Search: a283683 -id:a283683
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1, 2, 3, 1, 2, 2, 2, 3, 2, 4, 2, 1, 3, 2, 2, 3, 2, 1, 1, 3, 2, 2, 1, 3, 2, 3, 1, 3, 2, 1, 2, 1, 3, 2, 2, 2, 1, 3, 2, 3, 2, 1, 3, 2, 1, 2, 2, 1, 3, 2, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 3, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 1, 3, 2, 3, 2, 2, 2, 1, 3, 2, 4, 2, 2, 2, 1, 3, 2, 1, 3, 2, 2, 2, 1, 3, 2, 2, 3
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OFFSET
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1,2
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LINKS
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MATHEMATICA
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Take[Length /@ Most@Split@ Nest[Flatten@ Table[#[[n - i]], {n, Length[#] + 1}, {i, n - 1}] &, {0, 1}, 4], {1, -1, 2}] (* Ivan Neretin, Mar 17 2017 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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0, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 11, 11, 12, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 17, 17, 18, 18, 19, 19, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 23, 23, 23, 23
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OFFSET
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1,5
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LINKS
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FORMULA
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PROG
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(PARI) See Links section.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A283681
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Unique sequence with a(1)=1, a(2)=2, representing an array read by antidiagonals in which the i-th row is this sequence itself multiplied by i.
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+10
5
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1, 2, 2, 2, 4, 3, 2, 4, 6, 4, 4, 4, 6, 8, 5, 3, 8, 6, 8, 10, 6, 2, 6, 12, 8, 10, 12, 7, 4, 4, 9, 16, 10, 12, 14, 8, 6, 8, 6, 12, 20, 12, 14, 16, 9, 4, 12, 12, 8, 15, 24, 14, 16, 18, 10, 4, 8, 18, 16, 10, 18, 28, 16, 18, 20, 11, 4, 8, 12, 24, 20, 12, 21, 32, 18, 20, 22, 12, 6, 8
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OFFSET
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1,2
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COMMENTS
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Any integer greater than 1 appears infinitely many times.
In particular, any n appears at the position (n^2 + n)/2. For prime n > 2, this is its first appearance; for composite n, it is not the first.
2 appears at the positions 2, 3, 4, 7, 22, 232, 26797, ... (A007501(n) + 1).
When the sequence is considered as an array, any prime n appears only in the first row (infinitely many times) and in the first column (once).
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LINKS
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FORMULA
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a((n^2+n)/2)=n.
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EXAMPLE
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The sequence begins: 1, 2, 2, 2, 4, 3, 2, 4, 6, 4, ...
It represents a rectangular array read by downward antidiagonals. The first row of the array is this very sequence itself. The second row is this sequence multiplied by 2, and so on:
1 2 2 2 4 3 ...
2 4 4 4 8 ...
3 6 6 6 ...
4 8 8 ...
5 10 ...
6 ...
...
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MATHEMATICA
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Nest[Flatten@Table[#[[n - i]]*i, {n, Length[#] + 1}, {i, n - 1}] &, {1, 2}, 4]
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CROSSREFS
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Cf. A007501 (number of terms produced by the Mathematica code after n iterations).
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KEYWORD
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STATUS
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approved
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A283682
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Unique sequence with a(1)=0, a(2)=1, representing an array T(i,j) read by antidiagonals in which T(i,j) = a(i) + a(j).
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+10
5
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0, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 2, 2, 3, 1, 1, 2, 3, 2, 3, 2, 1, 2, 2, 2, 3, 3, 2, 2, 2, 2, 3, 2, 2, 4, 2, 2, 3, 2, 1, 3, 3, 2, 3, 3, 2, 3, 3, 1, 2, 2, 3, 3, 3, 2, 3, 3, 3, 2, 2, 2, 3, 2, 3, 4, 2, 2, 4, 3, 2, 3, 2, 2, 3, 3, 2, 4, 3, 2, 3, 4, 2, 3, 3
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,5
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COMMENTS
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Any positive integer appears infinitely many times.
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LINKS
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EXAMPLE
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The sequence begins: 0, 1, 1, 1, 2, 1, 1, 2, 2, 1, ...
It represents a rectangular array read by downward antidiagonals. The first row of the array is this sequence itself; so is the first column. Every term in the array is the sum of the initial terms of its row and column:
0 1 1 1 2 1...
1 2 2 2 3...
1 2 2 2...
1 2 2...
2 3...
1...
...
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MATHEMATICA
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Nest[Flatten@Table[#[[n - i]] + #[[i]], {n, Length[#] + 1}, {i, n - 1}] &, {0, 1}, 4]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A297359
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Array read by antidiagonals: Pascal-like recursion and self-referential boundaries.
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+10
4
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1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 2, 4, 6, 4, 2, 1, 6, 10, 10, 6, 1, 1, 7, 16, 20, 16, 7, 1, 3, 8, 23, 36, 36, 23, 8, 3, 3, 11, 31, 59, 72, 59, 31, 11, 3, 1, 14, 42, 90, 131, 131, 90, 42, 14, 1, 2, 15, 56, 132, 221, 262, 221, 132, 56, 15, 2, 4, 17, 71, 188, 353, 483, 483, 353, 188, 71, 17, 4, 6, 21, 88, 259, 541, 836, 966, 836, 541, 259
(list;
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graph;
refs;
listen;
history;
text;
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OFFSET
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1,5
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COMMENTS
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Array with recursion T(i,j) = T(i-1,j) + T(i,j-1), and boundaries T(0,n) = T(n,0) = a(n). Here a(n) is the array T read by antidiagonals. Require that a(0)=a(1)=1.
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LINKS
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EXAMPLE
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The array looks like
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 2, ...
1, 2, 3, 4, 6, 7, 8, 11, 14, 15, 17, ...
1, 3, 6, 10, 16, 23, 31, 42, 56, 71, 88, ...
1, 4, 10, 20, 36, 59, 90, 132, 188, 259, 347, ...
2, 6, 16, 36, 72, 131, 221, 353, 541, 800, ...
1, 7, 23, 59, 131, 262, 483, 836, 1377, ...
1, 8, 31, 90, 221, 483, 966, 1802, ...
3, 11, 42, 132, 353, 836, 1802, ...
3, 14, 56, 188, 541, 1377, ...
1, 15, 71, 259, 800, ...
2, 17, 88, 347, ...
The defining property is that when this array is read by antidiagonals we get 1,1,1,1,2,1,... which is both the sequence itself and the top row and first column of the array.
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MATHEMATICA
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t[a_, b_] := (t[a, b] = t[a, b - 1] + t[a - 1, b]);
t[0, x_] := a[x]; t[x_, 0] := a[x];
a[0] = 1; a[1] = 1;
a[x_] := With[{k = Floor[(Sqrt[8 x + 1] - 1)/2]},
t[x - k (k + 1)/2, (k + 1) (k + 2)/2 - x - 1]]
a /@ Range[60]
TableForm[ Table[t[i, j], {i, 0, 5}, {j, 0, 12}]]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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