A020990 -id:A020990 - cp's OEIS frontend

A360536 Analog of Rudin-Shapiro sum sequence A020990, based on counting patterns 00 instead of 11.

1, 0, 1, 0, -1, -2, -1, -2, -1, 0, 1, 0, -1, -2, -1, -2, -3, -4, -5, -4, -5, -6, -5, -6, -5, -4, -3, -4, -5, -6, -5, -6, -5, -4, -3, -4, -3, -2, -3, -2, -1, 0, 1, 0, -1, -2, -1, -2, -3, -4, -5, -4, -5, -6, -5, -6, -5, -4, -3, -4, -5, -6, -5, -6, -7, -8, -9

Offset: 0

Jeffrey Shallit, Feb 10 2023

sign

a(n) = Sum_{i=0..n} (-1)^(i+e(i)), where e(i) counts the (possibly overlapping) occurrences of 00 in the base-2 representation of n. Note that e(0) = 0. This is the analog of A020990, which is the same sum, but with e(n) replaced by the function that counts the (possibly overlapping) occurrences of 11 in the base-2 representation of n.

N. Rampersad and J. Shallit, Rudin-Shapiro sums via automata theory and logic, Arxiv preprint arXiv:2302.00405 [math.NT], February 1 2023.

Cf. A020985, A020990, A360535.

A020986 a(n) = n-th partial sum of Golay-Rudin-Shapiro sequence A020985.

Original entry on oeis.org

1, 2, 3, 2, 3, 4, 3, 4, 5, 6, 7, 6, 5, 4, 5, 4, 5, 6, 7, 6, 7, 8, 7, 8, 7, 6, 5, 6, 7, 8, 7, 8, 9, 10, 11, 10, 11, 12, 11, 12, 13, 14, 15, 14, 13, 12, 13, 12, 11, 10, 9, 10, 9, 8, 9, 8, 9, 10, 11, 10, 9, 8, 9, 8, 9, 10, 11, 10, 11, 12, 11, 12, 13, 14, 15, 14, 13, 12, 13, 12, 13, 14, 15, 14, 15, 16

Offset: 0

N. J. A. Sloane

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
John Brillhart and Patrick Morton, Über Summen von Rudin-Shapiroschen Koeffizienten, (German) Illinois J. Math. 22 (1978), no. 1, 126--148. MR0476686 (57 #16245). - From _N. J. A. Sloane_, Jun 06 2012
J. Brillhart and P. Morton, A case study in mathematical research: the Golay-Rudin-Shapiro sequence, Amer. Math. Monthly, 103 (1996) 854-869.
Philip Lafrance, Narad Rampersad, and Randy Yee, Some properties of a Rudin-Shapiro-like sequence, arXiv:1408.2277 [math.CO], 2014.
Narad Rampersad and Jeffrey Shallit, Rudin-Shapiro Sums Via Automata Theory and Logic, arXiv:2302.00405 [math.NT], February 1 2023.
Eric Weisstein's World of Mathematics, Rudin-Shapiro Sequence
Index entries for sequences related to coordinates of 2D curves

Cf. A020985, A020990.

a020986 n = a020986_list !! n
a020986_list = scanl1 (+) a020985_list
-- Reinhard Zumkeller, Jan 02 2012

a[n_] := 1 - 2 Mod[Length[FixedPointList[BitAnd[#, # - 1] &, BitAnd[n, Quotient[n, 2]]]], 2]; Accumulate@ Table[a@ n, {n, 0, 85}] (* Michael De Vlieger, Nov 30 2015, after Jan Mangaldan at A020985 *)
Table[RudinShapiro[n], {n, 0, 100}] // Accumulate (* Jean-François Alcover, Jun 30 2022 *)

def A020986(n): return sum(-1 if (m&(m>>1)).bit_count()&1 else 1 for m in range(n+1)) # Chai Wah Wu, Feb 11 2023

Brillhart and Morton (1978) list many properties.

Minor edits by N. J. A. Sloane, Jun 06 2012

A334577 a(n) is the Y-coordinate of the n-th point of the space filling curve P defined in Comments section; sequence A334576 gives X-coordinates.

