A134019
Let P(A) be the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are intersecting but for which x is not a subset of y and y is not a subset of x, or 1) x = y.
Original entry on oeis.org
1, 2, 4, 11, 46, 227, 1114, 5231, 23566, 102827, 438274, 1836551, 7601686, 31183427, 127084234, 515429471, 2083077406, 8396552027, 33779262994, 135696871991, 544528258726, 2183337968627, 8749031918554, 35043178292111, 140313885993646, 561679104393227, 2247987182714914, 8995761194057831
Offset: 0
a(3) = 11 because for P(A) = {{},{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}} we have for case 0 {{1,2},{1,3}}, {{1,2},{2,3}}, {{1,3},{2,3}} and we have for case 1 {{},{}}, {{1},{1}}, {{2},{2}}, {{3},{3}}, {{1,2},{1,2}}, {{1,3},{1,3}}, {{2,3},{2,3}}, {{1,2,3},{1,2,3}}.
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Table[3 StirlingS2[n + 1, 4] + StirlingS2[n + 1, 2] + 1, {n, 0, 27}] (* Michael De Vlieger, Nov 30 2015 *)
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a(n) = (4^n - 3^(n+1) + 5*2^n - 1)/2; \\ Michel Marcus, Nov 30 2015
A134045
Let P(A) be the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which either x is a subset of y or y is a subset of x, or 1) x and y are intersecting but for which x is not a subset of y and y is not a subset of x, or 2) x = y.
Original entry on oeis.org
1, 3, 7, 18, 61, 258, 1177, 5358, 23821, 103338, 439297, 1838598, 7605781, 31191618, 127100617, 515462238, 2083142941, 8396683098, 33779525137, 135697396278, 544529307301, 2183340065778, 8749036112857, 35043186680718, 140313902770861, 561679137947658, 2247987249823777, 8995761328275558
Offset: 0
a(2) = 7 because for P(A) = {{},{1},{2},{1,2}} we have for case 0 {{},{1}}, {{},{2}}, {{},{1,2}} and we have for case 2 {{},{}}, {{1},{1}}, {{2},{2}}, {{1,2},{1,2}}. There are 0 {x,y} of P(A) in this example that fall under case 1.
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Table[3 StirlingS2[n + 1, 4] + 2 StirlingS2[n + 1, 2] + 1, {n, 0, 27}] (* Michael De Vlieger, Nov 30 2015 *)
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a(n) = (4^n - 3^(n+1) + 7*2^n - 3)/2; \\ Michel Marcus, Nov 30 2015
A134064
Let P(A) be the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are intersecting but for which x is not a subset of y and y is not a subset of x, or 1) x and y are intersecting and for which either x is a proper subset of y or y is a proper subset of x, or 2) x = y.
Original entry on oeis.org
1, 2, 6, 23, 96, 407, 1716, 7163, 29616, 121487, 495276, 2009603, 8124936, 32761367, 131834436, 529712843, 2125993056, 8525430047, 34166159196, 136858084883, 548012945976, 2193794127527, 8780404589556, 35137304693723, 140596281975696, 562526325893807, 2250528914325516
Offset: 0
a(2) = 6 because for P(A) = {{},{1},{2},{1,2}} we have for case 1 {{1},{1,2}}, {{2},{1,2}} and we have for case 2 {{},{}}, {{1},{1}}, {{2},{2}}, {{1,2},{1,2}}. There are 0 {x,y} of P(A) in this example that fall under case 0.
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LinearRecurrence[{10,-35,50,-24},{1,2,6,23},30] (* Harvey P. Dale, Jul 04 2023 *)
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Vec((1-8*x+21*x^2-17*x^3)/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)) + O(x^30)) \\ Michel Marcus, Oct 30 2015