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a(n) = n*(5*n^2-8*n+5)/2.
8

%I #34 Sep 08 2022 08:46:05

%S 0,1,9,39,106,225,411,679,1044,1521,2125,2871,3774,4849,6111,7575,

%T 9256,11169,13329,15751,18450,21441,24739,28359,32316,36625,41301,

%U 46359,51814,57681,63975,70711,77904,85569,93721,102375,111546,121249,131499,142311,153700

%N a(n) = n*(5*n^2-8*n+5)/2.

%C Sequences of the type b(m)+m*b(m-1), where b is a polygonal number:

%C A006003(n) = A000217(n) + n*A000217(n-1) (b = triangular numbers);

%C A069778(n) = A000290(n+1) + (n+1)*A000290(n) (b = square numbers);

%C A143690(n) = A000326(n+1) + (n+1)*A000326(n) (b = pentagonal numbers);

%C A212133(n) = A000384(n) + n*A000384(n-1) (b = hexagonal numbers);

%C a(n) = A000566(n) + n*A000566(n-1) (b = heptagonal numbers);

%C A226450(n) = A000567(n) + n*A000567(n-1) (b = octagonal numbers);

%C A226451(n) = A001106(n) + n*A001106(n-1) (b = nonagonal numbers);

%C A204674(n) = A001107(n+1) + (n+1)*A001107(n) (b = decagonal numbers).

%H Bruno Berselli, <a href="/A226449/b226449.txt">Table of n, a(n) for n = 0..1000</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+5*x+9*x^2)/(1-x)^4.

%F a(n) - a(-n) = A008531(n) for n>0.

%t Table[n (5 n^2 - 8 n + 5)/2, {n, 0, 40}]

%t CoefficientList[Series[x (1 + 5 x + 9 x^2)/(1 - x)^4, {x, 0, 45}], x] (* _Vincenzo Librandi_, Aug 18 2013 *)

%t LinearRecurrence[{4,-6,4,-1},{0,1,9,39},50] (* _Harvey P. Dale_, May 19 2017 *)

%o (Magma) [n*(5*n^2-8*n+5)/2: n in [0..40]];

%o (Magma) I:=[0,1,9,39]; [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*(5*n^2-8*n+5)/2 \\ _Charles R Greathouse IV_, Oct 07 2015

%Y Cf. (see the comment) A000566, A006003, A069778, A143690, A204674, A212133, A226450, A226451.

%K nonn,easy

%O 0,3

%A _Bruno Berselli_, Jun 07 2013