A007584 9-gonal (or enneagonal) pyramidal numbers: a(n) = n*(n+1)*(7*n-4)/6. (Formerly M4695)

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

0,3

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).

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

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 *)

Prog

(Maxima) A007584[n]:=n*(n+1)*(7*n-4)/6$
makelist(A007584[n], n, 0, 30); /* Martin Ettl, Oct 29 2012 */
(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
(PARI) a(n) = n*(n+1)*(7*n-4)/6; \\ Michel Marcus, Mar 04 2014

Crossrefs

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

Cf. similar sequences listed in A237616.

Sequence in context: A225276 A008527 A366415 * A218329 A009924 A297721

Adjacent sequences: A007581 A007582 A007583 * A007585 A007586 A007587

Keyword

nonn,easy

Author

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

Status

approved