A350143
a(n) = Sum_{k=1..n} floor(n/(2*k-1))^2.
Original entry on oeis.org
1, 4, 10, 17, 27, 41, 55, 70, 93, 115, 137, 167, 193, 223, 267, 298, 332, 381, 419, 465, 525, 571, 617, 679, 738, 792, 868, 930, 988, 1080, 1142, 1205, 1297, 1367, 1459, 1560, 1634, 1712, 1820, 1914, 1996, 2120, 2206, 2300, 2450, 2544, 2638, 2764, 2875, 2996, 3136, 3246
Offset: 1
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a[n_] := Sum[Floor[n/(2*k - 1)]^2, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Dec 17 2021 *)
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a(n) = sum(k=1, n, (n\(2*k-1))^2);
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a(n) = sum(k=1, n, sumdiv(k, d, k/d%2*(2*d-1)));
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my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (2*k-1)*x^k/(1-x^(2*k)))/(1-x))
A350144
a(n) = Sum_{k=1..n} floor(n/(2*k-1))^3.
Original entry on oeis.org
1, 8, 28, 65, 127, 225, 353, 522, 759, 1037, 1369, 1803, 2273, 2827, 3539, 4260, 5078, 6095, 7123, 8301, 9709, 11103, 12623, 14449, 16312, 18270, 20614, 22920, 25358, 28338, 31130, 34107, 37627, 41001, 44761, 48976, 52974, 57200, 62136, 66986, 71908, 77720, 83140, 88854
Offset: 1
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a[n_] := Sum[Floor[n/(2*k - 1)]^3, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Dec 17 2021 *)
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a(n) = sum(k=1, n, (n\(2*k-1))^3);
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a(n) = sum(k=1, n, sumdiv(k, d, k/d%2*(d^3-(d-1)^3)));
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my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (k^3-(k-1)^3)*x^k/(1-x^(2*k)))/(1-x))
A350145
a(n) = Sum_{k=1..n} floor(n/(2*k-1))^n.
Original entry on oeis.org
1, 4, 28, 257, 3127, 46721, 823673, 16777474, 387440175, 10000060075, 285311849809, 8916117229571, 302875173709313, 11112007094026243, 437893920912819179, 18446744226340554502, 827240262649405488542, 39346408176856882188621
Offset: 1
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a[n_] := Sum[Floor[n/(2*k - 1)]^n, {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Dec 17 2021 *)
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a(n) = sum(k=1, n, (n\(2*k-1))^n);
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a(n) = sum(k=1, n, sumdiv(k, d, k/d%2*(d^n-(d-1)^n)));
A350161
Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} (-1)^(j+1) * floor(n/(2*j-1))^k.
Original entry on oeis.org
1, 1, 2, 1, 4, 2, 1, 8, 8, 3, 1, 16, 26, 15, 5, 1, 32, 80, 63, 25, 5, 1, 64, 242, 255, 125, 33, 5, 1, 128, 728, 1023, 625, 209, 45, 6, 1, 256, 2186, 4095, 3125, 1281, 335, 60, 7, 1, 512, 6560, 16383, 15625, 7745, 2385, 504, 73, 9, 1, 1024, 19682, 65535, 78125, 46593, 16775, 4080, 703, 95, 9
Offset: 1
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
2, 4, 8, 16, 32, 64, 128, ...
2, 8, 26, 80, 242, 728, 2186, ...
3, 15, 63, 255, 1023, 4095, 16383, ...
5, 25, 125, 625, 3125, 15625, 78125, ...
5, 33, 209, 1281, 7745, 46593, 279809, ...
5, 45, 335, 2385, 16775, 117585, 823415, ...
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T[n_, k_] := Sum[(-1)^(j + 1) * Floor[n/(2*j - 1)]^k, {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Dec 18 2021 *)
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T(n, k) = sum(j=1, n, (-1)^(j+1)*(n\(2*j-1))^k);
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T(n, k) = sum(j=1, n, sumdiv(j, d, kronecker(-4, j/d)*(d^k-(d-1)^k)));
Showing 1-4 of 4 results.