A007584 9-gonal (or enneagonal) pyramidal numbers: a(n) = n*(n+1)*(7*n-4)/6.

0, 1, 10, 34, 80, 155, 266, 420, 624, 885, 1210, 1606, 2080, 2639, 3290, 4040, 4896, 5865, 6954, 8170, 9520, 11011, 12650, 14444, 16400, 18525, 20826, 23310, 25984, 28855, 31930, 35216, 38720, 42449, 46410, 50610, 55056, 59755, 64714, 69940, 75440, 81221

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

For n > 1, the digital roots of this sequence A010888(A007584(n)) form the purely periodic 27-cycle 1, 1, 7, 8, 2, 5, 6, 3, 3, 4, 4, 1, 2, 5, 8, 9, 6, 6, 7, 7, 4, 5, 8, 2, 3, 9, 9. For n > 1, the units digits of this sequence A010879(A007584(n)) form the purely periodic 20-cycle 1, 0, 4, 0, 5, 6, 0, 4, 5, 0, 6, 0, 9, 0, 0, 6, 5, 4, 0, 0. - Ant King, Oct 30 2012

Partial sums of A001106. - Joerg Arndt, Jun 10 2013

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
• Index to sequences related to pyramidal numbers
• Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

Crossrefs

Cf. A093564 ((7, 1) Pascal, column m=3).

Cf. similar sequences listed in A237616.

Programs

Magma

I:=[0, 1, 10, 34, 80]; [n le 5 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jun 10 2013

Maple

a:=n->sum((n+j)^2-(n+j), j=0..n): seq(a(n)/2, n=0..30); # Zerinvary Lajos, May 26 2008

Mathematica

Table[n*(n+1)(7n-4)/6, {n, 0,100}] (* Vladimir Joseph Stephan Orlovsky, Jun 25 2009 *)
LinearRecurrence[{4,-6,4,-1},{1,10,34,80},30] (* Ant King, Oct 27 2012 *)
CoefficientList[Series[x (1 + 6 x) / (1 - x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 10 2013 *)

Maxima

A007584[n]:=n*(n+1)*(7*n-4)/6$
makelist(A007584[n],n,0,30); /* Martin Ettl, Oct 29 2012 */

PARI

a(n) = n*(n+1)*(7*n-4)/6; \\ Michel Marcus, Mar 04 2014

Formula

a(n) = (7*n-4)*binomial(n+1, 2)/3.

G.f.: x*(1+6*x)/(1-x)^4.

From Ant King, Oct 27 2012: (Start)

a(n) = a(n-1) + n*(7*n-5)/2.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 7.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).

a(n) = (n+1)*(2*A001106(n)+n)/6.

a(n) = A000292(n) + 6*A000292(n-1).

a(n) = A002414(n) + A000292(n-1).

a(n) = A000217(n) + 7*A000292(n-1).

a(n) = binomial(n+2,3) + 6*binomial(n+1,3). (End)

a(n) = Sum_{i = 0..n-1} (n-i)*(7*i+1) for n>0. - Bruno Berselli, Feb 10 2014

a(n) = A080851(7,n-1). - R. J. Mathar, Jul 28 2016

E.g.f.: (x/6)*(6 + 24*x + 7*x^2)*exp(x). - G. C. Greubel, Oct 29 2017