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A296776 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) + 2*n, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences. 4
1, 3, 13, 28, 56, 102, 179, 305, 511, 846, 1391, 2274, 3705, 6022, 9773, 15844, 25669, 41568, 67295, 108924, 176283, 285274, 461627, 746974, 1208678, 1955732, 3164493, 5120311, 8284893, 13405296, 21690284, 35095678, 56786063, 91881845, 148668015, 240549970 (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) = 3, b(0) = 2, b(1) = 4, b(2) = 5

a(2) = a(0) + a(1) + b(2) + 4 = 13

Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, ...)

MATHEMATICA

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

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

j = 1; While[j < 16, 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}];  (* A296776 *)

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

CROSSREFS

Cf. A001622, A296245, A298171, A298172.

Sequence in context: A002304 A117516 A075726 * A074498 A041035 A042269

Adjacent sequences:  A296773 A296774 A296775 * A296777 A296778 A296779

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jan 06 2018

STATUS

approved

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Last modified October 25 08:40 EDT 2021. Contains 348239 sequences. (Running on oeis4.)