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%I #25 Sep 08 2022 08:46:05
%S 0,1,10,45,124,265,486,805,1240,1809,2530,3421,4500,5785,7294,9045,
%T 11056,13345,15930,18829,22060,25641,29590,33925,38664,43825,49426,
%U 55485,62020,69049,76590,84661,93280,102465,112234,122605,133596,145225,157510,170469
%N a(n) = n*(3*n^2 - 5*n + 3).
%C See the comment in A226449.
%C For n >= 3, also the detour index of the n-barbell graph. - _Eric W. Weisstein_, Dec 20 2017
%H Bruno Berselli, <a href="/A226450/b226450.txt">Table of n, a(n) for n = 0..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BarbellGraph.html">Barbell Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DetourIndex.html">Detour Index</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F G.f.: x*(1+6*x+11*x^2)/(1-x)^4.
%F a(n) = A000567(n) + n*A000567(n-1).
%t Table[n (3 n^2 - 5 n + 3), {n, 0, 40}]
%t CoefficientList[Series[x (1 + 6 x + 11 x^2)/(1 - x)^4, {x, 0, 40}], x] (* _Vincenzo Librandi_, Aug 18 2013 *)
%t LinearRecurrence[{4, -6, 4, -1}, {1, 10, 45, 124}, {0, 20}] (* _Eric W. Weisstein_, Dec 20 2017 *)
%o (Magma) [n*(3*n^2-5*n+3): n in [0..40]];
%o (Magma) I:=[0,1,10,45]; [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..45]]; // _Vincenzo Librandi_, Aug 18 2013
%o (PARI) a(n) = n*(3*n^2 - 5*n + 3); \\ _Altug Alkan_, Dec 20 2017
%Y Cf. A000567.
%Y Similar sequences of the type b(m)+m*b(m-1), where b is a polygonal number: A006003, A069778, A143690, A204674, A212133, A226449, A226451.
%K nonn,easy
%O 0,3
%A _Bruno Berselli_, Jun 07 2013
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