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 A007587 12-gonal (or dodecagonal) pyramidal numbers: n(n+1)(10n-7)/6. (Formerly M4895) 7

%I M4895

%S 0,1,13,46,110,215,371,588,876,1245,1705,2266,2938,3731,4655,5720,

%T 6936,8313,9861,11590,13510,15631,17963,20516,23300,26325,29601,33138,

%U 36946,41035,45415,50096,55088,60401,66045,72030,78366,85063,92131,99580,107420,115661

%N 12-gonal (or dodecagonal) pyramidal numbers: n(n+1)(10n-7)/6.

%C Binomial transform of [1, 12, 21, 10, 0, 0, 0,...] = (1, 13, 46, 110,...). - _Gary W. Adamson_, Nov 28 2007

%C 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

%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="/A007587/b007587.txt">Table of n, a(n) for n = 0..1000</a>

%H B. Berselli, A description of the recursive method in Comments lines: website <a href="http://www.lanostra-matematica.org/2008/12/sequenze-numeriche-e-procedimenti.html">Matem@ticamente</a> (in Italian), 2008.

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

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

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

%F a(n) = Sum_{k=0..n} k*(5*k-4). - _Klaus Brockhaus_, Nov 20 2008

%F a(n) = sum( (n-i)*(10*i+1), i=0..n-1 ), with a(0)=0. - _Bruno Berselli_, Feb 10 2014

%F a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - _Wesley Ivan Hurt_, Oct 23 2014

%p A007587:=n->n*(n+1)*(10*n-7)/6: seq(A007587(n), n=0..50); # _Wesley Ivan Hurt_, Oct 23 2014

%t CoefficientList[Series[x (1 + 9 x) / (1 - x)^4, {x, 0, 40}], x] (* _Vincenzo Librandi_, Jun 20 2013 *)

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

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

%o (PARI) a(n)=if(n,([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,4,-6,4]^n*[0;1;13;46])[1,1],0) \\ _Charles R Greathouse IV_, Oct 07 2015

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

%Y Cf. A000566.

%Y See similar sequences listed in A237616.

%K nonn,easy,nice

%O 0,3

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

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