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A296272 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)*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. 4
1, 2, 23, 55, 120, 231, 423, 744, 1277, 2153, 3586, 5921, 9717, 15878, 25867, 42051, 68260, 110691, 179371, 290524, 470423, 761547, 1232620, 1994869, 3228245, 5223926, 8453041, 13677897, 22131930, 35810883, 57943935, 93756008, 151701203, 245458543, 397161152 (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, Table of n, a(n) for n = 0..1000

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) + b(1)*b(2) = 23;

Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...)

MATHEMATICA

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;

a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] b[n];

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}];  (* A296272 *)

Table[b[n], {n, 0, 20}]  (* complement *)

CROSSREFS

Cf. A001622, A296245.

Sequence in context: A084237 A106928 A070934 * A031915 A247603 A102385

Adjacent sequences:  A296269 A296270 A296271 * A296273 A296274 A296275

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Dec 12 2017

STATUS

approved

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Last modified September 22 20:30 EDT 2018. Contains 315270 sequences. (Running on oeis4.)