">Table of n, a(n) for n = 1..10000</a>

a(60) onwards by Joseph Likar, Aug 25 2024

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0, 0, 0, 1, 1, 2, 4, 5, 7, 11, 16, 20, 28, 37, 50, 65, 84, 106, 140, 175, 222, 277, 350, 432, 539, 663, 819, 999, 1225, 1489, 1816, 2192, 2653, 3191, 3846, 4603, 5516, 6578, 7852, 9327, 11083, 13120, 15532, 18328, 21620, 25430, 29904, 35071, 41110, 48080

Joseph Likar, <a href="/A344654/b344654_1.txt">Table of n, a(n) for n = 0..1000</a>

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a(33) onwards from Joseph Likar, Sep 06 2023

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From Joseph Likar, Aug 21 2023: (Start)

This sequence can be generated by the following pseudo code:

outputMap[2 to n] = n

for usedNumbers = 2 to n

for medianSize = ceil(usedNumbers / 2) + 1 to usedNumbers - 1

for medianValue = 1 to floor((n - usedNumbers + medianSize) / medianSize)

for numbersAboveMedian = 0 to usedNumbers - medianSize

calculateState(usedNumbers, medianSize, medianValue, numbersAboveMedian)

next numbersAboveMedian

next medianValue

next medianSize

if (usedNumbers mod 2 == 1)

for medianValue = 2 to Math.floor((n - usedNumbers + ceil(usedNumbers / 2))/ceil(usedNumbers / 2))

for numbersAboveMedian = 1 to usedNumbers - ceil(usedNumbers / 2) - 1

calculateState(usedNumbers, ceil(usedNumbers / 2), medianValue, numbersAboveMedian)

next numbersAboveMedian

next medianValue

endif

next usedNumbers

function calculateState(usedNumbers, medianSize, medianValue, numbersAboveMedian)

numbersBelowMedian = usedNumbers - medianSize - numbersAboveMedian)

bottomQTerms = gaussianBinomial(numbersBelowMedian + (medianValue - 1) - 1, numbersBelowMedian)

topQTerms = gaussianBinomial(n - 1 - 2 * numbersAboveMedian, numbersAboveMedian)

for bottomOffset = 0 to length(bottomQTerms) - 1

for topOffset = 0 to length(topQTerms) - 1

partitionSum = bottomOffset + topOffset + numbersBelowMedian + (medianSize * medianValue) + (numbersAboveMedian * (medianValue + 1));

if (partitionSum <= n)

outputMap[partitionSum] += bottomQTerms[bottomOffset] * topQTerms[topOffset]

next topOffset

next bottomOffset

end

where gaussianBinomial(m, n) takes the Gaussian Binomial (m choose n)_q and returns a zero-indexed array where each value in the array is the coefficient of q^index in the Gaussian binomial expansion. (End)

Joseph Likar, <a href="/A345165/a345165.java.txt">Java Implementation</a> using QBinomials

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