A209434 -id:A209434 - cp's OEIS frontend

A209435 Table T(m,n), read by antidiagonals, is the number of subsets of {1,...,n} which do not contain two elements whose difference is m+1.

1, 1, 2, 1, 2, 3, 1, 2, 4, 5, 1, 2, 4, 6, 8, 1, 2, 4, 8, 9, 13, 1, 2, 4, 8, 12, 15, 21, 1, 2, 4, 8, 16, 18, 25, 34, 1, 2, 4, 8, 16, 24, 27, 40, 55, 1, 2, 4, 8, 16, 32, 36, 45, 64, 89, 1, 2, 4, 8, 16, 32, 48, 54, 75, 104, 144, 1, 2, 4, 8, 16, 32, 64, 72, 81

Offset: 0

David Nacin, Mar 09 2012

1st row is the Fibonacci sequence.

			Table begins:
1, 2, 3, 5,  8, 13, 21,  34,  55,  89,  144, ...
1, 2, 4, 6,  9, 15, 25,  40,  64, 104,  169, ...
1, 2, 4, 8, 12, 18, 27,  45,  75, 125,  200, ...
1, 2, 4, 8, 16, 24, 36,  54,  81, 135,  225, ...
1, 2, 4, 8, 16, 32, 48,  72, 108, 162,  243, ...
1, 2, 4, 8, 16, 32, 64,  96, 144, 216,  324, ...
1, 2, 4, 8, 16, 32, 64, 128, 192, 288,  432, ...
1, 2, 4, 8, 16, 32, 64, 128, 256, 384,  576, ...
1, 2, 4, 8, 16, 32, 64, 128, 256, 512,  768, ...
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...
................................................

G. C. Greubel, Table of n, a(n) for the first 100 antidiagonals, flattened
M. Tetiva, Subsets that make no difference d, Mathematics Magazine 84 (2011), no. 4, 300-301.

Cf. A209434, A209436, A209437.

a[n_, m_] := Product[Fibonacci[Floor[(n + i)/(m + 1) + 2]], {i, 0, m}]; Flatten[Table[a[i, j - i], {i, 0, 30}, {j, 0, i}]]

T(n,m) = Product_{i=0 to m} F(floor[(n + i)/(m + 1) + 2]) where F(n) is the n-th Fibonacci number.

A209436 Table of a(n,m) = number of subsets of {1,...,n} which contain two elements whose difference is m+1.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 3, 0, 0, 0, 8, 2, 0, 0, 0, 19, 7, 0, 0, 0, 0, 43, 17, 4, 0, 0, 0, 0, 94, 39, 14, 0, 0, 0, 0, 0, 201, 88, 37, 8, 0, 0, 0, 0, 0, 423, 192, 83, 28, 0, 0, 0, 0, 0, 0, 880, 408, 181, 74, 16, 0, 0, 0, 0, 0, 0, 1815, 855, 387, 175, 56, 0, 0, 0, 0

Offset: 0

David Nacin, Mar 09 2012

			Table begins:
0,   0,   0,   0,   0,   0,   0,   0,   0,   0, 0, ...
0,   0,   0,   0,   0,   0,   0,   0,   0,   0, 0, ...
1,   0,   0,   0,   0,   0,   0,   0,   0,   0, 0, ...
3,   2,   0,   0,   0,   0,   0,   0,   0,   0, 0, ...
8,   7,   4,   0,   0,   0,   0,   0,   0,   0, 0, ...
19,  17,  14,  8,   0,   0,   0,   0,   0,   0, 0, ...
43,  39,  37,  28,  16,  0,   0,   0,   0,   0, 0, ...
94,  88,  83,  74,  56,  32,  0,   0,   0,   0, 0, ...
201, 192, 181, 175, 148, 112, 64,  0,   0,   0, 0, ...
423, 408, 387, 377, 350, 296, 224, 128, 0,   0, 0, ...
880, 855, 824, 799, 781, 700, 592, 448, 256, 0, 0, ...
......................................................
a(3,1) is the number of subsets of {1,2,3} containing two elements whose difference is two.  There are 2 of these: {1,3} and {1,2,3} so a(1,3) = 2.

