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A296255
Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)^2, where a(0) = 2, a(1) = 4, b(0) = 1, b(1) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
2
2, 4, 15, 44, 95, 188, 347, 616, 1063, 1800, 3007, 4976, 8179, 13411, 21879, 35614, 57854, 93868, 152163, 246515, 399207, 646298, 1046130, 1693104, 2739963, 4433851, 7174655, 11609406, 18785022, 30395452, 49181563, 79578171, 128760959, 208340426, 337102754
OFFSET
0,1
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.
FORMULA
a(n) = H + R, where H = f(n-1)*a(0) + f(n)*a(1) and R = f(n-1)*b(1)^2 + f(n-2)*b(2)^2 + ... + f(2)*b(n-2)^2 + f(1)*b(n-1)^2, where f(n) = A000045(n), the n-th Fibonacci number.
EXAMPLE
a(0) = 2, a(1) = 4, b(0) = 1, b(1) = 3;
a(2) = a(0) + a(1) + b(1)^2 = 15;
Complement: (b(n)) = (1, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, ...)
MATHEMATICA
a[0] = 2; a[1] = 4; b[0] = 1; b[1] = 3;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n-1]^2;
j = 1; While[j < 6 , 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}] (* A296255 *)
Table[b[n], {n, 0, 20}] (* complement *)
CROSSREFS
Sequence in context: A148278 A284728 A072206 * A277508 A332234 A316726
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Dec 11 2017
STATUS
approved