Roland Kneer - cp's OEIS frontend

A365538 a(0) = 1; otherwise, for i >= 0, a(4i+0) = a(4i+1) = a(2i), a(4i+2) = 2*a(2i+1), a(4i+3) = 0.

1, 1, 2, 0, 2, 2, 0, 0, 2, 2, 4, 0, 0, 0, 0, 0, 2, 2, 4, 0, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 4, 0, 4, 4, 0, 0, 4, 4, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Roland Kneer, Oct 23 2023

Related to a model of X-chromosome inheritance:

The two X chromosomes of a female are inherited one from each parent, while the X chromosome of a male is always inherited from his mother. Thus, the probability distribution of inheritance from the parents (mother, father) is (0.5, 0.5) for a female and (1, 0) for a male. For the inheritance of any X-chromosome of a female from the 2^i ancestors of the i-th generation before (right to left on an ahnentafel), the distribution is given by the first 2^i terms of the sequence, divided by 2^i. For example, the X-chromosome of a man, which was inherited from his mother, was inherited from his mother's 16 great-great-grandparents with probabilities 1/16, 1/16, 1/8, 0, 1/8, 1/8, 0, 0, 1/8, 1/8, 1/4, 0, 0, 0, 0, 0.

Roland Kneer, Table of n, a(n) for n = 0..10000

Cf. A280873, A003754.

Positions of 0's: A004780 (complement of A003714).

This sequence regarded as a triangle with rows of lengths 1, 1, 2, 4, 8, 16, ...:
  1;
  1;
  2, 0;
  2, 2, 0, 0;
  2, 2, 4, 0, 0, 0, 0, 0;
  2, 2, 4, 0, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
...

A364578 a(n) is the smallest n-digit number whose sum of digits is n.

Original entry on oeis.org

1, 11, 102, 1003, 10004, 100005, 1000006, 10000007, 100000008, 1000000009, 10000000019, 100000000029, 1000000000039, 10000000000049, 100000000000059, 1000000000000069, 10000000000000079, 100000000000000089, 1000000000000000099, 10000000000000000199

Offset: 1

Roland Kneer, Aug 14 2023

Michael S. Branicky, Table of n, a(n) for n = 1..1000

Cf. A051885, A202270 (largest instead of smallest).

Table[Module[{t=10^n},While[Total[IntegerDigits[t]]!=n+1,t++];t],{n,0,20}] (* or *) Join[{1},10^Range[63]+Flatten[Table[(k+1) 10^n-1,{n,0,6},{k,9}]]] (* Harvey P. Dale, Mar 31 2024 *)

a(n) = my(k=10^(n-1)); while (sumdigits(k) != n, k++); k; \\ Michel Marcus, Aug 16 2023

def a(n): m=(n-1)//9; return int("1"+"0"*(n-m-2)+str((n-1)%9)*(n>1)+"9"*m)
print([a(n) for n in range(1,21)]) # Michael S. Branicky, Aug 16 2023

A227549 Numbers that contain their base-16 representation in their decimal representation.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 357440, 357441, 357442, 357443, 357444, 357445, 357446, 357447, 357448, 357449, 1079653, 1081713, 1122966, 1123079, 1123080, 2246166, 3369253, 3371313, 3412566, 4494393, 4494400, 4535653, 5658739, 5658740, 5660793, 5660800

Offset: 1

Roland Kneer, Aug 05 2013

			357440 = (57440)_16
1079653 = (107965)_16
23132273099720801084801040 = (1322730997208010848010)_16

Roland Kneer and Giovanni Resta, Table of n, a(n) for n = 1..139 (terms < 16^22, first 55 terms from Roland Kneer)

Subsequence of A102489.

Select[Range[0,5661000],SequenceCount[IntegerDigits[#],IntegerDigits[#,16]]>0&] (* Harvey P. Dale, Apr 21 2023 *)

A227290 Repeatedly map numbers to number of letters in English name (A005589); sequence gives first number that needs n steps to get to 4.

