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

Author

Barry E. Williams, Dec 23 1999

Keywords

• nonn
• easy
• nice

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

Examples

			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
		

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).

Crossrefs

Cf. A051865, A180223.

Similar sequences are listed in A237616.

Programs

GAP

List([0..40], n-> n*(n+1)*(11*n-8)/6); # G. C. Greubel, Aug 30 2019

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

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 *)

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

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