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. 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.
KEYWORD
nonn,easy,nice
AUTHOR
Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 22 1999
STATUS
approved