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A296285 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) = 4, and (a(n)) and (b(n)) are increasing complementary sequences. 2
1, 2, 11, 25, 56, 111, 209, 376, 657, 1123, 1900, 3166, 5234, 8595, 14053, 22903, 37244, 60470, 98074, 158943, 257457, 416883, 674868, 1092349, 1767865, 2860914, 4629533, 7491257, 12121658, 19613843, 31736491, 51351388, 83088999, 134441575, 217531832 (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) = 4, b(1) = 3, b(2) = 5
a(2) = a(0) + a(1) + 2*b(0) = 11
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, ...)
MATHEMATICA
a[0] = 1; a[1] = 2; b[0] = 4;
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}]; (* A296285 *)
Table[b[n], {n, 0, 20}] (* complement *)
CROSSREFS
Sequence in context: A012251 A084547 A241238 * A077482 A141428 A104085
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Dec 13 2017
STATUS
approved

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Last modified February 22 22:43 EST 2024. Contains 370265 sequences. (Running on oeis4.)