A213500
Rectangular array T(n,k): (row n) = b**c, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and ** = convolution.
Original entry on oeis.org
1, 4, 2, 10, 7, 3, 20, 16, 10, 4, 35, 30, 22, 13, 5, 56, 50, 40, 28, 16, 6, 84, 77, 65, 50, 34, 19, 7, 120, 112, 98, 80, 60, 40, 22, 8, 165, 156, 140, 119, 95, 70, 46, 25, 9, 220, 210, 192, 168, 140, 110, 80, 52, 28, 10, 286, 275, 255, 228, 196, 161, 125, 90
Offset: 1
Northwest corner (the array is read by southwest falling antidiagonals):
1, 4, 10, 20, 35, 56, 84, ...
2, 7, 16, 30, 50, 77, 112, ...
3, 10, 22, 40, 65, 98, 140, ...
4, 13, 28, 50, 80, 119, 168, ...
5, 16, 34, 60, 95, 140, 196, ...
6, 19, 40, 70, 110, 161, 224, ...
T(6,1) = (1)**(6) = 6;
T(6,2) = (1,2)**(6,7) = 1*7+2*6 = 19;
T(6,3) = (1,2,3)**(6,7,8) = 1*8+2*7+3*6 = 40.
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b[n_] := n; c[n_] := n
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}] (* A213500 *)
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t(n,k) = sum(i=0, k - 1, (k - i) * (n + i));
tabl(nn) = {for(n=1, nn, for(k=1, n, print1(t(k,n - k + 1),", ");); print(););};
tabl(12) \\ Indranil Ghosh, Mar 26 2017
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def t(n, k): return sum((k - i) * (n + i) for i in range(k))
for n in range(1, 13):
print([t(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Mar 26 2017
A213849
Rectangular array: (row n) = b**c, where b(h) = ceiling(h/2), c(h) = floor(n-1+h), n>=1, h>=1, and ** = convolution.
Original entry on oeis.org
1, 2, 1, 5, 3, 2, 8, 6, 4, 2, 14, 11, 9, 5, 3, 20, 17, 14, 10, 6, 3, 30, 26, 23, 17, 13, 7, 4, 40, 36, 32, 26, 20, 14, 8, 4, 55, 50, 46, 38, 32, 23, 17, 9, 5, 70, 65, 60, 52, 44, 35, 26, 18, 10, 5, 91, 85, 80, 70, 62, 50, 41, 29, 21, 11, 6
Offset: 1
Northwest corner (the array is read by falling antidiagonals):
1...2...5....8....14...20...30...40
1...3...6....11...17...26...36...50
2...4...9....14...23...32...46...60
2...5...10...17...26...38...52...70
3...6...13...20...32...44...62...80
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b[n_]:=Floor[(n+1)/2];c[n_]:=Floor[(n+1)/2];
t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]
TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]
Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]
r[n_]:=Table[t[n,k],{k,1,60}] (* A213849 *)
d=Table[t[n,n],{n,1,50}] (* A049778 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
s1=Table[s[n],{n,1,50}] (* A213850 *)
A228314
Triangle read by rows: T(p,q) (1<=q<=p) is the hyper-Wiener index of the Cartesian product of the cycles C(p) and C(q) (Torus Grid Graph).
Original entry on oeis.org
0, 1, 10, 3, 27, 72, 10, 72, 186, 448, 20, 135, 345, 810, 1450, 42, 258, 648, 1464, 2580, 4482, 70, 413, 1029, 2282, 3990, 6846, 10388, 120, 672, 1656, 3584, 6200, 10464, 15736, 23552, 180, 981, 2403, 5130, 8820, 14742, 22050, 32760, 45360, 275, 1450, 3525, 7400, 12625
Offset: 1
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HWWi := proc (p, q) if `mod`(p, 2) = 1 and `mod`(q, 2) = 1 then (1/96)*p*q*(3*p^2*q^2+2*p^3*q+2*p*q^3-4*p*q-3*p^2-3*q^2-6*p-6*q+6*p^2*q+6*p*q^2+3) elif `mod`(p, 2) = 0 and `mod`(q, 2) = 0 then (1/96)*p^2*q^2*(6*q+6*p+3*p*q+2*p^2+2*q^2+8) elif `mod`(p, 2) = 1 and `mod`(q, 2) = 0 then (1/96)*p*q^2*(3*p^2*q+2*p^3+2*p*q^2+2*p-3*q-6+6*p^2+6*p*q) else (1/96)*p^2*q*(3*p*q^2+2*q^3+2*p^2*q+2*q-3*p-6+6*q^2+6*p*q) end if end proc: for i to 10 do seq(HWWi(i, j), j = 1 .. i) end do; # yields sequence in triangular form
H := proc (p, q) local br, h: br := proc (n) options operator, arrow: sum(t^k, k = 0 .. n-1) end proc; h := proc (m) if `mod`(m, 2) = 0 then m*(br((1/2)*m)-1)+(1/2)*m*t^((1/2)*m) else m*t*br((1/2)*m-1/2) end if end proc: sort(expand(2*h(p)*h(q)+p*h(q)+q*h(p))) end proc: Wi := proc (p, q) options operator, arrow: subs(t = 1, diff(H(p, q), t)) end proc: for i to 10 do seq(Wi(i, j), j = 1 .. i) end do; # yields sequence in triangular form
Showing 1-3 of 3 results.
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