G. C. Greubel, Table of n, a(n) for the first 100 aintidiagonals, flattened
M. Tetiva, Subsets that make no difference d, Mathematics Magazine 84 (2011), no. 4, 300-301.

Cf. A209434, A209435, A209437.

a[n_, m_] := 2^n - Product[Fibonacci[Floor[(n + i)/(m + 1) + 2]], {i, 0, m}]; Flatten[Table[a[j - i, i], {j, 0, 20}, {i, 0, j}]]

a(n,m) = 2^n - Product_{i=0 to m} F(floor[(n + i)/(m + 1) + 2]) where F(n) is the n-th Fibonacci number.

A209437 Table of T(m,n), read by antidiagonals, is the number of subsets of {1,...,n} which contain two elements whose difference is m.

Original entry on oeis.org

1, 0, 3, 0, 2, 8, 0, 0, 7, 19, 0, 0, 4, 17, 43, 0, 0, 0, 14, 39, 94, 0, 0, 0, 8, 37, 88, 201, 0, 0, 0, 0, 28, 83, 192, 423, 0, 0, 0, 0, 16, 74, 181, 408, 880, 0, 0, 0, 0, 0, 56, 175, 387, 855, 1815, 0, 0, 0, 0, 0, 32, 148, 377, 824, 1775, 3719, 0, 0, 0, 0, 0

Offset: 1

David Nacin, Mar 09 2012

m offset is 1, n offset is 2 so 1st entry is T(1,2).

			Table begins:
1, 3, 8, 19, 43, 94, 201, 423, 880, ...
0, 2, 7, 17, 39, 88, 192, 408, 855, ...
0, 0, 4, 14, 37, 83, 181, 387, 824, ...
0, 0, 0,  8, 28, 74, 175, 377, 799, ...
0, 0, 0,  0, 16, 56, 148, 350, 781, ...
0, 0, 0,  0,  0, 32, 112, 296, 700, ...
0, 0, 0,  0,  0,  0,  64, 224, 592, ...
0, 0, 0,  0,  0,  0,   0, 128, 448, ...
0, 0, 0,  0,  0,  0,   0,   0, 256, ...
0, 0, 0,  0,  0,  0,   0,   0,   0, ...
0, 0, 0,  0,  0,  0,   0,   0,   0, ...
.......................................
T(2,3) is the number of subsets of {1,2,3} containing two elements whose difference is two. There are 2 of these: {1,3} and {1,2,3} so T(2,3) = 2.

G. C. Greubel, Table of n, a(n) for the first 100 antidiagonals, flattened
M. Tetiva, Subsets that make no difference d, Mathematics Magazine 84 (2011), no. 4, 300-301.

Cf. A209434, A209435, A209436.

T[m_, n_] := 2^n - Product[Fibonacci[Floor[(n + i)/m + 2]], {i, 0, m - 1}]; Table[T[i, j + 2], {i, 1, 10}, {j, 0, 10}]; Flatten[Table[T[i - j + 1, j + 2], {i, 0, 20}, {j, 0, i}]]

T(m,n) = 2^n - Product_{i=0,...,m-1} F(floor((n + i)/m + 2)) where F(n) is the n-th Fibonacci number.

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A209435 Table T(m,n), read by antidiagonals, is the number of subsets of {1,...,n} which do not contain two elements whose difference is m+1.

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A209436 Table of a(n,m) = number of subsets of {1,...,n} which contain two elements whose difference is m+1.

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A209437 Table of T(m,n), read by antidiagonals, is the number of subsets of {1,...,n} which contain two elements whose difference is m.

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Keywords

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Mathematica

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