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A296284 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences. 33
1, 2, 9, 23, 52, 105, 199, 360, 639, 1098, 1857, 3098, 5123, 8416, 13763, 22434, 36485, 59242, 96087, 155728, 252255, 408487, 661292, 1070377, 1732317, 2803394, 4536465, 7340669, 11878002, 19219599, 31098591, 50319244, 81418955, 131739387, 213159600 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622). See A296245 for a guide to related sequences.
LINKS
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
EXAMPLE
a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5
a(2) = a(0) + a(1) + 2*b(0) = 9
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, ...)
MATHEMATICA
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = a[n - 1] + a[n - 2] + n*b[n-2];
j = 1; While[j < 10, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
Table[a[n], {n, 0, k}]; (* A296284 *)
Table[b[n], {n, 0, 20}] (* complement *)
CROSSREFS
Sequence in context: A062445 A009304 A154118 * A115185 A349189 A091107
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Dec 13 2017
STATUS
approved

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Last modified April 24 12:28 EDT 2024. Contains 371937 sequences. (Running on oeis4.)