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A112851 a(n) = (n-1)*n*(n+1)*(n+2)*(2*n+1)/40. 3
0, 0, 3, 21, 81, 231, 546, 1134, 2142, 3762, 6237, 9867, 15015, 22113, 31668, 44268, 60588, 81396, 107559, 140049, 179949, 228459, 286902, 356730, 439530, 537030, 651105, 783783, 937251, 1113861, 1316136, 1546776, 1808664, 2104872, 2438667, 2813517, 3233097 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

A112851 is the fourth sequence in A112852.

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

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, Summation of Series (1961), eq (54) page 10]

a(n) = Sum_{i=0..n} A000217(i-1)*A000217(i), where A000217(-1)=0. - Bruno Berselli, Feb 05 2014

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, a(0)=a(1)=0, a(2)=3, a(3)=21, a(4)=81, a(5)=231. - 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

a(n) = Sum_{i=0..n} A000217(i)*A033428(n-i). - Bruno Berselli, Mar 06 2018

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

Partial sums of sequence A006011.

Cf. A000217, A033428, A112852.

Sequence in context: A067002 A110450 A102832 * A253943 A034490 A071351

Adjacent sequences:  A112848 A112849 A112850 * A112852 A112853 A112854

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

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Last modified February 18 00:19 EST 2019. Contains 320237 sequences. (Running on oeis4.)