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A346636
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a(n) is the number of quadruples (a_1, a_2, a_3, a_4) having all terms in {1,...,n} such that there exists a quadrilateral with these side lengths.
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3
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0, 1, 16, 77, 236, 565, 1156, 2121, 3592, 5721, 8680, 12661, 17876, 24557, 32956, 43345, 56016, 71281, 89472, 110941, 136060, 165221, 198836, 237337, 281176, 330825, 386776, 449541, 519652, 597661, 684140, 779681, 884896, 1000417, 1126896, 1265005, 1415436
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OFFSET
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0,3
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COMMENTS
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The existence of such a four-sided polygon implies that every element of the quadruple is less than the sum of the other elements.
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LINKS
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Table of n, a(n) for n=0..36.
Giovanni Corbelli, VB routine for generating number of four-sided polygons
Giovanni Corbelli Proof of the formula: Number of k-tuples with elements in {1,2,…,N} corresponding to k-sided polygons
Sean A. Irvine, Java program (github)
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FORMULA
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Formula: a(n) = n^4 - 4*binomial(n+1,4) = n^4 - (n+1)*binomial(n,3).
General formula for k-tuples: a_k(n) = n^k - k*binomial(n+1,k) = n^k - (n+1)*binomial(n,k-1).
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PROG
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(Visual Basic) See Links
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CROSSREFS
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Cf. A006003, A346637, A346638.
Sequence in context: A300919 A007834 A228741 * A200873 A082966 A198182
Adjacent sequences: A346633 A346634 A346635 * A346637 A346638 A346639
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KEYWORD
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nonn
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AUTHOR
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Giovanni Corbelli, Jul 26 2021
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STATUS
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approved
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