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 1, 0, 0, 0, 0, 1, 1, 1, 2, 3, 4, 4, 4, 5, 6, 6, 5, 4, 3, 3, 3, 2, 2, 2, 1, 0, 0, 0, 0, 1, 2, 2, 1, 0, 0, 0, 0, 1, 1, 1, 2, 3, 3, 3, 3, 2, 1, 1, 2, 3, 4, 4, 4, 5, 5, 5, 6, 7, 8, 8, 8, 9, 10, 10, 9, 8, 8, 8, 8, 9, 9, 9, 10, 11, 12, 12, 12, 13

Offset: 0

Rémy Sigrist, May 06 2020

nonn

The space filling curve P corresponds to the midpoint curve of the alternate paperfolding curve and can be built as follows:
- we define the family {P_k, k > 0}:
    - P_1 corresponds to the points (0, 0), (1, 0), (2, 0) and (2, 1), in that order:
                    +
                    |
                    |
          +----+----+
         O
    - for any k > 0, P_{n+1} is built from four copies of P_n as follows:
                                            +
                                            |A
                +                           |
               C|                   +----+  |
         A     B|    --->           |C  B|  |B  C
        +-------+                   +    |  +----+-+
       O                           C|    |        C|
                             A     B|   A|  A     B|
                            +-------+    +-+-------+
                           O
- the space filling curve P is the limit of P_k as k tends to infinity.
We can also describe the space filling curve P by mean of an L-system (see Links section).

			The first points of the space filling curve P are as follows:
      6|                                  20...21
       |                                  |    |
      5|                                  19   22
       |                                  |    |
      4|                        16...17...18   23
       |                        |              |
      3|                        15   26...25...24
       |                        |    |
      2|              4....5    14   27...28...29
       |              |    |    |              |
      1|              3    6    13...12...11   30
       |              |    |              |    |
      0|    0....1....2    7....8....9....10   31..
       |
    ---+----------------------------------------
    y/x|    0    1    2    3    4    5    6    7
- hence a(15) = a(24) = a(25) = a(26) = 3.

Rémy Sigrist, Table of n, a(n) for n = 0..4095
Joerg Arndt, L-system corresponding to P
Kevin Ryde, Iterations of the Alternate Paperfolding Curve
Rémy Sigrist, PARI program for A334577
Index entries for sequences related to coordinates of 2D curves

Cf. A020990, A334576.

PARI
```
See Links section.
```

a(n+1) = (A020990(n) + A020990(n+1) - 1)/2 for any n >= 0.

A360535 Analog of Rudin-Shapiro sum sequence A020986, based on counting patterns 00 instead of 11.

Original entry on oeis.org

1, 2, 3, 4, 3, 4, 5, 6, 7, 6, 7, 8, 7, 8, 9, 10, 9, 10, 9, 8, 7, 8, 9, 10, 11, 10, 11, 12, 11, 12, 13, 14, 15, 14, 15, 16, 17, 16, 15, 14, 15, 14, 15, 16, 15, 16, 17, 18, 17, 18, 17, 16, 15, 16, 17, 18, 19, 18, 19, 20, 19, 20, 21, 22, 21, 22, 21, 20, 19, 20

Offset: 0

Jeffrey Shallit, Feb 10 2023

nonn

a(n) = Sum_{i=0..n} (-1)^e(i), where e(i) counts the (possibly overlapping) occurrences of 00 in the base-2 representation of n. Note that e(0) = 0. This is the analog of A020986, which is the same sum, but with e(n) replaced by the function that counts the (possibly overlapping) occurrences of 11 in the base-2 representation of n.

			For n = 4, a(n) = 1+1+1+1+(-1) = 3.

N. Rampersad and J. Shallit, Rudin-Shapiro sums via automata theory and logic, Arxiv preprint arXiv:2302.00405 [math.NT], February 1 2023.

Cf. A020985, A020990, A360536.

cp's OEIS Frontend

A360536 Analog of Rudin-Shapiro sum sequence A020990, based on counting patterns 00 instead of 11.

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A020986 a(n) = n-th partial sum of Golay-Rudin-Shapiro sequence A020985.

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A334577 a(n) is the Y-coordinate of the n-th point of the space filling curve P defined in Comments section; sequence A334576 gives X-coordinates.

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Author

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Crossrefs

Programs

PARI

Formula

A360535 Analog of Rudin-Shapiro sum sequence A020986, based on counting patterns 00 instead of 11.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs