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A172082 a(n) = n*(n+1)*(6*n-5)/2. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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. [From 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).

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+17*x^2)/(x-1)^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]

MATHEMATICA

Table[(18n^3+3n^2-15n)/6, {n, 0, 50}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {0, 1, 21, 78}, 50] (* Harvey P. Dale, Jun 29 2011 *)

CoefficientList[Series[(x + 17 x^2)/(x - 1)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jan 02 2014 *)

PROG

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

CROSSREFS

Cf. A051682.

Cf. similar sequences listed in A237616.

Sequence in context: A045559 A144314 A010009 * A068085 A135945 A158493

Adjacent sequences:  A172079 A172080 A172081 * A172083 A172084 A172085

KEYWORD

nonn,easy

AUTHOR

Vincenzo Librandi, Jan 25 2010

STATUS

approved

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Last modified May 23 18:42 EDT 2017. Contains 286926 sequences.