This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
import Data.List ((\\)) a249407 n = a249407_list !! (n-1) a249407_list = f [2..] where f ws@(u:v:_) = u : v : f (ws \\ [u, v, u * v])
import Data.List ((\\)) a249408 n = a249408_list !! (n-1) a249408_list = filter ((== 0) . a005369) a249406_list
import Data.List ((\\)) a075326 n = a075326_list !! n a075326_list = 0 : f [1..] where f ws@(u:v:_) = y : f (ws \\ [u, v, y]) where y = u + v -- Reinhard Zumkeller, Oct 26 2014
# Maple code for M+1 terms of sequence, from N. J. A. Sloane, Oct 26 2014 c:=0; a:=[c]; t:=0; M:=100; for n from 1 to M do s:=t+1; if s in a then s:=s+1; fi; t:=s+1; if t in a then t:=t+1; fi; c:=s+t; a:=[op(a),c]; od: [seq(a[n],n=1..nops(a))];
(* Three sequences a,b,c as in Comments *)
z = 200;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {}; b = {}; c = {};
Do[AppendTo[a,
mex[Flatten[{a, b, c}], If[Length[a] == 0, 1, Last[a]]]];
AppendTo[b, mex[Flatten[{a, b, c}], Last[a]]];
AppendTo[c, Last[a] + Last[b]], {z}];
Take[a, 100] (* A075425 *)
Take[b, 100] (* A047215 *)
Take[c, 100] (* A075326 *)
Grid[{Join[{"n"}, Range[0, 20]], Join[{"a(n)"}, Take[a, 21]],
Join[{"b(n)"}, Take[b, 21]], Join[{"c(n)"}, Take[c, 21]]},
Alignment -> ".",
Dividers -> {{2 -> Red, -1 -> Blue}, {2 -> Red, -1 -> Blue}}]
(* Peter J. C. Moses, Apr 26 2018 *)
********
(* Sequence "a" via A035263 substitutions *)
Accumulate[Prepend[Flatten[Nest[Flatten[# /. {0 -> {1, 1}, 1 -> {1, 0}}] &, {0}, 7] /. Thread[{0, 1} -> {{5, 5}, {6, 4}}]], 3]]
(* Peter J. C. Moses, May 01 2018 *)
********
(* Sequence "a" via Hofstadter substitutions; see his 2014 link *)
morph = Rest[Nest[Flatten[#/.{1->{3},3->{1,1,3}}]&,{1},6]]
hoff = Accumulate[Prepend[Flatten[morph/.Thread[{1,3}->{{6,4,5,5},{6,4,6,4,6,4,5,5}}]],3]]
(* Peter J. C. Moses, May 01 2018 *)
def aupton(nn):
alst, disallowed, mink = [0], {0}, 1
for n in range(1, nn+1):
nextk = mink + 1
while nextk in disallowed: nextk += 1
an = mink + nextk
alst.append(an)
disallowed.update([mink, nextk, an])
mink = nextk + 1
while mink in disallowed: mink += 1
return alst
print(aupton(57)) # Michael S. Branicky, Jan 31 2022
def A075326(n): return 5*n-1-int((n|(~((m:=n-1>>1)+1)&m).bit_length())&1) if n else 0 # Chai Wah Wu, Sep 11 2024
a000304 n = a000304_list !! (n-2) a000304_list = 2 : 3 : zipWith (*) a000304_list (tail a000304_list) -- Reinhard Zumkeller, Jul 06 2014
A000304 := proc(n) option remember; if n <=3 then n else A000304(n-1)*A000304(n-2); fi; end;
nxt[{a_,b_}]:={b,a*b}; Transpose[NestList[nxt,{2,3},12]][[1]] (* Harvey P. Dale, Nov 16 2014 *)
Distribution of oblong numbers on A249406 and its complement, m*(m+1) factorizations are shown in respective columns: . n | A002378 | A249406 | A249407 n | A002378 | A249406 | A249407 . ---+---------+---------+-------- ---+---------+---------+-------- . 1 | 2 | | 1*2 13 | 182 | 13*14 | . 2 | 6 | 2*3 | 14 | 210 | | 14*15 . 3 | 12 | | 3*4 15 | 240 | 15*16 | . 4 | 20 | 4*5 | 16 | 272 | | 16*17 . 5 | 30 | | 5*6 17 | 306 | 17*18 | . 6 | 42 | | 6*7 18 | 342 | | 18*19 . 7 | 56 | 7*8 | 19 | 380 | | 19*20 . 8 | 72 | | 8*9 20 | 420 | | 20*21 . 9 | 90 | 9*10 | 21 | 462 | | 21*22 . 10 | 110 | | 10*11 22 | 506 | 22*23 | . 11 | 132 | 11*12 | 23 | 552 | | 23*24 . 12 | 156 | | 12*13 24 | 600 | 24*25 | .
a249411 n = a249411_list !! (n-1) a249411_list = filter ((== 1) . a005369) a249407_list
Start with a(1) = 0. The missing numbers are 1 2 3 4 5 6 ... Multiply the first two, and we get 2, which is therefore a(2). Cross 1, 2, and 1*2 = 2 off the missing list. The first two missing numbers are now 3 and 4, so a(3) = 3*4 = 12. Cross off 3,4,12 from the missing list. Repeat!
M:=50; A:=[0]; miss:=[seq(n,n=1..M^2)]:
for n from 1 to M do t1:=miss[1]*miss[2]; A:=[op(A),t1];
miss:=[seq(miss[i],i=3..nops(miss))];
miss:=remove('x->x=t1',miss);
od:
A;
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