OFFSET
0,2
COMMENTS
This sequence is also the sums of a partition of the integers into groups of (n+1)^2 integers starting at 1 and not repeating or skipping any: a(0)=1, a(1)=2+3+4+5=14, a(2)=6+7+8+9+10+11+12+13+14=90, etc.
LINKS
Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).
FORMULA
Let SP(m) be the m-th square pyramidal number m*(m+1)*(2*m+1)/6 and let T(k) be the k-th Triangular number k*(k+1)/2; then a(n) = T(SP(n+1))-T(SP(n)) = ((n+1)^2*(n+2)*(2*n^2+2*n+3))/6.
G.f.: (1+8*x+21*x^2+10*x^3)/(1-x)^6. [Colin Barker, Apr 30 2012]
EXAMPLE
SP(3)=14, SP(4)=30, T(14)=105 and T(30)=465, so a(3)=465-105=360.
MATHEMATICA
nn=30; Join[{1}, With[{tr=Accumulate[Range[(nn(nn+1)(2nn+1))/6]]}, Differences[ Table[tr[[n]], {n, Accumulate[Range[nn]^2]}]]]](* or *) LinearRecurrence[ {6, -15, 20, -15, 6, -1}, {1, 14, 90, 360, 1075, 2646}, 30] (* Harvey P. Dale, Mar 07 2013 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Amarnath Murthy, Nov 09 2002
EXTENSIONS
More terms and better description from Bruce Corrigan (scentman(AT)myfamily.com), Nov 14 2002
STATUS
approved