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%I #66 Apr 18 2023 17:41:47
%S 1,2,5,14,19,44,51,100,109,190,201,322,335,504,519,744,761,1050,1069,
%T 1430,1451,1892,1915,2444,2469,3094,3121,3850,3879,4720,4751,5712,
%U 5745,6834,6869,8094,8131,9500,9539,11060,11101,12782,12825,14674,14719,16744
%N a(n) = 1*2 + 3*4 + 5*6 + 7*8 + 9*10 + 11*12 + 13*14 + ... + (up to n).
%C In general, for sequences that multiply the first k natural numbers, and then add the product of the next k natural numbers (preserving the order of operations up to n), we have a(n) = Sum_{i=1..floor(n/k)} (k*i)!/(k*i-k)! + Sum_{j=1..k-1} (1-sign((n-j) mod k)) * (Product_{i=1..j} n-i+1). Here, k=2. - _Wesley Ivan Hurt_, Sep 10 2018
%C a(2n) is the total area of the family of n rectangles, where the k-th rectangle has dimensions (2k) X (2k-1). - _Wesley Ivan Hurt_, Oct 01 2018
%H Colin Barker, <a href="/A228958/b228958.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-3,-3,3,1,-1).
%F a(n) = (1/12)*(2*n^3+4*n+3/2+(3*n^2-6*n-3/2)*(-1)^n). [based on Alcover program]
%F G.f.: x*(x^5 - x^4 + 6*x^3 + x + 1)/((x-1)^4*(x+1)^3). [_Joerg Arndt_, Sep 13 2013]
%F E.g.f.: (x*(9 + 9*x + 2*x^2)*cosh(x) + (3 + 3*x + 3*x^2 + 2*x^3)*sinh(x))/12. - _Stefano Spezia_, Apr 18 2023
%e 1 = 1
%e 1*2 = 2
%e 1*2 + 3 = 5
%e 1*2 + 3*4 = 14
%e 1*2 + 3*4 + 5 = 19
%e 1*2 + 3*4 + 5*6 = 44
%e 1*2 + 3*4 + 5*6 + 7 = 51
%e 1*2 + 3*4 + 5*6 + 7*8 = 100
%e 1*2 + 3*4 + 5*6 + 7*8 + 9 = 109
%e 1*2 + 3*4 + 5*6 + 7*8 + 9*10 = 190
%e ...
%t a[n_?OddQ] := (2*n^3-3*n^2+10*n+3)/12; a[n_?EvenQ] := n*(n+2)*(2*n-1)/12; Table[a[n], {n, 1, 40}] (* _Jean-François Alcover_, Sep 10 2013 *)
%t CoefficientList[Series[x(x^5 - x^4 + 6*x^3 + x + 1)/((x-1)^4*(x+1)^3), {x, 0, 40}], x] (* _Stefano Spezia_, Sep 23 2018 *)
%t LinearRecurrence[{1,3,-3,-3,3,1,-1},{1,2,5,14,19,44,51},50] (* _Harvey P. Dale_, Mar 11 2023 *)
%o (PARI) Vec( x*(x^5 - x^4 + 6*x^3 + x + 1)/((x-1)^4*(x+1)^3) + O(x^66) ) \\ _Joerg Arndt_, Sep 17 2013
%o (Magma) [(1/12)*(2*n^3+4*n+3/2+(3*n^2-6*n-3/2)*(-1)^n): n in [1..50]]; // _Vincenzo Librandi_, Sep 11 2018
%Y Cf. A093361, (k=1) A000217, (k=2) this sequence, (k=3) A319014, (k=4) A319205, (k=5) A319206, (k=6) A319207, (k=7) A319208, (k=8) A319209, (k=9) A319211, (k=10) A319212.
%K nonn,easy
%O 1,2
%A _Robert Pfister_, Sep 09 2013
%E Definition corrected by _Ivan Panchenko_, Dec 02 2013
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