A279610
a(n) = concatenate n consecutive integers, starting with the last number of the previous batch.
Original entry on oeis.org
1, 12, 234, 4567, 7891011, 111213141516, 16171819202122, 2223242526272829, 293031323334353637, 37383940414243444546, 4647484950515253545556, 565758596061626364656667, 67686970717273747576777879, 7980818283848586878889909192
Offset: 1
a(4) is the concatenation of 4 numbers beginning with the last number (4) that was used to build a(3), so a(4) = 4 5 6 7 = 4567. Then a(5) is the concatenation of 5 numbers beginning with the last number of a(4), which is 7, so a(5) = 7 8 9 10 11 = 7891011. And so on.
For n = 3, n^2/2 - n/2 + 1 = 4; a(3) = 4 + 3*10^1 + 2*10^(1+1) = 234.
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Table[FromDigits[Flatten[IntegerDigits /@ Range[(n(n - 1))/2 + 1, (n(n + 1))/2 + 1 ]]], {n, 0, 20}]
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from _future_ import division
def A279610(n):
return int(''.join(str(d) for d in range((n-1)*(n-2)//2+1,n*(n-1)//2+2))) # Chai Wah Wu, Dec 17 2016
A284652
Number T(n,k) of self-avoiding planar walks of length k starting at (0,0), ending at (n,0), remaining in the first quadrant and using steps (0,1), (1,0), (1,1), (-1,1), and (1,-1) with the restriction that (0,1) is never used below the diagonal and (1,0) is never used above the diagonal; triangle T(n,k), k>=0, floor((sqrt(1+8*k)-1)/2)<=n<=k, read by columns.
Original entry on oeis.org
1, 1, 1, 2, 1, 4, 1, 4, 9, 1, 4, 8, 21, 7, 16, 22, 51, 3, 21, 54, 54, 127, 1, 17, 87, 178, 142, 323, 1, 15, 87, 269, 565, 370, 835, 10, 116, 370, 896, 1766, 983, 2188, 9, 99, 499, 1473, 2776, 5446, 2627, 5798, 4, 91, 536, 2290, 5528, 8657, 16655, 7086, 15511
Offset: 0
Triangle T(n,k) begins:
1;
. 1, 1;
. . 2, 1, 1, 1;
. . . 4, 4, 4, 7, 3, 1, 1;
. . . . 9, 8, 16, 21, 17, 15, 10, 9, ... ;
. . . . . 21, 22, 54, 87, 87, 116, 99, ... ;
. . . . . . 51, 54, 178, 269, 370, 499, ... ;
. . . . . . . 127, 142, 565, 896, 1473, ... ;
. . . . . . . . 323, 370, 1766, 2776, ... ;
. . . . . . . . . 835, 983, 5446, ... ;
. . . . . . . . . . 2188, 2627, ... ;
A219356
Triangle read by rows: A219274 with rows reversed.
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 3, 1, 4, 5, 1, 5, 9, 16, 1, 6, 14, 49, 1, 7, 20, 92, 70, 1, 8, 27, 153, 204, 168, 1, 9, 35, 235, 405, 738, 768, 1, 10, 44, 341, 715, 1815, 3300, 1, 11, 54, 474, 1166, 3630, 9460, 7887, 1, 12, 65, 637, 1794, 6578, 21307, 28743, 15015
Offset: 0
A219274 with rows reversed begins:
1;
1;
1;
1, 2;
1, 3;
1, 4, 5;
1, 5, 9, 16;
1, 6, 14, 49;
1, 7, 20, 92, 70;
1, 8, 27, 153, 204, 168;
1, 9, 35, 235, 405, 738, 768;
...
Last elements of rows give:
A219339.
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h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
end:
g:= proc(n, i, l) local s; s:=i*(i+1)/2;
`if`(n=s, h([l[], seq(i-j, j=0..i-1)]), `if`(n>s, 0,
g(n, i-1, l)+ `if`(i>n, 0, g(n-i, i-1, [l[], i]))))
end:
T:= (n, k)-> `if`(k>n, 0, g(n-k, k-1, [k])):
seq(seq(T(n, n-k), k=0..(n-floor(sqrt(2*n)+1/2))), n=0..14);
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