A294285 -id:A294285 - cp's OEIS frontend

A335436 Triangle read by rows: T(n,k) = 2*n+1 for k = 0 and otherwise T(n,k) = Sum_{i=n-k..n, j=0..i-n+k, i<>n or j<>k} T(i,j).

1, 3, 4, 5, 8, 21, 7, 12, 35, 96, 9, 16, 49, 144, 410, 11, 20, 63, 192, 574, 1680, 13, 24, 77, 240, 738, 2240, 6692, 15, 28, 91, 288, 902, 2800, 8604, 26112, 17, 32, 105, 336, 1066, 3360, 10516, 32640, 100296, 19, 36, 119, 384, 1230, 3920, 12428, 39168, 122584, 380480

Offset: 0

Oboifeng Dira, Jul 14 2020

			Triangle begins:
  1;
  3,  4;
  5,  8, 21;
  7, 12, 35,  96;
  9, 16, 49, 144, 410;
  ...
T(3,2) = ((2+sqrt(2))^3-(2-sqrt(2))^3)*(6-2+1)/(4*sqrt(2)) = (28*sqrt(2))*(5)/(4*sqrt(2)) = 35.

Oboifeng Dira, Table of n, a(n) for n = 0..54

Cf. A053199, A294285, A064339.

T := proc (n, k) if k = 0 and 0 <= n then 2*n+1 elif 1 <= k and k <= n then round((((2+sqrt(2))^(k+1)-(2-sqrt(2))^(k+1))*(2*n-k+1)/(4*sqrt(2)))) else 0 end if end proc:seq(print(seq(T(n, k), k=0..n)), n=0..9);

T(n,k) = if (k==0, 2*n+1, if (k<=n, sum(i=n-k, n, sum(j=0, i-n+k, if ((i==n) && (j==k), 0, T(i,j)), 0))));
matrix(10, 10, n, k, T(n-1, k-1)) \\ Michel Marcus, Sep 08 2020

T(n, k) = if (k==0, 2*n+1, if (k>n, 0, my(w=quadgen(8, 'w)); ((2+w)^(k+1)-(2-w)^(k+1))*(2*n-k+1)/(4*w)));
matrix(10, 10, n, k, T(n-1, k-1)) \\ Michel Marcus, Sep 10 2020

T(n,0) = 2*n+1 for k=0;

T(n,k) = ((2+sqrt(2))^(k+1)-(2-sqrt(2))^(k+1))*(2*n-k+1)/(4*sqrt(2)) for 1<=k<=n.

A294284 Sum of the smaller parts of the partitions of n into two distinct parts with larger part squarefree.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 3, 6, 9, 7, 10, 8, 11, 8, 12, 17, 22, 28, 34, 31, 37, 33, 40, 48, 56, 51, 59, 53, 60, 53, 61, 70, 79, 72, 82, 93, 104, 97, 109, 122, 135, 128, 142, 135, 149, 140, 154, 169, 184, 199, 214, 204, 219, 235, 251, 268, 285, 274, 292, 281

Offset: 1

Wesley Ivan Hurt, Oct 26 2017

Sum of the widths of the distinct rectangles with squarefree length and positive integer width such that L + W = n, W < L. For example, a(13) = 11; the rectangles are 2 X 11, 3 X 10, 6 X 7. The sum of the widths is then 2 + 3 + 6 = 11. - Wesley Ivan Hurt, Nov 12 2017

Index entries for sequences related to partitions

Cf. A008683, A294145, A294285.

Table[Sum[i*MoebiusMu[n - i]^2, {i, Floor[(n-1)/2]}], {n, 60}]

a(n) = sum(i=1, (n-1)\2, i*moebius(n-i)^2); \\ Michel Marcus, Nov 05 2017

a(n) = Sum_{i=1..floor((n-1)/2)} i * mu(n-i)^2, where mu is the Möbius function (A008683).

cp's OEIS Frontend

A335436 Triangle read by rows: T(n,k) = 2*n+1 for k = 0 and otherwise T(n,k) = Sum_{i=n-k..n, j=0..i-n+k, i<>n or j<>k} T(i,j).

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A294284 Sum of the smaller parts of the partitions of n into two distinct parts with larger part squarefree.

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