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 A172073 a(n) = (4*n^3 + n^2 - 3*n)/2. 7

%I

%S 0,1,15,54,130,255,441,700,1044,1485,2035,2706,3510,4459,5565,6840,

%T 8296,9945,11799,13870,16170,18711,21505,24564,27900,31525,35451,

%U 39690,44254,49155,54405,60016,66000,72369,79135,86310,93906,101935,110409,119340

%N a(n) = (4*n^3 + n^2 - 3*n)/2.

%C 14-gonal (or tetradecagonal) pyramidal numbers generated by the formula n*(n+1)*(2*d*n-(2*d-3))/6 for d=6.

%C In fact, the sequence is related to A000567 by a(n) = n*A000567(n) - Sum_{i=0..n-1} A000567(i) and this is the case d=6 in the identity n*(n*(d*n-d+2)/2) - Sum_{k=0..n-1} k*(d*k-d+2)/2 = n*(n+1)*(2*d*n-2*d+3)/6. - _Bruno Berselli_, Nov 29 2010

%C Except for the initial 0, this is the principal diagonal of the convolution array A213761. - _Clark Kimberling_, Jul 04 2012

%C Starting (1, 15, 54,...), this is the binomial transform of (1, 14, 25, 12, 0, 0, 0,...). - _Gary W. Adamson_, Jul 29 2015

%D E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93. - _Bruno Berselli_, Feb 13 2014

%H Vincenzo Librandi, <a href="/A172073/b172073.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/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) = n*(n+1)*(4*n-3)/2.

%F G.f.: x*(1+11*x)/(1-x)^4. - _Bruno Berselli_, Dec 15 2010

%F a(n) = Sum_{i=0..n} A051866(i). - _Bruno Berselli_, Dec 15 2010

%F a(0)=0, a(1)=1, a(2)=15, a(3)=54; for n>3, a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - _Harvey P. Dale_, Jan 29 2013

%F a(n) = Sum_{i=0..n-1} (n-i)*(12*i+1), with a(0)=0. - _Bruno Berselli_, Feb 10 2014

%t f[n_] := n (n + 1) (4 n - 3)/2; Array[f, 40, 0]

%t LinearRecurrence[{4,-6,4,-1},{0,1,15,54},40] (* _Harvey P. Dale_, Jan 29 2013 *)

%t CoefficientList[Series[x (1 + 11 x)/(1 - x)^4, {x, 0, 40}], x] (* _Vincenzo Librandi_, Jan 01 2014 *)

%o (MAGMA) [(4*n^3+n^2-3*n)/2: n in [0..50]]; // _Vincenzo Librandi_, Jan 01 2014

%o (PARI) a(n)=(4*n^3+n^2-3*n)/2 \\ _Charles R Greathouse IV_, Oct 07 2015

%Y Cf. similar sequences listed in A237616.

%Y Cf. A194454.

%K nonn,easy

%O 0,3

%A _Vincenzo Librandi_, Jan 25 2010

%E Edited by _Bruno Berselli_, Dec 14 2010

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Last modified March 26 12:43 EDT 2019. Contains 321497 sequences. (Running on oeis4.)