A036059 The summarize Fibonacci sequence: summarize the previous two terms!.

1, 1, 21, 1221, 3231, 233231, 533221, 15534221, 3514334231, 3534533241, 3544832231, 183544733221, 28172544634231, 2827162554535241, 2827265554337241, 2837267544338231, 3847264544637221, 3847362564636221, 2837662564536221, 2827863534537221, 3837564524538221

Offset: 0

Author

Floor van Lamoen

Keywords

• nonn
• base
• nice
• easy

Comments

After the 23rd term the sequence goes into a cycle of 16 terms.

Examples

			a(20) = 3837564524538221;
a(21) = 4837265534637221;
a(22+16*k) = 3837365544636221, k >= 0;
a(36) = a(20+16) = 3837265554834221 <> a(20);
a(37) = a(21+16) = 3837266544735221 <> a(21);
a(38) = a(22+16) = 3837365544636221 = a(22). - _Reinhard Zumkeller_, Aug 10 2014
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..181, 10 periods.
• Index to sequences related to say what you see

Crossrefs

Cf. A036058, A036066.

Programs

Haskell

import Data.List (sort, group)
a036059 n = a036059_list !! n
a036059_list = map (read . concatMap show) fss :: [Integer] where
   fss = [1] : [1] : zipWith h (tail fss) fss where
         h vs ws = concatMap (\us -> [length us, head us]) $
                   group $ reverse $ sort $ vs ++ ws
-- Reinhard Zumkeller, Aug 10 2014

Maple

a:= proc(n) option remember; `if`(n<2, 1, (p-> parse(cat(seq((c->
     `if`(c=0, [][], [c, 9-i][]))(coeff(p, x, 9-i)), i=0..9))))(
      add(x^i, i=map(x-> convert(x, base, 10)[], [a(n-1),a(n-2)]))))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jun 18 2022

Mathematica

a[0] = a[1] = 1; a[n_] := a[n] = FromDigits @ Flatten @ Reverse @ Select[ Transpose @ { DigitCount[a[n-1]] + DigitCount[a[n-2]], Append[ Range[9], 0]}, #[[1]] > 0&];
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Dec 30 2017 *)

Python

def aupton(nn):
  alst = [1, 1]
  for n in range(2, nn+1):
    prev2, anstr = sorted(str(alst[-2]) + str(alst[-1])), ""
    for d in sorted(set(prev2), reverse=True):
      anstr += str(prev2.count(d)) + d
    alst.append(int(anstr))
  return alst
print(aupton(20)) # Michael S. Branicky, Feb 02 2021