Joseph Likar, <a href="/A345165/b345165_1.txt">Table of n, a(n) for n = 0..1000</a>

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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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0, 0, 1, 1, 2, 2, 5, 5, 8, 11, 17, 20, 29, 37, 51, 65, 85, 106, 141, 175, 223, 277, 351, 432, 540, 663, 820, 999, 1226, 1489, 1817, 2192, 2654, 3191, 3847, 4603, 5517, 6578, 7853, 9327, 11084, 13120, 15533, 18328, 21621, 25430, 29905, 35071, 41111, 48080, 56206, 65554, 76420, 88918, 103394, 120015, 139214, 161222, 186593, 215632, 249006, 287165, 330938, 380888, 438075, 503255, 577713, 662459

From Joseph Likar, Aug 21 2023: (Start)

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)

Terms a(26)-a(1000) onwards by Joseph Likar, Aug 21 2023

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0, 0, 1, 1, 2, 2, 5, 5, 8, 11, 17, 20, 29, 37, 51, 65, 85, 106, 141, 175, 223, 277, 351, 432, 540, 663, 820, 999, 1226, 1489, 1817, 2192, 2654, 3191, 3847, 4603, 5517, 6578, 7853, 9327, 11084, 13120, 15533, 18328, 21621, 25430, 29905, 35071, 41111, 48080, 56206, 65554, 76420, 88918, 103394, 120015, 139214, 161222, 186593, 215632, 249006, 287165, 330938, 380888, 438075, 503255, 577713, 662459

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.

Joseph Likar, <a href="/A345165/b345165_1.txt">Table of n, a(n) for n = 0..1000</a>

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Terms a(26)-a(1000) by Joseph Likar, Aug 21 2023

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