A295067
Solution of the complementary equation a(n) = 2*a(n-2) + b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.
Original entry on oeis.org
1, 3, 4, 11, 14, 29, 36, 67, 82, 146, 177, 307, 370, 631, 758, 1281, 1536, 2583, 3094, 5189, 6212, 10403, 12450, 20833, 24928, 41696, 49887, 83424, 99807, 166882, 199649, 333801, 399336, 667641, 798712, 1335323, 1597466, 2670689, 3194976, 5341423, 6389998
Offset: 0
a(0) = 1, a(1) = 3, a(2) = 3, b(0) = 2, b(1) = 5
a(2) = 2*a(0) + b(0) = 4
Complement: (b(n)) = (2, 5, 6, 7, 8, 9, 10, 12, 13, 15, ... )
-
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2; b[1]=5;
a[n_] := a[n] = 2 a[n - 2] + b[n - 2];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}] (* A295067 *)
Table[b[n], {n, 0, 10}]
A295068
Solution of the complementary equation a(n) = 2*a(n-2) - b(n-1) + n, where a(0) = 4, a(1) = 5, b(0) = 1, and (a(n)) and (b(n)) are increasing complementary sequences.
Original entry on oeis.org
4, 5, 8, 10, 14, 18, 25, 32, 46, 60, 87, 115, 169, 224, 332, 442, 658, 878, 1310, 1749, 2613, 3491, 5219, 6975, 10431, 13942, 20854, 27876, 41700, 55744, 83392, 111480, 166776, 222952, 333544, 445896, 667080, 891784, 1334151, 1783559, 2668293, 3567109
Offset: 0
a(0) = 4, a(1) = 5, b(0) = 1
a(2) = 2*a(0) - b(1) + 2 = 8
Complement: (b(n)) = (1, 2, 3, 6, 7, 9, 11, 12, 13, 15, 16, 17, 19, ... )
-
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 4; a[1] = 5; b[0] = 1;
a[n_] := a[n] = 2 a[n - 2] - b[n - 1] + n;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}] (* A295068 *)
Table[b[n], {n, 0, 10}]
A295069
Solution of the complementary equation a(n) = 2*a(n-2) - b(n-2) + n, where a(0) = 3, a(1) = 4, b(0) = 1, and (a(n)) and (b(n)) are increasing complementary sequences.
Original entry on oeis.org
3, 4, 7, 9, 13, 17, 24, 31, 45, 59, 86, 114, 168, 223, 331, 441, 657, 877, 1309, 1748, 2612, 3490, 5218, 6974, 10430, 13941, 20853, 27875, 41699, 55743, 83391, 111479, 166775, 222951, 333543, 445895, 667079, 891783, 1334150, 1783558, 2668292, 3567108
Offset: 0
a(0) = 3, a(1) = 4, b(0) = 1
a(2) = 2*a(0) - b(0) + 2 = 7
Complement: (b(n)) = (1, 2, 5, 6, 8, 10, 11, 12, 14, 15, 16, 18, ... )
-
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 3; a[1] = 4; b[0] = 1;
a[n_] := a[n] = 2 a[n - 2] + b[n - 1] + n;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}] (* A295069 *)
Table[b[n], {n, 0, 10}]
A295054
Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(1) + b(2) + ... + b(n-1), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
Original entry on oeis.org
1, 2, 7, 18, 40, 81, 153, 276, 482, 823, 1383, 2298, 3788, 6209, 10137, 16505, 26821, 43526, 70569, 114340, 185178, 299812, 485310, 785469, 1271154, 2057027, 3328615, 5386107, 8715219, 14101856, 22817639, 36920094, 59738368, 96659134
Offset: 0
a(0) = 1, a(1) = 2, b(0) = 3
b(1) = 4 (least "new number")
a(2) = a(1) + a(0) + b(1) = Complement: (b(n)) = (3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, ...)
-
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = a[n - 1] + a[n - 2] + Sum[b[k], {k, 1, n - 1}];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}] (* A295054 *)
Table[b[n], {n, 0, 10}]
A295055
Solution of the complementary equation a(n) = a(n-2) + b(0) + b(1) + ... + b(n-1), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
Original entry on oeis.org
1, 2, 8, 14, 26, 39, 60, 83, 115, 150, 195, 245, 306, 373, 452, 538, 637, 744, 865, 995, 1140, 1295, 1467, 1650, 1851, 2064, 2296, 2541, 2806, 3085, 3385, 3700, 4037, 4390, 4767, 5161, 5580, 6017, 6480, 6962, 7471, 8000, 8557, 9135, 9742, 10371, 11030, 11712
Offset: 0
a(0) = 1, a(1) = 2, b(0) = 3
b(1) = 4 (least "new number")
a(2) = a(0) + b(0) + b(1) = 8
Complement: (b(n)) = (3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 15, ...)
