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%I #29 Sep 08 2022 08:45:41
%S 1,33,173,505,1113,2081,3493,5433,7985,11233,15261,20153,25993,32865,
%T 40853,50041,60513,72353,85645,100473,116921,135073,155013,176825,
%U 200593,226401,254333,284473,316905,351713,388981,428793,471233,516385,564333,615161,668953
%N a(n) = 14*n^3 - 30*n^2 + 24*n - 7.
%C Previous name was: The sequence gives the three-dimensional forms of the centered hexagonal numbers. Two examples: its third term 173 is built 19 + 37 + 61 + 37 + 19 and its fourth term 505 is built 37 + 61 + 91 + 127 + 91 + 61 + 37.
%C The sequence's digital roots run through 1, 6, 2.
%H Vincenzo Librandi, <a href="/A155883/b155883.txt">Table of n, a(n) for n = 1..1000</a>
%H David Z. Crookes, <a href="https://www.jstor.org/stable/30214492">De Pulchritudine Numerorum Figuratorum (On the Beauty of Figurate Numbers)</a>, Mathematics in School (May, 1988), 38-39.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = 14*n^3 - 30*n^2 + 24*n - 7.
%F G.f.: x*(1+29*x+47*x^2+7*x^3)/(1-x)^4. [_Colin Barker_, Jun 16 2012]
%F a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4). - _Vincenzo Librandi_, Jun 30 2012
%t CoefficientList[Series[(1+29*x+47*x^2+7*x^3)/(1-x)^4,{x,0,40}],x] (* _Vincenzo Librandi_, Jun 30 2012 *)
%o (Magma) I:=[1, 33, 173, 505]; [n le 4 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 30 2012
%K nonn,easy,less
%O 1,2
%A _David Z. Crookes_, Jan 29 2009
%E More terms from _Colin Barker_, Jun 16 2012
%E New name using explicit formula from _Joerg Arndt_, Jan 15 2021
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