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Revision History for A294480

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Solution of the complementary equation a(n) = a(n-2) + b(n-1) + 2n, where a(0) = 1, a(1) = 3, b(0) = 2.
(history; published version)
#5 by Susanna Cuyler at Wed Nov 01 12:27:23 EDT 2017
STATUS

proposed

approved

#4 by Clark Kimberling at Wed Nov 01 09:36:44 EDT 2017
STATUS

editing

proposed

#3 by Clark Kimberling at Wed Nov 01 09:34:34 EDT 2017
DATA

1, 3, 9, 14, 23, 31, 43, 55, 70, 85, 103, 122, 143, 165, 189, 214, 41, 241, 269, 299, 331, 364, 399, 435, 473, 512, 553, 596, 640, 686, 733, 782, 832, 884, 937, 992, 1048, 1106, 1166, 1227, 1290, 1354, 1420, 1487, 1556, 1626, 1698, 1771, 1846, 1923, 2001, 2081

LINKS

Clark Kimberling, <a href="/A294480/b294480.txt">Table of n, a(n) for n = 0..1000</a>

#2 by Clark Kimberling at Wed Nov 01 09:24:42 EDT 2017
NAME

allocated for Clark KimberlingSolution of the complementary equation a(n) = a(n-2) + b(n-1) + 2n, where a(0) = 1, a(1) = 3, b(0) = 2.

DATA

1, 3, 9, 14, 23, 31, 43, 55, 70, 85, 103, 122, 143, 165, 189, 214, 41, 269, 299, 331, 364, 399, 435, 473, 512, 553, 596, 640, 686, 733, 782, 832, 884, 937, 992, 1048, 1106, 1166, 1227, 1290, 1354, 1420, 1487, 1556, 1626, 1698, 1771, 1846, 1923, 2001, 2081

OFFSET

0,2

COMMENTS

The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294476 for a guide to related sequences.

LINKS

Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.html">Complementary equations</a>, J. Int. Seq. 19 (2007), 1-13.

EXAMPLE

a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that

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

Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 10, 12, 13, 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 - 2] + b[n - 1] + 2n;

b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];

Table[a[n], {n, 0, 40}] (* A294480 *)

Table[b[n], {n, 0, 10}]

CROSSREFS
KEYWORD

allocated

nonn,easy

AUTHOR

Clark Kimberling, Nov 01 2017

STATUS

approved

editing

#1 by Clark Kimberling at Tue Oct 31 14:28:40 EDT 2017
NAME

allocated for Clark Kimberling

KEYWORD

allocated

STATUS

approved