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A081441 a(n) = (2*n^3 - n^2 - n + 2)/2.

%I A081441 #33 Jun 25 2025 18:04:53
%S A081441 1,1,6,22,55,111,196,316,477,685,946,1266,1651,2107,2640,3256,3961,
%T A081441 4761,5662,6670,7791,9031,10396,11892,13525,15301,17226,19306,21547,
%U A081441 23955,26536,29296,32241,35377,38710,42246,45991,49951,54132,58540,63181
%N A081441 a(n) = (2*n^3 - n^2 - n + 2)/2.
%C A081441 Diagonal in array of n-gonal numbers A081422.
%H A081441 Harvey P. Dale, <a href="/A081441/b081441.txt">Table of n, a(n) for n = 0..1000</a>
%H A081441 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A081441 G.f.: (1 -4*x +11*x^2 -8*x^3)/(1-x)^5.
%F A081441 a(n) = (n + 1)*(2*n^2 - 3*n + 2)/2 = (n-1)*A005564(n+1) - n*A005564(n), where A005564(0..2) = 0, -1, 0. - _Bruno Berselli_, May 19 2015
%F A081441 E.g.f.: (2 + 5*x^2 + 2*x^3)*exp(x)/2. - _G. C. Greubel_, Aug 14 2019
%p A081441 a:= n-> (2*n^3-n^2-n+2)/2: seq(a(n), n=0..50); # _Zerinvary Lajos_, Sep 13 2006
%t A081441 Table[(2n^3-n^2-n+2)/2,{n,0,40}] (* _Harvey P. Dale_, May 29 2012 *)
%t A081441 CoefficientList[Series[(1 - 4 x + 11 x^2 - 8 x^3) / (1 - x)^5, {x, 0, 50}],x] (* _Vincenzo Librandi_, Aug 08 2013 *)
%o A081441 (Magma) [(2*n^3-n^2-n+2)/2: n in [0..50]]; // _Vincenzo Librandi_, Aug 08 2013
%o A081441 (PARI) vector(40, n, n--; (2*n^3-n^2-n+2)/2) \\ _G. C. Greubel_, Aug 14 2019
%o A081441 (Sage) [(2*n^3-n^2-n+2)/2 for n in (0..40)] # _G. C. Greubel_, Aug 14 2019
%o A081441 (GAP) List([0..40], n-> (2*n^3-n^2-n+2)/2); # _G. C. Greubel_, Aug 14 2019
%Y A081441 Cf. A005564, A081422, A081436, A081437, A081438.
%K A081441 nonn,easy
%O A081441 0,3
%A A081441 _Paul Barry_, Mar 21 2003