A050409 Truncated square pyramid numbers: a(n) = Sum_{k = n..2*n} k^2.

0, 5, 29, 86, 190, 355, 595, 924, 1356, 1905, 2585, 3410, 4394, 5551, 6895, 8440, 10200, 12189, 14421, 16910, 19670, 22715, 26059, 29716, 33700, 38025, 42705, 47754, 53186, 59015, 65255, 71920, 79024, 86581, 94605, 103110, 112110, 121619

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From N. J. A. Sloane, Feb 13 2013

M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

a(n) = n*(n+1)*(14*n+1)/6.

a(n) = A132121(n,4) for n>3. - Reinhard Zumkeller, Aug 12 2007

From Bruno Berselli, Feb 11 2011: (Start)

G.f.: x*(5+9*x)/(1-x)^4.

a(n) = A129371(2*n). (End)

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 22 2012

E.g.f.: x*(30 + 57*x + 14*x^2)*exp(x)/6. - G. C. Greubel, Oct 30 2019

Maple

seq(add((n+k)^2, k=0..n), n=0..40); # Zerinvary Lajos, Dec 01 2006

Mathematica

LinearRecurrence[{4, -6, 4, -1}, {0, 5, 29, 86}, 40] (* Vincenzo Librandi, Jun 22 2012 *)
Table[(n(n+1)(14n+1))/6, {n, 0, 40}] (* Harvey P. Dale, Mar 08 2020 *)

Prog

(Magma) [&+[k^2: k in [n..2*n]]: n in [0..40]]; // Bruno Berselli, Feb 11 2011
(Magma) I:=[0, 5, 29, 86]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jun 22 2012
(PARI) a(n)=sum(k=n, n+n, k^2)
(PARI) vector(40, n, n*(n-1)*(14*n-13)/6) \\ G. C. Greubel, Oct 30 2019
(Sage) [n*(n+1)*(14*n+1)/6 for n in (0..40)] # G. C. Greubel, Oct 30 2019
(GAP) List([0..40], n-> n*(n+1)*(14*n+1)/6); # G. C. Greubel, Oct 30 2019

Crossrefs

Cf. A000330, A033994, A129371, A132112, A132121, A132124.

Cf. A225144. - Bruno Berselli, Jun 06 2013

Cf. A045943: Sum_{k = n..2*n} k.

Cf. A304993: Sum_{k = n..2*n} k*(k+1)/2.

Sequence in context: A339935 A236075 A272650 * A111937 A215850 A308396

Adjacent sequences: A050406 A050407 A050408 * A050410 A050411 A050412

Keyword

nonn,easy,nice

Author

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 22 1999

Status

approved