A380853 Number of ways to place six distinct positive integers on a triangle, three on the corners and three on the sides such that the sum of the three values on each side is n.

0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 5, 13, 14, 25, 37, 47, 58, 89, 98, 126, 159, 188, 219, 276, 303, 362, 423, 478, 536, 633, 688, 781, 881, 973, 1068, 1211, 1301, 1443, 1589, 1724, 1866, 2066, 2202, 2396, 2598, 2790, 2986, 3250, 3439, 3699, 3967, 4219, 4480, 4819, 5071

Offset: 1

Author

Derek Holton and Alex Holton, Feb 06 2025

Keywords

• nonn
• easy

Comments

Solutions differing by only rotation or reflections are not counted separately.

If the numbers do not need to be distinct and rotations and reflections are counted separately we get A019298(n-2). If the numbers do not need to be distinct but rotations and reflections do not count separately we get A006918(n-2). If the six numbers must be distinct and reflections and rotations count separately we get 6*a(n). - R. J. Mathar, Feb 27 2025

Examples

			The a(9) = 1 solution is:
       1
     5   6
   3   4   2
		

Links

• R. J. Mathar, Illustrations - Examples (2025)
• Index entries for linear recurrences with constant coefficients, signature (-1,0,2,3,2,0,-3,-4,-3,0,2,3,2,0,-1,-1).

Crossrefs

Cf. A342467, A380105.

Programs

Maple

A380853 := proc(n)
    -3412+2353*n+30*n^3-480*n^2+(135*n-900)*(-1)^n ;
    %-144*(2*A010891(n)+A010891(n-1)+2*A010891(n-2)) ;
    %-160*(17*A049347(n)+8*A049347(n-1)) ;
    %-360*A057077(n) ;
    %+480*(-1)^n*(A099254(n)-A099254(n-1)) ;
    %/720 ;
end proc:
seq(A380853(n),n=1..40) ; # R. J. Mathar, Feb 27 2025

PARI

a(n) = my(c=0, t); for(x=3, n-5, t=n-x; for(y=2, min(x-1, t-1), for(z=1, y-1, if(#Set([x, y, z, t-y, t-z, n-y-z])==6, c++)))); c; \\ Jinyuan Wang, Feb 07 2025

Formula

G.f.: x^9*(1 + 4*x + 8*x^2 + 16*x^3 + 18*x^4 + 18*x^5 + 15*x^6 + 10*x^7)/((1 - x)^4*(1 + 2*x + 2*x^2 + x^3)^2*(1 + x + 2*x^2 + 2*x^3 + 2*x^4 + x^5 + x^6)). - Stefano Spezia, Feb 08 2025

A380105(n) = a(n)-a(n-3). - R. J. Mathar, Mar 13 2025