OFFSET
0,3
COMMENTS
Also the Wiener index of the (n-1)-triangular grid graph (indexed so the 0-triangular grid graph is the singleton). - Eric W. Weisstein, Sep 08 2017
REFERENCES
L. B. W. Jolley, Summation of Series, Dover. N.Y., 1961, eq. (54), page 10.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Triangular Grid Graph.
Eric Weisstein's World of Mathematics, Wiener Index.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).
FORMULA
4*a(n+1) = 1*2^2*3 + 2*3^2*4 + 3*4^2*5 + ... (n terms). [Jolley]
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>5. - Harvey P. Dale, Oct 28 2014
G.f.: 3*x^2*(1 + x) / (1 - x)^6. - Colin Barker, Sep 08 2017
a(n) = (1/2) * Sum_{k=0..n} C(k^2,2). - Wesley Ivan Hurt, Sep 23 2017
From Amiram Eldar, Feb 15 2022: (Start)
Sum_{n>=2} 1/a(n) = 40*(16*log(2) - 11)/9.
Sum_{n>=2} (-1)^n/a(n) = 20*(8*Pi - 25)/9. (End)
MAPLE
a:=n->sum(j^4-j^2, j=0..n)/4: seq(a(n), n=0..36); # Zerinvary Lajos, May 08 2008
MATHEMATICA
Table[(n - 1) n (n + 1)(n + 2)(2 n + 1)/40, {n, 0, 30}] (* Josh Locker *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 0, 3, 21, 81, 231}, 40] (* Harvey P. Dale, Oct 28 2014 *)
PROG
(Magma) [(n-1)*n*(n+1)*(n+2)*(2*n+1)/40: n in [0..40]]; // Vincenzo Librandi, Feb 06 2014
(PARI) for(n=0, 50, print1((n-1)*n*(n+1)*(n+2)*(2*n+1)/40, ", ")) \\ G. C. Greubel, Jul 23 2017
(PARI) concat(vector(2), Vec(3*x^2*(1 + x) / (1 - x)^6 + O(x^30))) \\ Colin Barker, Sep 08 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Alford Arnold, Sep 24 2005
EXTENSIONS
More terms from Josh Locker (jlocker(AT)mail.rochester.edu) and Michael W. Motily (mwm5036(AT)psu.edu), Oct 04 2005
STATUS
approved