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A172117 a(n) = n*(n+1)*(20*n-17)/6. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

B. 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).

FORMULA

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

a(0)=0, a(1)=1, a(2)=23, a(3)=86; for n>3, a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). [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]

MATHEMATICA

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

PROG

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

(MAGMA) [n*(n+1)*(20*n-17)/6: n in [0..60]]; // Vincenzo Librandi, Aug 01 2015

CROSSREFS

Cf. A051624.

Cf. similar sequences listed in A237616.

Sequence in context: A060456 A056580 A010011 * A217529 A284711 A193018

Adjacent sequences:  A172114 A172115 A172116 * A172118 A172119 A172120

KEYWORD

nonn,easy

AUTHOR

Vincenzo Librandi, Jan 26 2010

STATUS

approved

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Last modified May 30 06:51 EDT 2017. Contains 287302 sequences.