Jean-Paul Delahaye - cp's OEIS frontend

User: Jean-Paul Delahaye

Jean-Paul Delahaye has authored 5 sequences.

A337658 Let M_k denote the addition table for the first k terms of A337655. M_k contains exactly k*(k+1)/2 distinct numbers, and these numbers are a subset of the entries in M_{k+1}. The present sequence consists of the numbers that never appear in any M_k.

1, 5, 11, 13, 15, 18, 19, 21, 25, 26, 28, 31, 34, 35, 39, 40, 41, 42, 43, 45, 47, 48, 49, 50, 54, 56, 58, 59, 60, 61, 63, 64, 66, 67, 68, 71, 74, 76, 77, 78, 79, 80, 82, 84, 85, 86, 87, 88, 89, 93, 94, 96, 98, 101, 104, 107, 109, 110, 111, 113, 114, 115, 117, 119, 120, 122, 124, 127

Offset: 1

Jean-Paul Delahaye, Oct 01 2020

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C++ program for A337658

Cf. A337655, A337656, A337657.

A337655 a(1)=1; thereafter, a(n) is the smallest number such that both the addition and multiplication tables for (a(1),...,a(n)) contain n*(n+1)/2 different entries (the maximum possible).

Original entry on oeis.org

1, 2, 5, 7, 15, 22, 31, 50, 68, 90, 101, 124, 163, 188, 215, 253, 322, 358, 455, 486, 527, 631, 702, 780, 838, 920, 1030, 1062, 1197, 1289, 1420, 1500, 1689, 1765, 1886, 2114, 2353, 2410, 2570, 2686, 2857, 3063, 3207, 3477, 3616, 3845, 3951, 4150, 4480, 4595, 4746, 5030, 5286, 5698, 5999, 6497, 6624, 6938, 7219, 7661, 7838, 8469, 8665, 9198, 9351, 9667, 9966

Offset: 1

Jean-Paul Delahaye, Sep 30 2020

nonn

If one specifies that not only are there n(n+1)/2 distinct numbers in the addition and multiplication tables, but that all n(n+1) numbers are distinct, then the sequence is A337946 - David A. Corneth, Oct 02 2020

Rémy Sigrist, Table of n, a(n) for n = 1..3000 (a(1)-a(101) from Jean-Paul Delahaye, a(102)-a(1000) from Peter Kagey)
Rémy Sigrist, C++ program for A337655

See A337659 and A337660 (for the addition table), and A337661 and A337662 (for the multiplication table).
Cf. A337656, A337657, A337658.
For similar sequences that focus just on the addition or multiplication tables, see A005282 and A066720.
Cf. also A337946.

terms=67;a[1]=b[1]=1;a1=b1={1};Do[k=a[n-1]+1;While[a2=Union@Join[{2k},Array[a@#+k&,n-1]];b2=Union@Join[{k^2},Array[b@#*k&,n-1]];Intersection[a2,a1]!={}||Intersection[b2,b1]!={},k++];a[n]=b[n]=k;a1=Union[a1,a2];b1=Union[b1,b2],{n,2,terms}];Array[a,terms] (* Giorgos Kalogeropoulos, Nov 15 2021 *)

A319879 a(n) = minimal number m of unit squares needed to make an figure formed from squares (joined edge to edge) which has n holes.

Original entry on oeis.org

1, 8, 13, 18, 21, 26, 29, 34, 37, 40, 45, 48, 51, 56, 59, 62, 65, 70, 73, 76, 79, 84, 87, 90, 93, 96, 101, 104, 107, 110, 113, 118, 121, 124, 127, 130, 133, 138, 141, 144, 147, 150, 153, 158, 161, 164, 167, 170, 173, 176, 181

Offset: 0

Jean-Paul Delahaye, Sep 30 2018

nonn

The holes must not touch each other.

			Examples from _N. J. A. Sloane_, Jan 30 2019 for n=1, 2, 3:
XXX
XOX
XXX
---
XXXXX
XOXOX
XXXXX
-----
XXXXXXX
XOXOXOX
XXXXXXX

Jean-Paul Delahaye, Des trous à entourer avec parcimonie, Pour las science, No. 501, Juillet 2019, pp. 82-87.