Original entry on oeis.org

4, 0, 3, 1, 11, 23, 323, 1103323373373373373373373373373

Offset: 0

Roland Kneer, Jul 05 2013

Based on the observation by Diane Karloff that the trajectory of A005589 always converges to 4.

40311123323 (concatenation of the first seven terms) is the first time prime is obtained when concatenating k first initial terms of the sequence. - Jonathan Vos Post, Jul 06 2013, edited by Antti Karttunen, Jul 25 2013

			"eleven" has 6 letters, "six" has 3 letters, "three" has 5 letters, "five" has 4 letters.
So a(4)=11, as A005589^4(11)=4 and 11 is the first such number.
But a(3)=1, because "one" is the same length as "six".

Cf. A016037.

A227093 Decimal expansion of 1/9899.

Original entry on oeis.org

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

Offset: 0

Roland Kneer, Jul 01 2013

Group the terms 2 by 2 to get the first 11 Fibonacci numbers (A000045): 00 01 01 02 03 05 08 13 21 34 55 (89, 144, 233, ...).

			0.00010102030508132134559046368320032326497626022830588...

Paolo Xausa, Table of n, a(n) for n = 0..1000

Cf. A000045, A021093, A034948.

Maple
```
Digits := 140; evalf(1/9899);
```

First[RealDigits[1/9899, 10, 100, -1]] (* Paolo Xausa, Jun 16 2024 *)

Equals Sum_{i>=0} Fibonacci(i)/100^(i+1).

A227036 Expansion of 2(1+x^2)/((1-x)(1-x-2*x^3)).

Original entry on oeis.org

2, 4, 8, 16, 28, 48, 84, 144, 244, 416, 708, 1200, 2036, 3456, 5860, 9936, 16852, 28576, 48452, 82160, 139316, 236224, 400548, 679184, 1151636, 1952736, 3311108, 5614384, 9519860, 16142080, 27370852, 46410576, 78694740, 133436448, 226257604, 383647088, 650519988, 1103035200, 1870329380

Offset: 0

Roland Kneer, Jun 28 2013

Conjecture: The perimeter of the n-th iteration of the Harter-Heighway dragon is a(n) segments or a(n)/2^(n/2) base units.
a(n) = 2^(n+1)-4*A003230(n-4) (two times the number of segments, minus four times the number of squares)
The first differences 2, 2, 4, 8, 12, 20,.. are twice the (empirical) A203175. - R. J. Mathar, Jul 02 2013

			For the 4th iteration, take two 3rd iteration dragons (2*16); put together, they will make one square, so subtract the inner perimeter 4.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Kevin Ryde, Iterations of the Dragon Curve, see index "B" and "R".
Helena Verrill, On the Boundary of the Harter-Heighway dragon curve, arXiv:2407.17326 [math.CO], 2024.
Wikipedia, Dragon curve
Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-2).

Cf. A014577.

LinearRecurrence[{2, -1, 2, -2}, {2, 4, 8, 16}, 40] (* T. D. Noe, Jul 02 2013 *)
CoefficientList[Series[2 (1 + x^2) / ((1 - x) (1 - x - 2 x^3)), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 17 2013 *)

Vec(2*(1+x^2)/((1-x)*(1-x-2*x^3))+O(x^66)) \\ Joerg Arndt, Jul 01 2013

cp's OEIS Frontend

User: Roland Kneer

A365538 a(0) = 1; otherwise, for i >= 0, a(4i+0) = a(4i+1) = a(2i), a(4i+2) = 2*a(2i+1), a(4i+3) = 0.

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A364578 a(n) is the smallest n-digit number whose sum of digits is n.

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A227549 Numbers that contain their base-16 representation in their decimal representation.

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A227290 Repeatedly map numbers to number of letters in English name (A005589); sequence gives first number that needs n steps to get to 4.

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A227093 Decimal expansion of 1/9899.

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Maple

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A227036 Expansion of 2*(1+x^2)/((1-x)*(1-x-2*x^3)).

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A227036 Expansion of 2(1+x^2)/((1-x)(1-x-2*x^3)).