A296227 Solution of the complementary equation a(n) = a(0)*b(n-1) + a(1)*b(n-2) + ... + a(n-1)*b(0) - n, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
1, 2, 8, 34, 146, 628, 2703, 11632, 50057, 215415, 927016, 3989317, 17167612, 73879038, 317930779, 1368182139, 5887829959, 25337665679, 109038016813, 469233798454, 2019298993572, 8689843823858, 37395841786394, 160929127296116, 692541811472532
Offset: 0
Examples
a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4 a(2) = a(0)*b(1) + a(1)*b(0) - 2 = 8 Complement: (b(n)) = (3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, ...)
Links
- Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
Programs
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Mathematica
mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &]; a[0] = 1; a[1] = 2; b[0] = 3; a[n_] := a[n] = - n + Sum[a[k]*b[n - k - 1], {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, 200}] (* A296227 *) Table[b[n], {n, 0, 20}] N[Table[a[n]/a[n - 1], {n, 1, 200, 10}], 200]; RealDigits[Last[t], 10][[1]] (* A296228 *)
Comments