A172117 a(n) = n*(n+1)*(20*n-17)/6.

0, 1, 23, 86, 210, 415, 721, 1148, 1716, 2445, 3355, 4466, 5798, 7371, 9205, 11320, 13736, 16473, 19551, 22990, 26810, 31031, 35673, 40756, 46300, 52325, 58851, 65898, 73486, 81635, 90365, 99696, 109648, 120241, 131495, 143430, 156066

Offset: 0

Author

Vincenzo Librandi, Jan 26 2010

Keywords

• nonn
• easy

Comments

Generated by the formula n*(n+1)*(2*d*n-2*d+3)/6 for d=10.

This sequence is related to A051624 by a(n) = n*A051624(n) - Sum_{i=0..n-1} A051624(i) = n*(n+1)*(20*n-17)/2; in fact, this is the case d=10 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, Aug 26 2010

Also, a(n) = n*A190816(n) - Sum_{i=0..n-1} A190816(i) for n>0. - Bruno Berselli, Dec 18 2013

Starting with offset 1, the sequence is the binomial transform of (1, 22, 41, 20, 0, 0, 0, ...). - Gary W. Adamson, Jul 31 2015

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 entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A051624.

Cf. similar sequences listed in A237616.

Programs

Magma

[n*(n+1)*(20*n-17)/6: n in [0..50]]; // Vincenzo Librandi, Aug 01 2015

Mathematica

Table[(20n^3+3n^2-17n)/6,{n,0,50}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,1,23,86},40] (* Harvey P. Dale, May 15 2011 *)

PARI

a(n)=n*(20*n^2+3*n-17)/6 \\ Charles R Greathouse IV, Jan 11 2012

SageMath

[sum( (-1)^j*(20-j)*binomial(n+2-j, 3-j) for j in (0..1)) for n in (0..50)] # G. C. Greubel, Apr 15 2022

Formula

G.f.: x*(1+19*x)/(1-x)^4. - Bruno Berselli, Aug 26 2010

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 3. - Harvey P. Dale, May 15 2011

a(n) = Sum_{i=0..n-1} (n-i)*(20*i+1), with a(0)=0. - Bruno Berselli, Feb 11 2014

E.g.f.: (1/6)*x*(6 + 63*x + 20*x^2)*exp(x). - G. C. Greubel, Apr 15 2022