A294532 - OEIS

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A294532

Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3.

1, 2, 6, 12, 23, 42, 73, 124, 207, 342, 562, 918, 1495, 2429, 3941, 6388, 10348, 16756, 27125, 43903, 71052, 114980, 186058, 301065, 487151, 788245, 1275426, 2063702, 3339160, 5402895, 8742089, 14145019, 22887144, 37032200, 59919382, 96951621, 156871043 (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, which, for the sequences in the following guide, are a(0) = 1, a(1) = 2, b(0) = 3:` `a(n) = a(n-1) + a(n-2) + b(n-2) A294532` `a(n) = a(n-1) + a(n-2) + b(n-2) + 1 A294533` `a(n) = a(n-1) + a(n-2) + b(n-2) + 2 A294534` `a(n) = a(n-1) + a(n-2) + b(n-2) + 3 A294535` `a(n) = a(n-1) + a(n-2) + b(n-2) - 1 A294536` `a(n) = a(n-1) + a(n-2) + b(n-2) + n A294537` `a(n) = a(n-1) + a(n-2) + b(n-2) + 2n A294538` `a(n) = a(n-1) + a(n-2) + b(n-2) + n - 1 A294539` `a(n) = a(n-1) + a(n-2) + b(n-2) + 2n - 1 A294540` `a(n) = a(n-1) + a(n-2) + b(n-1) A294541` `a(n) = a(n-1) + a(n-2) + b(n-1) + 1 A294542` `a(n) = a(n-1) + a(n-2) + b(n-1) + 2 A294543` `a(n) = a(n-1) + a(n-2) + b(n-1) + 3 A294544` `a(n) = a(n-1) + a(n-2) + b(n-1) - 1 A294545` `a(n) = a(n-1) + a(n-2) + b(n-1) + n A294546` `a(n) = a(n-1) + a(n-2) + b(n-1) + 2n A294547` `a(n) = a(n-1) + a(n-2) + b(n-1) + n - 1 A294548` `a(n) = a(n-1) + a(n-2) + b(n-1) + n + 1 A294549` `a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) A294550` `a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 1 A294551` `a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + n A294552` `a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - n A294553` `a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 2 A294554` `a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 3 A294555` `a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + n + 1 A294556` `a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + n - 1 A294557` `a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 2n A294558` `a(n) = a(n-1) + a(n-2) + b(n-1) + 2b(n-2) A294559` `a(n) = a(n-1) + a(n-2) + 2b(n-1) + 2b(n-2) A294560` `a(n) = a(n-1) + a(n-2) + 2b(n-1) + b(n-2) A294561` `a(n) = a(n-1) + a(n-2) + b(n-1) - b(n-2) + 1 A294562` `a(n) = a(n-1) + a(n-2) + b(n-1) - b(n-2) + n A294563` `a(n) = a(n-1) + a(n-2) + 2b(n-1) - b(n-2) - 1 A294564` `a(n) = a(n-1) + a(n-2) + 2b(n-1) - b(n-2) - 3 A294565` `Conjecture: for every sequence listed here, a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).`
	LINKS	`Table of n, a(n) for n=0..36.` `Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.`
	EXAMPLE	`a(0) = 1, a(1) = 2, b(0) = 3, so that` `b(1) = 4 (least "new number")` `a(2) = a(0) + a(1) + b(0) = 6` `Complement: (b(n)) = (3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, ...)`
	MATHEMATICA	`mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;` `a[0] = 1; a[1] = 3; b[0] = 2;` `a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 2];` `b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];` `Table[a[n], {n, 0, 40}] (* A294532 *)` `Table[b[n], {n, 0, 10}]`
	CROSSREFS	`Cf. A001622, A293076, A294413.` `Sequence in context: A062476 A192703 A192969 * A323950 A291014 A257479` `Adjacent sequences: A294529 A294530 A294531 * A294533 A294534 A294535`
	KEYWORD	`nonn,easy`
	AUTHOR	`Clark Kimberling, Nov 03 2017`
	STATUS	`approved`

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Last modified January 29 23:01 EST 2023. Contains 359939 sequences. (Running on oeis4.)