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A077538 First differences of triangular numbers with square pyramidal indices. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified September 23 08:04 EDT 2021. Contains 347610 sequences. (Running on oeis4.)