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A297011 Solution of the complementary equation a(n) = 2*a(n-1) + a(n-2) - b(n), where a(0) = 3, a(1) = 5, b(0) = 1, b(1) = 2, b(2) = 4, and (a(n)) and (b(n)) are increasing complementary sequences. 3
3, 5, 9, 17, 36, 81, 188, 446, 1068, 2569, 6192, 14938, 36052, 87024, 210081, 507166, 1224392, 2955928, 7136225, 17228354, 41592908, 100414144, 242421169, 585256454, 1412934048, 3411124520, 8235183057, 19881490602, 47998164228, 115877819024, 279753802241 (list; graph; refs; listen; history; text; internal format)
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(2). 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) = 3, a(1) = 5, b(0) = 1, b(1) = 2, b(2) = 4

a(2) = 2*a(1) + a(0) - b(2) = 9

Complement: (b(n)) = (1, 2, 4, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 18, 19, ...)

MATHEMATICA

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

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

j = 1; While[j < 9, 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}]; (* A297011 *)

Table[b[n], {n, 0, 25}] (* complement *)

Take[u, 30]

CROSSREFS

Cf. A297012, A297013.

Sequence in context: A251705 A018095 A003217 * A178717 A006723 A217097

Adjacent sequences:  A297008 A297009 A297010 * A297012 A297013 A297014

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jan 13 2018

STATUS

approved

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Last modified September 21 23:55 EDT 2019. Contains 327286 sequences. (Running on oeis4.)