OFFSET
0,3
COMMENTS
This sequence is related to A180223 by 2*a(n) = n*A180223(n) - Sum_{i=0..n-1} A180223(i). Also, 13-gonal (or tridecagonal) pyramidal numbers. - Bruno Berselli, Dec 14 2010
REFERENCES
A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 189-196.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93.
LINKS
Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian), 2008.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
a(n) = n*(n+1)*(11*n-8)/6.
G.f.: x*(1+10*x)/(1-x)^4. - Bruno Berselli, Aug 19 2010
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Bruno Berselli, Aug 19 2010
a(n) = Sum_{i=0..n-1} (n-i)*(11*i+1), with a(0)=0. - Bruno Berselli, Feb 10 2014
E.g.f.: exp(x)*x*(6 + 36*x + 11*x^2)/6. - Stefano Spezia, May 04 2022
EXAMPLE
After 0, the sequence is provided by the row sums of the triangle (see above, fourth formula):
1;
2, 12;
3, 24, 23;
4, 36, 46, 34;
5, 48, 69, 68, 45; ... - Vincenzo Librandi, Feb 12 2014
MAPLE
seq(n*(n+1)*(11*n-8)/6, n=0..40); # G. C. Greubel, Aug 30 2019
MATHEMATICA
Accumulate[Table[n (11n-9)/2, {n, 0, 40}]] (* or *) LinearRecurrence[ {4, -6, 4, -1}, {0, 1, 14, 50}, 40] (* Harvey P. Dale, Nov 14 2011 *)
CoefficientList[Series[x (1 + 10 x)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Feb 12 2014 *)
PROG
(Magma) I:=[0, 1, 14, 50]; [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..50]]; // Vincenzo Librandi, Feb 12 2014
(PARI) a(n)=n*(n+1)*(11*n-8)/6 \\ Charles R Greathouse IV, Oct 07 2015
(Sage) [n*(n+1)*(11*n-8)/6 for n in (0..40)] # G. C. Greubel, Aug 30 2019
(GAP) List([0..40], n-> n*(n+1)*(11*n-8)/6); # G. C. Greubel, Aug 30 2019
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
Barry E. Williams, Dec 23 1999
STATUS
approved