A294366
Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + 2n, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.
Original entry on oeis.org
1, 3, 12, 26, 52, 95, 167, 285, 478, 792, 1303, 2131, 3473, 5646, 9164, 14858, 24073, 38985, 63115, 102160, 165338, 267564, 432971, 700608, 1133655, 1834342, 2968079, 4802506, 7770673, 12573270, 20344037, 32917404, 53261541, 86179048, 139440695, 225619852
Offset: 0
a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
a(2) = a(1) + a(0) + b(1) + 4 = 12;
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, ...)
-
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + 2n;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}] (* A294366 *)
Table[b[n], {n, 0, 10}]
A294367
Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + n - 1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.
Original entry on oeis.org
1, 3, 9, 19, 37, 67, 117, 200, 335, 555, 912, 1491, 2429, 3948, 6407, 10387, 16829, 27253, 44121, 71415, 115579, 187039, 302665, 489753, 792469, 1282275, 2074799, 3357131, 5431989, 8789181, 14221233, 23010479, 37231779, 60242328, 97474179, 157716581
Offset: 0
a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
a(2) = a(1) + a(0) + b(1) + 1 = 12;
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, ...)
-
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 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, 40}] (* A294367 *)
Table[b[n], {n, 0, 10}]
A294368
Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + n + 1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.
Original entry on oeis.org
1, 3, 11, 23, 45, 81, 141, 239, 399, 660, 1083, 1769, 2880, 4679, 7591, 12304, 19931, 32273, 52244, 84559, 136848, 221454, 358351, 579856, 938260, 1518171, 2456488, 3974718, 6431267, 10406048, 16837380, 27243495, 44080944, 71324510, 115405527, 186730112
Offset: 0
a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
a(2) = a(1) + a(0) + b(1) + 3 = 11;
b(2) is the first positive integer not already seen, namely 5.
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, ...)
-
A[0]:= 1: B[0]:= 2:
A[1]:= 3: B[1]:= 4:
Av:= {$5..200}:
for n from 2 to 100 do
A[n]:= A[n-1]+A[n-2]+B[n-1]+n+1;
B[n]:= min(Av minus {A[n]});
Av:= Av minus {A[n],B[n]};
od:
seq(A[i],i=0..100); # Robert Israel, Oct 29 2017
-
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 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, 40}] (* A294368 *)
Table[b[n], {n, 0, 10}]
Comments