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.
a261392 = maximum . a261019_row
G.f. = x + 3*x^2 + 6*x^3 + 11*x^4 + 19*x^5 + 32*x^6 + 53*x^7 + 87*x^8 + ...
a001911 n = a001911_list !! n a001911_list = 0 : 1 : map (+ 2) (zipWith (+) a001911_list $ tail a001911_list) -- Reinhard Zumkeller, Jun 18 2013
[(Fibonacci(n+3))-2: n in [0..85]]; // Vincenzo Librandi, Apr 23 2011
a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+a[n-2]+2 od: seq(a[n],n=0..50); # Miklos Kristof, Mar 09 2005 A001911:=(1+z)/(z-1)/(z**2+z-1); # Simon Plouffe in his 1992 dissertation with another offset a:= n-> (Matrix([[0,-1,1]]). Matrix([[1,1,0], [1,0,0], [2,0,1]])^n)[1,1]: seq(a(n), n=0..50); # Alois P. Heinz, Jul 24 2008
Table[Fibonacci[n+3] -2, {n,0,50}] (* Vladimir Joseph Stephan Orlovsky, Nov 19 2010 *)
LinearRecurrence[{2,0,-1}, {0,1,3}, 40] (* Harvey P. Dale, Jun 06 2011 *)
Fibonacci[Range[3,40]]-2 (* Harvey P. Dale, Jun 28 2015 *)
a(n)=fibonacci(n+3)-2 \\ Charles R Greathouse IV, Mar 14 2012
[fibonacci(n+3)-2 for n in range(60)] # G. C. Greubel, Oct 21 2024
Begin with a(1)=1, in binary, "1". This contains the string "1" but not "10", so we add 2. Thus a(2)=1+2=3. This also contains "1" but not "10", so we move to a(3)=3+2=5. This contains "1" and "10" but not "11", so we add 3. Thus a(4)=5+3=8. (See A261018 for the successive numbers that are added. - _N. J. A. Sloane_, Aug 17 2015)
a260273 n = a260273_list !! (n-1) a260273_list = iterate (\x -> x + a261461 x) 1 -- Reinhard Zumkeller, Aug 30 2015, Aug 17 2015
public static void main(String[] args) {
int a=1;
for(int iter=0;iter<100;iter++){
System.out.print(a+", ");
int inc;
for(inc=1; contains(a,inc); inc++);
a+=inc;
}
}
static boolean contains(int a,int test){
int mask=(Integer.highestOneBit(test)<<1)-1;
while(a >= test){
if((a & mask) == test) return true;
a >>= 1;
}
return false;
}
sublistQ[L1_, L2_] := Module[{l1 = Length[L1], l2 = Length[L2], k}, If[l2 <= l1, For[k = 1, k <= l1 - l2 + 1, k++, If[L1[[k ;; k + l2 - 1]] == L2, Return[True]]]]; False];
a[1] = 1; a[n_] := a[n] = Module[{bb = IntegerDigits[a[n-1], 2], k}, For[k = 1, sublistQ[bb, IntegerDigits[k, 2]], k++]; a[n-1] + k]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Apr 01 2016 *)
NestList[Function[k, k + FromDigits[#, 2] &@ SelectFirst[IntegerDigits[Range[2^8], 2], Length@ SequencePosition[IntegerDigits[k, 2], #] == 0 &]], 1, 64] (* Michael De Vlieger, Apr 01 2016, Version 10.1 *)
A260273_list, a = [1], 1 for i in range(10**3): b, s = 1, format(a,'b') while format(b,'b') in s: b += 1 a += b s = format(a,'b') A260273_list.append(a) # Chai Wah Wu, Aug 26 2015
Triangle begins: 1,1, 1,1,1,1, 1,1,2,3,1,0,0,0, 1,1,3,6,4,1,0,0,0,0,0,0,0,0,0,0, ... For row 3, here are the 8 strings of length 3 and for each one, the smallest missing number k: 000 1 001 2 010 3 011 2 100 3 101 3 110 4 111 0
notVis[bits_] := For[i = 0, True, i++, If[SequencePosition[bits, IntegerDigits[i, 2]] == {}, Return[i]]];
T[n_, k_] := Select[Rest[IntegerDigits[#, 2]]& /@ Range[2^n, 2^(n+1)-1], notVis[#] == k&] // Length;
Table[T[n, k], {n, 1, 6}, {k, 0, 2^n-1}] // Flatten (* Jean-François Alcover, Aug 02 2018 *)
a261016 = sum . zipWith (*) [0..] . a261019_row' -- Reinhard Zumkeller, Aug 18 2015
(* This program is not suitable to compute more than a dozen terms. *)
notVis[bits_] := For[i = 0, True, i++, If[SequencePosition[bits, IntegerDigits[i, 2]] == {}, Return[i]]];
T[n_, k_] := Select[Rest[IntegerDigits[#, 2]] & /@ Range[2^n, 2^(n+1) - 1], notVis[#] == k &] // Length;
a[n_] := Sum[k*T[n, k], {k, 0, 2^n - 1}];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 12}] (* Jean-François Alcover, Aug 02 2018 *)
a261017 = subtract 1 . length . a261019_row -- Reinhard Zumkeller, Aug 18 2015
(* This program is not suitable to compute more than a dozen terms. *)
notVis[bits_] := For[i = 0, True, i++, If[SequencePosition[bits, IntegerDigits[i, 2]] == {}, Return[i]]];
T[n_, k_] := Select[Rest[IntegerDigits[#, 2]] & /@ Range[2^n, 2^(n+1) - 1], notVis[#] == k &] // Length;
a[n_] := Do[If[T[n, k] > 0, Return[k]], {k, 2^n - 1, 0, -1}];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 12}] (* Jean-François Alcover, Aug 02 2018 *)
a261441 0 = 0 a261441 n = a261019' (n + 3) 5
a261442 n = a261019' (n + 6) 6
a261443 n = a261019' (n + 5) 7
CoefficientList[Series[(x^16+5x^15+11x^14+10x^13-9x^12-40x^11-52x^10-19x^9+36x^8+61x^7+31x^6-13x^5-26x^4- 14x^3+ 4x^2+ 2x+2)x/((x+1)(x^2+x+1)(x^3+x^2-1)(x^2+x-1)(x^3+x^2+x-1)(x^4+x^3-1)(x^3+x-1)(x-1)^3),{x,0,40}],x] (* or *) LinearRecurrence[{4,-2,-5,-6,10,21,0,-29,-33,11,44,30,-16,-36,-17,9,16,6,-2,-3,-1},{0,2,10,40,116,296,699,1557,3325,6893,13964,27789,54536,105854,203645,388970,738596,1395718,2626914,4927664,9217604},40] (* Harvey P. Dale, Oct 30 2024 *)
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