The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A296284 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences. 33

%I #4 Dec 13 2017 18:40:31

%S 1,2,9,23,52,105,199,360,639,1098,1857,3098,5123,8416,13763,22434,

%T 36485,59242,96087,155728,252255,408487,661292,1070377,1732317,

%U 2803394,4536465,7340669,11878002,19219599,31098591,50319244,81418955,131739387,213159600

%N Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

%C 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.

%H Clark Kimberling, <a href="/A296284/b296284.txt">Table of n, a(n) for n = 0..1000</a>

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

%e a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5

%e a(2) = a(0) + a(1) + 2*b(0) = 9

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

%t a[0] = 1; a[1] = 2; b[0] = 3;

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

%t j = 1; While[j < 10, k = a[j] - j - 1;

%t While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];

%t Table[a[n], {n, 0, k}]; (* A296284 *)

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

%Y Cf. A001622, A296245.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Dec 13 2017

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 19 13:25 EDT 2024. Contains 372694 sequences. (Running on oeis4.)