login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A296257 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)^2, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences. 4
1, 2, 12, 30, 67, 133, 249, 446, 776, 1322, 2219, 3710, 6125, 10060, 16441, 26790, 43555, 70706, 114661, 185808, 300953, 487290, 788819, 1276734, 2066229, 3343692, 5410705, 8755238, 14166904, 22923166, 37091159, 60015481, 97107865, 157124642, 254233876 (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. 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.

FORMULA

a(n) = H + R, where H = f(n-1)*a(0) + f(n)*a(1) and R = f(n-1)*b(0)^2 + f(n-2)*b(1)^2 + ... + f(2)*b(n-3)^2 + f(1)*b(n-2)^2, where f(n) = A000045(n), the n-th Fibonacci number.

EXAMPLE

a(0) = 1, a(1) = 2, b(0) = 3;

a(2) = a(0) + a(1) + b(0)^2 = 12;

Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, ...)

MATHEMATICA

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

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

j = 1; While[j < 6 , 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}]     (* A296257 *)

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

CROSSREFS

Cf. A001622, A296245.

Sequence in context: A249055 A127118 A259127 * A301774 A286230 A083175

Adjacent sequences:  A296254 A296255 A296256 * A296258 A296259 A296260

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Dec 11 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 August 18 13:53 EDT 2019. Contains 326100 sequences. (Running on oeis4.)