-
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = a[n - 2] + Sum[b[k], {k, 0, n - 1}];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}] (* A295055 *)
Table[b[n], {n, 0, 10}]
A295056
Solution of the complementary equation a(n) = 2*a(n-1) + b(n-1), where a(0) = 1, a(1) = 4, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.
Original entry on oeis.org
1, 4, 11, 27, 60, 127, 262, 533, 1076, 2164, 4341, 8696, 17407, 34830, 69677, 139372, 278763, 557546, 1115113, 2230248, 4460519, 8921062, 17842149, 35684324, 71368676, 142737381, 285474792, 570949615
Offset: 0
a(0) = 1, a(1) = 4, b(0) = 2
b(1) = 3 (least "new number")
a(2) = 2*a(1) + b(1) = 11
Complement: (b(n)) = (2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, ...)
-
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 4; b[0] = 2;
a[n_] := a[n] = 2 a[n - 1] + b[n - 1];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}] (* A295056 *)
Table[b[n], {n, 0, 10}]
A295057
Solution of the complementary equation a(n) = 2*a(n-1) + b(n-1), where a(0) = 2, a(1) = 5, b(0) = 1, and (a(n)) and (b(n)) are increasing complementary sequences.
Original entry on oeis.org
2, 5, 13, 30, 66, 139, 286, 581, 1172, 2355, 4722, 9458, 18931, 37878, 75773, 151564, 303147, 606314, 1212649, 2425320, 4850663, 9701350, 19402725, 38805476, 77610979, 155221986, 310444001, 620888033
Offset: 0
a(0) = 2, a(1) = 5, b(0) = 1
b(1) = 3 (least "new number")
a(2) = 2*a(1) + b(1) = 13
Complement: (b(n)) = (1, 3, 4, 6, 7, 8, 9, 10, 11, 12, 14, ...)
-
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 2; a[1] = 5; b[0] = 1;
a[n_] := a[n] = 2 a[n - 1] + b[n - 1];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}] (* A295057 *)
Table[b[n], {n, 0, 10}]
A295058
Solution of the complementary equation a(n) = 2*a(n-1) - b(n-1), where a(0) = 3, a(1) = 5, b(0) = 1, and (a(n)) and (b(n)) are increasing complementary sequences.
Original entry on oeis.org
3, 5, 8, 12, 18, 29, 49, 88, 165, 317, 620, 1225, 2434, 4851, 9683, 19346, 38671, 77320, 154617, 309210, 618395, 1236764, 2473501, 4946974, 9893918, 19787805, 39575578, 79151123, 158302212, 316604389, 633208742
Offset: 0
a(0) = 3, a(1) = 5, b(0) = 1
b(1) = 2 (least "new number")
a(2) = 2*a(1) - b(1) = 8
Complement: (b(n)) = (1, 2, 4, 6, 7, 9, 10, 11, 13, 14, 15, ...)
-
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 3; a[1] = 5; b[0] = 1;
a[n_] := a[n] = 2 a[n - 1] - b[n - 1];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}] (* A295058 *)
Table[b[n], {n, 0, 10}]
A295059
Solution of the complementary equation a(n) = 2*a(n-1) + b(n-2), where a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
Original entry on oeis.org
1, 4, 10, 23, 51, 108, 223, 454, 917, 1845, 3702, 7417, 14848, 29711, 59438, 118893, 237804, 475627, 951274, 1902569, 3805160, 7610344, 15220713, 30441452, 60882931, 121765890, 243531809, 487063648
Offset: 0
a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3
a(2) = 2*a(1) + b(0) = 10
Complement: (b(n)) = (2, 3, 5, 6, 7, 8, 9, 11, 12, 13, 14, ...)
-
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 2; a[1] = 5; b[0] = 1;
a[n_] := a[n] = 2 a[n - 1] + b[n - 1];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}] (* A295059 *)
Table[b[n], {n, 0, 10}]
A295060
Solution of the complementary equation a(n) = 2*a(n-1) - b(n-2), where a(0) = 3, a(1) = 5, b(0) = 1, b(1) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.
Original entry on oeis.org
3, 5, 9, 16, 28, 50, 93, 178, 346, 681, 1350, 2687, 5360, 10705, 21393, 42768, 85517, 171014, 342007, 683992, 1367961, 2735898, 5471771, 10943516, 21887005, 43773981, 87547932, 175095833, 350191634
Offset: 0
a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3
a(2) = 2*a(1) - b(0) = 9
Complement: (b(n)) = (1, 2, 4, 6, 7, 8, 10, 11, 12, 13, 14, ...)
-
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 3; a[1] = 5; b[0] = 1; b[1]=2;
a[n_] := a[n] = 2 a[n - 1] - b[n - 2];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}] (* A295060 *)
Table[b[n], {n, 0, 10}]
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