A077538 First differences of triangular numbers with square pyramidal indices.

1, 14, 90, 360, 1075, 2646, 5684, 11040, 19845, 33550, 53966, 83304, 124215, 179830, 253800, 350336, 474249, 630990, 826690, 1068200, 1363131, 1719894, 2147740, 2656800, 3258125, 3963726, 4786614, 5740840, 6841535, 8104950

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

Sequence in context: A186257 A241305 A195267 * A114242 A054487 A200191

Adjacent sequences: A077535 A077536 A077537 * A077539 A077540 A077541

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