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A007584 9-gonal (or enneagonal) pyramidal numbers: a(n) = n*(n+1)*(7*n-4)/6.
(Formerly M4695)
15

%I M4695

%S 0,1,10,34,80,155,266,420,624,885,1210,1606,2080,2639,3290,4040,4896,

%T 5865,6954,8170,9520,11011,12650,14444,16400,18525,20826,23310,25984,

%U 28855,31930,35216,38720,42449,46410,50610,55056,59755,64714,69940,75440,81221

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

%C 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

%C Partial sums of A001106. - _Joerg Arndt_, Jun 10 2013

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

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

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

%H Vincenzo Librandi, <a href="/A007584/b007584.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4, -6, 4, -1).

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

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

%F From _Ant King_, Oct 27 2012: (Start)

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

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

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

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

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

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

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

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

%F a(n) = Sum_{i = 0..n-1} (n-i)*(7*i+1) for n>0. - _Bruno Berselli_, Feb 10 2014

%F a(n) = A080851(7,n-1). - _R. J. Mathar_, Jul 28 2016

%F E.g.f.: (x/6)*(6 + 24*x + 7*x^2)*exp(x). - _G. C. Greubel_, Oct 29 2017

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

%t Table[n*(n+1)(7n-4)/6, {n, 0,100}] (* _Vladimir Joseph Stephan Orlovsky_, Jun 25 2009 *)

%t LinearRecurrence[{4,-6,4,-1},{1,10,34,80},30] (* _Ant King_, Oct 27 2012 *)

%t CoefficientList[Series[x (1 + 6 x) / (1 - x)^4, {x, 0, 50}], x] (* _Vincenzo Librandi_, Jun 10 2013 *)

%o (Maxima) A007584[n]:=n*(n+1)*(7*n-4)/6$

%o makelist(A007584[n],n,0,30); /* _Martin Ettl_, Oct 29 2012 */

%o (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

%o (PARI) a(n) = n*(n+1)*(7*n-4)/6; \\ _Michel Marcus_, Mar 04 2014

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

%Y Cf. similar sequences listed in A237616.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_, _R. K. Guy_

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Last modified March 22 17:25 EDT 2019. Contains 321422 sequences. (Running on oeis4.)