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A346636 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. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The existence of such a four-sided polygon implies that every element of the quadruple is less than the sum of the other elements.

LINKS

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)

FORMULA

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).

PROG

(Visual Basic) See Links

CROSSREFS

Cf. A006003, A346637, A346638.

Sequence in context: A300919 A007834 A228741 * A200873 A082966 A198182

Adjacent sequences:  A346633 A346634 A346635 * A346637 A346638 A346639

KEYWORD

nonn

AUTHOR

Giovanni Corbelli, Jul 26 2021

STATUS

approved

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Last modified July 7 12:47 EDT 2022. Contains 355148 sequences. (Running on oeis4.)