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 A296141 Sum of the smaller parts of the partitions of n into two distinct parts with the larger part even. 0
 0, 0, 1, 0, 1, 2, 4, 2, 4, 6, 9, 6, 9, 12, 16, 12, 16, 20, 25, 20, 25, 30, 36, 30, 36, 42, 49, 42, 49, 56, 64, 56, 64, 72, 81, 72, 81, 90, 100, 90, 100, 110, 121, 110, 121, 132, 144, 132, 144, 156, 169, 156, 169, 182, 196, 182, 196, 210, 225, 210, 225, 240 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS a(n+1) is the sum of the smaller parts in the partitions of n into two parts with the larger part odd. For example, a(11) = 9; the partitions of 10 into two parts are (9,1), (8,2), (7,3), (6,4) and (5,5). Three of these partitions have an odd number as their larger part, namely (9,1), (7,3) and (5,5). Adding the smaller parts of these partitions gives 1 + 3 + 5 = 9. LINKS FORMULA a(n) = Sum_{i=1..floor((n-1)/2)} i * ((n-i+1) mod 2). Conjectures from Colin Barker, Dec 06 2017: (Start) G.f.: x^3*(1 - x + x^2 + x^3) / ((1 - x)^3*(1 + x)^2*(1 + x^2)^2). a(n) = a(n-1) + 2*a(n-4) - 2*a(n-5) - a(n-8) + a(n-9) for n > 9. (End) a(n) = floor((n+1)/4)^2*(n mod 2)+(1+floor((n-2)/4))*floor((n-2)/4)*((n+1) mod 2). - Wesley Ivan Hurt, Dec 08 2017 EXAMPLE a(10) = 6; the partitions of 10 into two parts are (9,1), (8,2), (7,3), (6,4) and (5,5). Two of these partitions have an even number as their larger part, namely (8,2) and (6,4). Adding the smaller parts of these partitions gives 2 + 4 = 6. MATHEMATICA Table[Sum[i Mod[n - i + 1, 2], {i, Floor[(n - 1)/2]}], {n, 80}] PROG (PARI) a(n) = sum(i=1, floor((n-1)/2), i*lift(Mod(n-i+1, 2))) \\ Iain Fox, Dec 06 2017 CROSSREFS Cf. A295287, A295293. Sequence in context: A047975 A112791 A227346 * A286536 A318768 A166242 Adjacent sequences:  A296138 A296139 A296140 * A296142 A296143 A296144 KEYWORD nonn,easy AUTHOR Wesley Ivan Hurt, Dec 05 2017 STATUS approved

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Last modified July 13 11:46 EDT 2020. Contains 335687 sequences. (Running on oeis4.)