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A007587 12-gonal (or dodecagonal) pyramidal numbers: n(n+1)(10n-7)/6.
(Formerly M4895)
6
0, 1, 13, 46, 110, 215, 371, 588, 876, 1245, 1705, 2266, 2938, 3731, 4655, 5720, 6936, 8313, 9861, 11590, 13510, 15631, 17963, 20516, 23300, 26325, 29601, 33138, 36946, 41035, 45415, 50096, 55088, 60401, 66045, 72030, 78366, 85063, 92131, 99580, 107420, 115661 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Binomial transform of [1, 12, 21, 10, 0, 0, 0,...] = (1, 13, 46, 110,...). - Gary W. Adamson, Nov 28 2007

This sequence is related to A000566 by a(n) = n*A000566(n)-sum(A000566(i), i=0..n-1) and this is the case d=5 in the identity n*(n*(d*n-d+2)/2)-sum(k*(d*k-d+2)/2, k=0..n-1) = n*(n+1)*(2*d*n-2*d+3)/6. - Bruno Berselli, Oct 18 2010

REFERENCES

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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.

FORMULA

a(n) = (10*n-7)*binomial(n+1, 2)/3.

G.f.: x*(1+9*x)/(1-x)^4.

a(n) = Sum_{k=0..n} k*(5*k-4). [Klaus Brockhaus, Nov 20 2008]

a(n) = sum( (n-i)*(10*i+1), i=0..n-1 ), with a(0)=0. [Bruno Berselli, Feb 10 2014]

MATHEMATICA

CoefficientList[Series[x (1 + 9 x) / (1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 20 2013 *)

Table[n(n+1)(10n-7)/6, {n, 0, 50}] (* Harvey P. Dale, Nov 13 2013 *)

PROG

(MAGMA) [ n eq 1 select 0 else Self(n-1)+(n-1)*(5*n-9): n in [1..35] ]; // Klaus Brockhaus, Nov 20 2008

CROSSREFS

Cf. A093645 ((10, 1) Pascal, column m=3). Partial sums of A051624.

Cf. A000566.

Cf. similar sequences listed in A237616.

Sequence in context: A121964 A147208 A010003 * A219905 A034462 A116476

Adjacent sequences:  A007584 A007585 A007586 * A007588 A007589 A007590

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, R. K. Guy.

STATUS

approved

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Last modified April 25 04:22 EDT 2014. Contains 240994 sequences.