A050441 Partial sums of A051865.

0, 1, 14, 50, 120, 235, 406, 644, 960, 1365, 1870, 2486, 3224, 4095, 5110, 6280, 7616, 9129, 10830, 12730, 14840, 17171, 19734, 22540, 25600, 28925, 32526, 36414, 40600, 45095, 49910, 55056, 60544, 66385, 72590, 79170, 86136, 93499, 101270

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 to sequences related to pyramidal numbers

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

Cf. A051865, A180223.

Similar sequences are listed in A237616.

Sequence in context: A005914 A009960 A009928 * A225921 A350850 A205354

Adjacent sequences: A050438 A050439 A050440 * A050442 A050443 A050444

Keyword

nonn,easy,nice

Author

Barry E. Williams, Dec 23 1999

Status

approved