cp's OEIS Frontend

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.

Showing 1-5 of 5 results.

A261646 Row lengths in A261644.

Original entry on oeis.org

1, 1, 1, 3, 5, 7, 15, 27, 49, 90, 171, 326, 613, 1174, 2255, 4333, 8386, 16213, 31457, 61097, 118577, 230322, 447767, 870570, 1693887, 3297343, 6423347, 12523562, 24435171, 47707012, 93215393, 182260604, 356622869, 698284551, 1368195878, 2682527954, 5262729874, 10330704931
Offset: 1

Views

Author

Reinhard Zumkeller, Aug 31 2015

Keywords

Comments

First differences of A261396;
also row lengths in A261712.

Crossrefs

Cf. A261644, A261712, A261396.

Programs

  • Haskell
    a261646 = length . a261644_row

Formula

a(n) = A261396(n) - A261396(n-1).

Extensions

a(21)-a(38) from Chai Wah Wu, Aug 31 2015

A261712 Reversed rows of A261644, seen as table read by rows.

Original entry on oeis.org

1, 1, 3, 1, 5, 8, 1, 5, 9, 12, 15, 3, 8, 13, 20, 25, 28, 31, 3, 8, 13, 17, 24, 29, 33, 37, 41, 44, 47, 52, 57, 60, 63, 2, 6, 10, 15, 24, 29, 33, 40, 47, 52, 57, 62, 66, 70, 77, 81, 85, 88, 91, 99, 105, 108, 111, 116, 121, 124, 127, 4, 12, 20, 25, 30, 34, 38
Offset: 1

Views

Author

Reinhard Zumkeller, Aug 31 2015

Keywords

Comments

T(n+1,k) = T(n,k) for k = 1 .. A261727(n).

Examples

			.   n |    T(n,*)                                            | A261646(n)
. ----+------------------------------------------------------+-----------
.   1 | 1                                                    |         1
.   2 | 1                                                    |         1
.   3 | 3                                                    |         1
.   4 | 1,5,8                                                |         3
.   5 | 1,5,9,12,15                                          |         5
.   6 | 3,8,13,20,25,28,31                                   |         7
.   7 | 3,8,13,17,24,29,33,37,41,44,47,52,57,60,63           |        15
.   8 | 2,6,10,15,24,29,33,40,47,52,57,62,66,70,77,81,85,... |        27
.   9 | 4,12,20,25,30,34,38,47,52,57,62,66,70,77,84,91,...   |        49
.  10 | 2,6,10,15,24,29,33,37,45,50,55,60,68,76,84,93,98,... |        90
.  11 | 2,6,10,15,24,29,33,37,45,50,55,60,68,76,84,93,98,... |       171
.  12 | 2,6,10,15,24,29,33,37,45,50,55,60,68,76,84,93,98,... |       326
.  13 | 2,6,10,15,24,29,33,37,45,50,55,60,68,76,84,93,98,... |       613
.  14 | 2,6,10,15,24,29,33,37,45,50,55,60,68,76,84,93,98,... |      1174
.  15 | 2,6,10,15,24,29,33,37,45,50,55,60,68,76,84,93,98,... |      2255
.  16 | 2,6,10,15,24,29,33,37,45,50,55,60,68,76,84,93,98,... |      4333 .

Links

Crossrefs

Cf. A261644, A261646 (row lengths), A261727.

Programs

  • Haskell
    a261712 n k = a261712_tabf !! (n-1) !! (k-1)
a261712_row n = a261712_tabf !! (n-1)
a261712_tabf = map reverse a261644_tabf

A261645 First differences of A261644, seen as flattened list.

Original entry on oeis.org

0, 2, 5, -3, -4, 14, -3, -3, -4, -4, 30, -3, -3, -5, -7, -5, -5, 60, -3, -3, -5, -5, -3, -3, -4, -4, -4, -5, -7, -4, -5, -5, 124, -3, -3, -5, -5, -3, -3, -6, -8, -3, -3, -4, -4, -7, -4, -4, -5, -5, -5, -7, -7, -4, -5, -9, -5, -4, -4, 252, -3, -3, -3, -6, -3
Offset: 1

Views

Author

Reinhard Zumkeller, Aug 31 2015

Keywords

Links

Crossrefs

Cf. A261644, A261018.

Programs

  • Haskell
    a261645 n = a261645_list !! (n-1)
a261645_list = zipWith (-) (tail a261644_list) a261644_list

A260273 Successively add the smallest nonzero binary number that is not a substring.

Original entry on oeis.org

1, 3, 5, 8, 11, 15, 17, 20, 23, 27, 31, 33, 36, 39, 44, 51, 56, 61, 65, 68, 71, 76, 81, 84, 87, 91, 95, 99, 104, 111, 115, 120, 125, 129, 132, 135, 140, 145, 148, 151, 157, 165, 168, 171, 175, 179, 186, 190, 194, 199, 204, 209, 216, 223, 227, 232, 241, 246
Offset: 1

Views

Author

Alex Meiburg, Jul 22 2015

Keywords

Comments

a(n) is at least Omega(n), at most O(n*log(n)).
The empirical approximation n*(log(n)/2 + exp(1)) is startlingly close to tight, compared with many increasing upper bounds.
A261644(n) = A062383(a(n)) - a(n). - Reinhard Zumkeller, Aug 30 2015

Examples

			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)

Links

Crossrefs

See A261922 and A261461 for the smallest missing number function; also A261923, A262279, A261281.
Cf. A030308, A261015, A261016, A261017, A261019.
See also A261396 (when a(n) just passes a power of 2), A261416 (the limiting behavior just past a power of 2).
First differences are A261018.
A262288 is the decimal analog.
Cf. A261644, A062383.

Programs

  • Haskell
    a260273 n = a260273_list !! (n-1)
a260273_list = iterate (\x -> x + a261461 x) 1
-- Reinhard Zumkeller, Aug 30 2015, Aug 17 2015
  • Java
    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;
}
  • Mathematica
    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 *)
  • Python
    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

Formula

a(n+1) = a(n) + A261461(a(n)). - Reinhard Zumkeller, Aug 30 2015

A261396 a(n) = smallest k such that A260273(k) >= 2^n.

Original entry on oeis.org

1, 2, 3, 4, 7, 12, 19, 34, 61, 110, 200, 371, 697, 1310, 2484, 4739, 9072, 17458, 33671, 65128, 126225, 244802, 475124, 922891, 1793461, 3487348, 6784691, 13208038, 25731600, 50166771, 97873783, 191089176, 373349780, 729972649, 1428257200, 2796453078, 5478981032, 10741710906, 21072415837
Offset: 0

Views

Author

N. J. A. Sloane, Aug 17 2015

Keywords

Comments

The sequence indicates the first time a term in A260273 has binary length n+1.
A261646 = first differences = row lengths of tables A261644 and A261712. - Reinhard Zumkeller, Aug 30 2015

Crossrefs

Cf. A260273.
Cf. A261646, A261644, A261712.

Programs

  • Haskell
    a261396 n = a261396_list !! (n-1)
a261396_list = f 1 1 a260273_list where
   f z k (x:xs) | x >= z    = k : f (2 * z) (k + 1) xs
                | otherwise = f z (k + 1) xs
-- Reinhard Zumkeller, Aug 30 2015

Extensions

a(18)-a(23) from Alois P. Heinz, Aug 19 2015
a(24)-a(34) from Chai Wah Wu, Aug 26 2015
a(35)-a(38) from Chai Wah Wu, Aug 31 2015
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.