A329404
Interleave 2*n*(3*n-1), (2*n+1)*(6*n+1) for n >= 0.
Original entry on oeis.org
0, 1, 4, 21, 20, 65, 48, 133, 88, 225, 140, 341, 204, 481, 280, 645, 368, 833, 468, 1045, 580, 1281, 704, 1541, 840, 1825, 988, 2133, 1148, 2465, 1320, 2821, 1504, 3201, 1700, 3605, 1908, 4033, 2128, 4485, 2360, 4961
Offset: 0
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LinearRecurrence[{0,3,0,-3,0,1},{0,1,4,21,20,65},100] (* Paolo Xausa, Nov 13 2023 *)
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concat(0, Vec(x*(1 + 4*x + 18*x^2 + 8*x^3 + 5*x^4) / ((1 - x)^3*(1 + x)^3) + O(x^45))) \\ Colin Barker, Nov 13 2019
A180855
Square array read by antidiagonals: T(m,n) is the Wiener index of the banana tree B(n,k) (n>=1, k>=2). B(n,k) is the graph obtained by taking n copies of a star graph on k nodes and connecting with an edge one leaf of each of these n stars with an additional node.
Original entry on oeis.org
4, 20, 10, 48, 56, 18, 88, 138, 108, 28, 140, 256, 270, 176, 40, 204, 410, 504, 444, 260, 54, 280, 600, 810, 832, 660, 360, 70, 368, 826, 1188, 1340, 1240, 918, 476, 88, 468, 1088, 1638, 1968, 2000, 1728, 1218, 608, 108, 580, 1386, 2160, 2716, 2940, 2790, 2296, 1560, 756, 130
Offset: 1
T(1,2)=4 because the banana tree B(1,2) reduces to a path on 3 nodes, where the distances are 1, 1, and 2.
Square array T(n,k) begins:
4,10,18,28,40,54,70;
20,56,108,176,260,360,476;
48,138,270,444,660,918,1218;
88,256,504,832,1240,1728,2296;
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T := proc (n, k) options operator, arrow: n*(k-1)*(3*n*k-2*k+2) end proc: for n to 10 do seq(T(n+2-j, j), j = 2 .. n+1) end do; # yields sequence in triangular form
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T[n_, k_] := n*(k - 1)*(3*n*k - 2*k + 2);
Table[T[n - k + 2, k], {n, 1, 10}, {k, 2, n + 1}] // Flatten (* Jean-François Alcover, Aug 26 2024 *)
A192030
Square array read by antidiagonals: W(n,p) (n>=1, p>=1) is the Wiener index of the graph G(n,p) obtained in the following way: consider n copies of a star tree with p-1 edges, add a vertex to their union, and connect this vertex with the roots of the star trees.
Original entry on oeis.org
1, 4, 4, 9, 20, 9, 16, 48, 48, 16, 25, 88, 117, 88, 25, 36, 140, 216, 216, 140, 36, 49, 204, 345, 400, 345, 204, 49, 64, 280, 504, 640, 640, 504, 280, 64, 81, 368, 693, 936, 1025, 936, 693, 368, 81, 100, 468, 912, 1288, 1500, 1500, 1288, 912, 468, 100, 121, 580, 1161, 1696, 2065, 2196, 2065, 1696, 1161, 580, 121
Offset: 1
W(2,2)=20 because G(2,2) is the path graph with 4 edges; its Wiener index is 4*1+3*2+2*3+1*4=20.
The square array starts:
1,4,9,16,25,36,49,...;
4,20,48,88,140,204,280,...;
9,48,117,216,345,504,693,...;
16,88,216,400,640,936,1288,...;
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W := proc (n, p) options operator, arrow; n*p*(2*n*p-n-p+1) end proc: for n to 11 do seq(W(n-i, i+1), i = 0 .. n-1) end do; # yields sequence in triangular form
W := proc (n, p) options operator, arrow; n*p*(2*n*p-n-p+1) end proc: for n to 7 do seq(W(n, p), p = 1 .. 10) end do; # yields the first 10 entries in each of the first 7 rows
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