login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A296292 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-1), where a(0) = 2, a(1) = 4, b(0) = 1, b(1) = 3, and (a(n)) and (b(n)) are increasing complementary sequences. 4
2, 4, 12, 31, 67, 133, 248, 444, 772, 1315, 2217, 3686, 6083, 9977, 16298, 26545, 43147, 70032, 113557, 184007, 298024, 482535, 781109, 1264242, 2045999, 3310941, 5357694, 8669445, 14028035, 22698437, 36727492, 59427014, 96155658, 155583893, 251740843 (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(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) = 2, a(1) = 4, b(0) = 1, b(1) = 3, b(2) = 5

a(2) = a(0) + a(1) + 2*b(1) = 12

Complement: (b(n)) = (1, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, ...)

MATHEMATICA

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

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

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

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

Table[a[n], {n, 0, k}]; (* A296292 *)

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

CROSSREFS

Cf. A001622, A296245.

Sequence in context: A148189 A148190 A151434 * A287966 A148191 A141312

Adjacent sequences:  A296289 A296290 A296291 * A296293 A296294 A296295

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Dec 14 2017

STATUS

approved

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 27 16:20 EDT 2020. Contains 337383 sequences. (Running on oeis4.)