OFFSET
0,2
COMMENTS
This sequence gives the number of triangles of all sizes in (3*n^2+2*n)-polyiamonds in a pentagonal or heptagonal configuration.
Also sum of 2*n*(n+1)*(n+2)/3 triangles oriented in one direction and n*(n+1)^2/2 oriented in the opposite direction.
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
a(n) = (1/2)*( Sum_{j=0..n} (n+1-j)*(3*n-j) + Sum_{j=0..n-1} (n-j)*(3*n+1-3*j) ).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Colin Barker, Mar 02 2015
G.f.: x*(x + 6) / (x - 1)^4. - Colin Barker, Mar 02 2015
a(n) = -A007584(-n-1). [Bruno Berselli, Mar 02 2015]
MAPLE
MATHEMATICA
Table[n (n + 1) (7 n + 11)/6, {n, 0, 50}] (* Bruno Berselli, Mar 02 2015 *)
LinearRecurrence[{4, -6, 4, -1}, {0, 6, 25, 64}, 50] (* Harvey P. Dale, Jul 17 2015 *)
PROG
(PARI) vector(50, n, n--; n*(n+1)*(7*n+11)/6)
(PARI) concat(0, Vec(x*(x+6)/(x-1)^4 + O(x^100))) \\ Colin Barker, Mar 02 2015
(Magma) [n*(n+1)*(7*n+11)/6: n in [0..50]]; // Bruno Berselli, Mar 02 2015
(Sage) [n*(n+1)*(7*n+11)/6 for n in (0..50)] # Bruno Berselli, Mar 02 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Luce ETIENNE, Mar 02 2015
STATUS
approved