A298875 -id:A298875 - cp's OEIS frontend

A298872 Solution (b(n)) of the system of 3 complementary equations in Comments.

2, 6, 11, 18, 26, 35, 45, 57, 70, 84, 99, 116, 135, 155, 176, 198, 221, 245, 270, 298, 327, 357, 388, 420, 453, 487, 523, 560, 598, 637, 677, 718, 760, 804, 850, 897, 945, 994, 1044, 1095, 1147, 1200, 1254, 1309, 1365, 1423, 1482, 1543, 1605, 1668, 1732

Offset: 0

Clark Kimberling, Apr 18 2018

Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2:
a(n) = least new;
b(n) = least new k >= a(n) + b(n-1);
c(n) = a(n) + 2 b(n);
where "least new k" means the least positive integer not yet placed. The sequences a,b,c partition the positive integers.

			n:   0    1    2    3    4    5    6    7    8   9
a:   1    4    5    7    8    9   10   12   13   14
b:   2    6   11   18   26   35   45   57   70   84
c:   3   16   27   43   60   30   79  100  126  153

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A299634, A298871, A298873, A298875.

z = 400;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {3};
Do[{AppendTo[a, mex[Flatten[{a, b, c}], 1]],
   AppendTo[b, mex[Flatten[{a, b, c}], Last[a] + Last[b]]],
   AppendTo[c, Last[a] + 2 Last[b]]}, {z}];
Take[a, 100]  (* A298871 *)
Take[b, 100]  (* A298872 *)
Take[c, 100]  (* A298873 *)

A298874 Solution (a(n)) of the system of 3 equations in Comments.

Original entry on oeis.org

1, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 24, 25, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81

Offset: 0

Clark Kimberling, Apr 19 2018

Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2:
a(n) = least new;
b(n) = a(n) + b(n-1);
c(n) = a(n) + 2 b(n);
where "least new k" means the least positive integer not yet placed.
***
Do these sequences a,b,c partition the positive integers? They differ from the corresponding partitioning sequences A298871, A298872, and A298872. For example, A298872(56) = 2139, whereas A298875(56) = 2138.

			n:   0    1    2    3    4    5    6    7    8    9
a:   1    4    5    7    8    9   10   12   13   14
b:   2    6   11   18   26   35   45   57   70   84
c:   3   16   27   43   60   30   79  100  126  153

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A299634, A298871, A298875, A298876.

z = 200;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {3};
Do[{AppendTo[a, mex[Flatten[{a, b, c}], 1]],
   AppendTo[b, Last[a] + Last[b]],
   AppendTo[c, Last[a] + 2 Last[b]]}, {z}];
Take[a, 100]  (* A298874 *)
Take[b, 100]  (* A298875 *)
Take[c, 100]  (* A298876 *)

A298876 Solution (c(n)) of the system of 3 equations in Comments.

Original entry on oeis.org

3, 16, 27, 43, 60, 79, 100, 126, 153, 182, 213, 249, 289, 330, 373, 418, 465, 514, 565, 624, 683, 744, 807, 872, 939, 1008, 1082, 1157, 1234, 1313, 1394, 1477, 1562, 1652, 1746, 1841, 1938, 2037, 2138, 2241, 2346, 2453, 2562, 2673, 2786, 2904, 3023, 3147

Offset: 0

Clark Kimberling, Apr 19 2018

Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2:
a(n) = least new;
b(n) = a(n) + b(n-1);
c(n) = a(n) + 2 b(n);
where "least new k" means the least positive integer not yet placed.
***
Do these sequences a,b,c partition the positive integers?  They differ from the corresponding partitioning sequences A298871, A298872, and A298872.  For example, A298872(56) = 2139, whereas A298875(56) = 2138.
Differs from A298873 first at n=56. - Georg Fischer, Oct 10 2018

			n:   0    1    2    3    4    5    6    7    8    9
a:   1    4    5    7    8    9   10   12   13   14
b:   2    6   11   18   26   35   45   57   70   84
c:   3   16   27   43   60   30   79  100  126  153

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A299634, A298871, A298874, A298875.

z = 200;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {3};
Do[{AppendTo[a, mex[Flatten[{a, b, c}], 1]],
   AppendTo[b, Last[a] + Last[b]],
   AppendTo[c, Last[a] + 2 Last[b]]}, {z}];
Take[a, 100]  (* A298874 *)
Take[b, 100]  (* A298875 *)
Take[c, 100]  (* A298876 *)

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A298872 Solution (b(n)) of the system of 3 complementary equations in Comments.

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A298874 Solution (a(n)) of the system of 3 equations in Comments.

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Mathematica

A298876 Solution (c(n)) of the system of 3 equations in Comments.

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Mathematica