Entry 123 changed to 124 by the author, Jul 01 2019. - N. J. A. Sloane, Jul 01 2019

A030300 Runs have lengths 2^n, n >= 0.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Jean-Paul Delahaye

An example of a sequence with property that the fraction of 1's in the first n terms does not converge to a limit. - N. J. A. Sloane, Sep 24 2007
Image, under the coding sending a,d,e -> 1 and b,c -> 0, of the fixed point, starting with a, of the morphism a -> ab, b -> cd, c -> ee, d -> eb, e -> cc. - Jeffrey Shallit, May 14 2016
This sequence taken as digits of a base-b fraction is g(1/b) = Sum_{n>=1} a(n)/b^n = b/(b-1) * Sum_{k>=0} (-1)^k/b^(2^k) per the generating function below.  With initial 0, it is binary expansion .01001111 = A275975.  With initial 0 and digits 2*a(n), it is ternary expansion .02002222 = A160386.  These and in general g(1/b) for any integer b>=2 are among forms which Kempner showed are transcendental. - Kevin Ryde, Sep 07 2019

Antti Karttunen, Table of n, a(n) for n = 1..65537
Aubrey J. Kempner, On Transcendental Numbers, Transactions of the American Mathematical Society, volume 17, number 4, October 1916, pages 476-482.
Kevin Ryde, Plot of A079947(n)/n, illustrating proportion of 1s in the first n terms here does not converge (but oscillates with rises and falls by hyperbolas)
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Ralf Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.
Index entries for sequences related to binary expansion of n
Index entries for characteristic functions

Cf. A030301. Partial sums give A079947.
Cf. A065359, A083905.
Characteristic function of A053738.

f0 := n->[seq(0,i=1..2^n)]; f1 := n->[seq(1,i=1..2^n)]; s := []; for i from 0 to 4 do s := [op(s), op(f1(2*i)), op(f0(2*i+1))]; od: A030300 := s;

nMax = 6; Table[1 - Mod[n, 2], {n, 0, nMax}, {2^n}] // Flatten (* Jean-François Alcover, Oct 20 2016 *)

a(n) = if(n, !(logint(n,2)%2)); /* Kevin Ryde, Aug 02 2019 */

def A030300(n): return n.bit_length()&1 # Chai Wah Wu, Jan 30 2023

a(n) = A065359(n) + A083905(n).
a(n) = (1/2)*(1+(-1)^floor(log_2(n))). - Benoit Cloitre, Feb 22 2003
G.f.: 1/(1-x) * Sum_{k>=0} (-1)^k*x^2^k. - Ralf Stephan, Jul 12 2003
a(n) = 1 - a(floor(n/2)). - Vladeta Jovovic, Aug 04 2003
a(n) = A115253(2n, n) mod 2. - Paul Barry, Jan 18 2006
a(n) = 1 - A030301(n). - Antti Karttunen, Oct 10 2017

A030301 n-th run has length 2^(n-1).

Original entry on oeis.org

0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Jean-Paul Delahaye

Cf. A000523, A029837.
Cf. A030300. Partial sums give A079954.
Characteristic function of A053754 (after its initial 0).

[Floor(Log(n)/Log(2)) mod 2: n in [1..100]]; // Vincenzo Librandi, Jun 23 2015

nMax = 7; Table[1 - Mod[n, 2], {n, nMax}, {2^(n-1)}] // Flatten (* Jean-François Alcover, Oct 20 2016 *)
Table[{PadRight[{},2^(n-1),0],PadRight[{},2^n,1]},{n,1,8,2}]//Flatten (* Harvey P. Dale, Apr 12 2023 *)

PARI
```
a(n)=if(n<1,0,1-length(binary(n))%2)
```

a(n)=if(n<1,0,if(n%2==0,-a(n/2)+1,-a((n-1)/2)+1-(((n-1)/2)==0))) /* Ralf Stephan */

def A030301(n): return n.bit_length()&1^1 # Chai Wah Wu, Jan 30 2023

a(n) = A000523(n) mod 2 = (A029837(n+1)+1) mod 2.
a(n) = 0 iff n has an odd number of digits in binary, = 1 otherwise. - Henry Bottomley, Apr 06 2000
a(n) = (1/2)*{1-(-1)^floor(log(n)/log(2))}. - Benoit Cloitre, Nov 22 2001
a(n) = 1-a(floor(n/2)). - Vladeta Jovovic, Aug 04 2003
a(n) = 1 - A030300(n). - Antti Karttunen, Oct 10 2017

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User: Jean-Paul Delahaye

A337658 Let M_k denote the addition table for the first k terms of A337655. M_k contains exactly k*(k+1)/2 distinct numbers, and these numbers are a subset of the entries in M_{k+1}. The present sequence consists of the numbers that never appear in any M_k.

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A337655 a(1)=1; thereafter, a(n) is the smallest number such that both the addition and multiplication tables for (a(1),...,a(n)) contain n*(n+1)/2 different entries (the maximum possible).

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A319879 a(n) = minimal number m of unit squares needed to make an figure formed from squares (joined edge to edge) which has n holes.

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A030300 Runs have lengths 2^n, n >= 0.

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A030301 n-th run has length 2^(n-1).

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Magma

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