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 A319866 a(n) = 2*1 + 4*3 + 6*5 + 8*7 + 10*9 + 12*11 + ... + (up to the n-th term). 9
 2, 2, 6, 14, 20, 44, 52, 100, 110, 190, 202, 322, 336, 504, 520, 744, 762, 1050, 1070, 1430, 1452, 1892, 1916, 2444, 2470, 3094, 3122, 3850, 3880, 4720, 4752, 5712, 5746, 6834, 6870, 8094, 8132, 9500, 9540, 11060, 11102, 12782, 12826, 14674, 14720, 16744 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS For similar multiply/add sequences in descending blocks of k natural numbers, we have: a(n) = Sum_{j=1..k-1} (floor((n-j)/k)-floor((n-j-1)/k)) * (Product_{i=1..j} n-i-j+k+1) + Sum_{j=1..n} (floor(j/k)-floor((j-1)/k)) * (Product_{i=1..k} j-i+1). Here, k=2. The denominators of the generating functions for these sequences are (1 + x)*(1 - x^k)^(k+1). - Georg Fischer and Andrew Howroyd, Mar 07 2020 LINKS Colin Barker, Table of n, a(n) for n = 1..1000 Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1). FORMULA G.f.: 2*x/((-1 + x)^2*(1 + x)^2) + 2*(x^2 + 3*x^4)/((-1 + x)^4 (1 + x)^3). - Stefano Spezia, Sep 30 2018 From Colin Barker, Sep 30 2018: (Start) a(n) = (4*n - 6*n + 3*n^2 + 2*n^3) / 12 for n even. a(n) = (15 + 4*n + 6*n - 3*n^2 + 2*n^3) / 12 for n odd. a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7) for n>7. (End) EXAMPLE a(1) = 2; a(2) = 2*1 = 2; a(3) = 2*1 + 4 = 6; a(4) = 2*1 + 4*3 = 14; a(5) = 2*1 + 4*3 + 6 = 20; a(6) = 2*1 + 4*3 + 6*5 = 44; a(7) = 2*1 + 4*3 + 6*5 + 8 = 52; a(8) = 2*1 + 4*3 + 6*5 + 8*7 = 100; a(9) = 2*1 + 4*3 + 6*5 + 8*7 + 10 = 110; a(10) = 2*1 + 4*3 + 6*5 + 8*7 + 10*9 = 190; a(11) = 2*1 + 4*3 + 6*5 + 8*7 + 10*9 + 12 = 202; etc. MAPLE a:=(n, k)->add((floor((n-j)/k)-floor((n-j-1)/k))*(mul(n-i-j+k+1, i=1..j)), j=1..k-1) + add((floor(j/k)-floor((j-1)/k))*(mul(j-i+1, i=1..k)), j=1..n): seq(a(n, 2), n=1..50); # Muniru A Asiru, Sep 30 2018 MATHEMATICA k:=2; a[n_]:= Sum[(Floor[(n-j)/k]-Floor[(n-j-1)/k])* Product[n-i-j+k+1, {i, 1, j }] , {j, 1, k-1} ]  + Sum[(Floor[j/k]-Floor[(j-1)/k])* Product[j-i+1, {i, 1, k} ], {j, 1, n}]; Array[a, 50] (* Stefano Spezia, Sep 30 2018 *) CoefficientList[Series[2/((-1 + x)^2 (1 + x)^2) + ( 2 (x + 3 x^3))/((-1 + x)^4 (1 + x)^3), {x, 0, 50}], x] (* Stefano Spezia, Sep 30 2018 *) PROG (PARI) Vec(2*x*(1 - x^2 + 4*x^3) / ((1 - x)^4*(1 + x)^3) + O(x^50)) \\ Colin Barker, Sep 30 2018 CROSSREFS For similar sequences, see: A000217 (k=1), this sequence (k=2), A319867 (k=3), A319868 (k=4), A319869 (k=5), A319870 (k=6), A319871 (k=7), A319872 (k=8), A319873 (k=9), A319874 (k=10). Sequence in context: A019101 A233230 A320070 * A266007 A051890 A071109 Adjacent sequences:  A319863 A319864 A319865 * A319867 A319868 A319869 KEYWORD nonn,easy AUTHOR Wesley Ivan Hurt, Sep 29 2018 STATUS approved

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Last modified August 7 10:19 EDT 2022. Contains 355985 sequences. (Running on oeis4.)