A172082 a(n) = n*(n+1)*(6*n-5)/2.

0, 1, 21, 78, 190, 375, 651, 1036, 1548, 2205, 3025, 4026, 5226, 6643, 8295, 10200, 12376, 14841, 17613, 20710, 24150, 27951, 32131, 36708, 41700, 47125, 53001, 59346, 66178, 73515, 81375, 89776, 98736, 108273, 118405, 129150, 140526

Offset: 0

Author

Vincenzo Librandi, Jan 25 2010

Keywords

• nonn
• easy

Comments

Generated by formula: n*(n+1)*(2*d*n-2*d+3)/6 with d=9.

This sequence is related to A051682 by a(n) = n*A051682(n) - Sum_{i=0..n-1} A051682(i); in fact this is the case d=9 in the identity n*(n*(d*n-d+2)/2) - Sum_{i=0..n-1} i*(d*i-d+2)/2 = n*(n+1)*(2*d*n -2*d + 3)/6. - Bruno Berselli, Apr 16 2012

References

• E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93. - Bruno Berselli, Feb 13 2014

Links

• Vincenzo Librandi, 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. A051682.

Cf. similar sequences listed in A237616.

Programs

GAP

List([0..40], n-> n*(n+1)*(6*n-5)/2); # G. C. Greubel, Aug 30 2019

Magma

[(18*n^3+3*n^2-15*n)/6: n in [0..40]]; // Vincenzo Librandi, Jan 02 2014

Maple

seq(n*(n+1)*(6*n-5)/2, n=0..40); # G. C. Greubel, Aug 30 2019

Mathematica

Table[(18n^3+3n^2-15n)/6,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1}, {0,1,21,78}, 40] (* Harvey P. Dale, Jun 29 2011 *)
CoefficientList[Series[x*(1+17*x)/(1-x)^4, {x,0,40}], x] (* Vincenzo Librandi, Jan 02 2014 *)

PARI

vector(40, n, n*(n-1)*(6*n-11)/2) \\ G. C. Greubel, Aug 30 2019

Sage

[n*(n+1)*(6*n-5)/2 for n in (0..40)] # G. C. Greubel, Aug 30 2019

Formula

a(0)=0, a(1)=1, a(2)=21, a(3)=78; for n>3, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Jun 29 2011

G.f.: x*(1+17*x)/(1-x)^4. - Harvey P. Dale, Jun 29 2011

a(n) = Sum_{i=0..n-1} (n-i)*(18*i+1), with a(0)=0. - Bruno Berselli, Feb 10 2014

E.g.f.: x*(2 + 19*x + 6*x^2)*exp(x)/2. - G. C. Greubel, Aug 30 2019

From Amiram Eldar, Jan 10 2022: (Start)

Sum_{n>=1} 1/a(n) = 2*(3*sqrt(3)*Pi + 9*log(3) + 12*log(2) - 5)/55.

Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(6*Pi + 6*sqrt(3)*log(sqrt(3)+2) - 16*log(2) + 5)/55. (End)