A172078 a(n) = n*(16*n^2 + 3*n - 13)/6.

0, 1, 19, 70, 170, 335, 581, 924, 1380, 1965, 2695, 3586, 4654, 5915, 7385, 9080, 11016, 13209, 15675, 18430, 21490, 24871, 28589, 32660, 37100, 41925, 47151, 52794, 58870, 65395, 72385, 79856, 87824, 96305, 105315, 114870, 124986, 135679

Offset: 0

Author

Vincenzo Librandi, Jan 25 2010

Keywords

• nonn
• easy

Comments

Generated by the formula n*(n+1)*(2*d*n-(2*d-3))/6 for d=8.

In fact, the sequence is related to A001107 by a(n) = n*A001107(n) - Sum_{k=0..n-1} A001107(k), and this is the case d=8 in the identity n*(n*(d*n-d+2)/2) - Sum_{k=0..n-1} k*(d*k-d+2)/2 = n*(n+1)*(2*d*n-2*d+3)/6. - Bruno Berselli, Dec 14 2010

Inverse binomial transform of this sequence: 0, 1, 17, 16, 0, 0 (0 continued). - Bruno Berselli, Dec 14 2010

Principal diagonal of the convolution array A213835. - Clark Kimberling, Jul 04 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
• B. 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. A001107.

Cf. similar sequences listed in A237616.

Programs

GAP

List([0..40], n-> n*(n+1)*(16*n-13)/6); # G. C. Greubel, Aug 30 2019

Magma

[n*(n+1)*(16*n-13)/6: n in [0..40]]; // G. C. Greubel, Aug 30 2019

Maple

A172078:=n->(16*n^3+3*n^2-13*n)/6: seq(A172078(n), n=0..50); # Wesley Ivan Hurt, Jan 21 2017

Mathematica

LinearRecurrence[{4,-6,4,-1}, {0,1,19,70}, 50] (* Vincenzo Librandi, Mar 01 2012 *)
Table[n (16n^2+3n-13)/6,{n,0,40}] (* Harvey P. Dale, Aug 14 2023 *)

PARI

a(n)=n*(16*n^2+3*n-13)/6 \\ Charles R Greathouse IV, Oct 07 2015

Sage

[n*(n+1)*(16*n-13)/6 for n in (0..40)] # G. C. Greubel, Aug 30 2019

Formula

a(n) = n*(n+1)*(16*n-13)/6.

G.f.: x*(1+15*x)/(1-x)^4. - Bruno Berselli, Dec 14 2010

a(n) = Sum_{i=0..n-1} (n-i)*(16*i+1), with a(0)=0. - Bruno Berselli, Feb 10 2014

E.g.f.: x*(6 +51*x +16*x^2)*exp(x)/6. - G. C. Greubel, Aug 30 2019