A296849 Solution of the complementary equation a(n) = 2*a(n-1) + a(n-2) + b(n), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.
1, 2, 10, 28, 73, 182, 446, 1085, 2628, 6354, 15350, 37069, 89504, 216094, 521710, 1259533, 3040796, 7341146, 17723110, 42787389, 103297912, 249383238, 602064414, 1453512093, 3509088629, 8471689381, 20452467422, 49376624257, 119205715969, 287788056229
Offset: 0
Examples
a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5 a(2) = 2*a(1) + a(0) + b(2) = 10 Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, ...)
Links
- Clark Kimberling, Table of n, a(n) for n = 0..1000
- Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
Programs
-
Mathematica
a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5; a[n_] := a[n] = 2*a[n - 1] + a[n - 2] + b[n]; j = 1; While[j < 7, k = a[j] - j - 1; While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; u = Table[a[n], {n, 0, k}]; (* A296849 *) Table[b[n], {n, 0, 20}] (* complement *) Take[u, 30]
Comments