A122245 -id:A122245 - cp's OEIS frontend

A122229 a(n) = A014486(A122228(n)).

0, 2, 12, 56, 228, 920, 3684, 14744, 58980, 235928, 943716, 3774872, 15099492, 60397976, 241591908, 966367640, 3865470564, 15461882264, 61847529060, 247390116248, 989560464996, 3958241859992, 15832967439972

Offset: 0

Antti Karttunen, Sep 14 2006

A simple formula exists, cf. A080675.

A. Karttunen, Python program for computing this sequence and the associated image.
A. Karttunen, Terms a(1)-a(128) drawn as binary strings, in Wolframesque fashion.

A122230 shows the same sequence in binary. Compare to similar Wolframesque plots given in A080070, A122232, A122235, A122239, A122242, A122245.

A122232 a(n) = A014486(A122231(n)).

Original entry on oeis.org

42, 212, 992, 3876, 15448, 64644, 252056, 989988, 4108676, 16147220, 63393540, 266083460, 1047285272, 4245874244, 16903342544, 67034166420, 274274527940, 1068738181764, 4246566244100, 17369295361736, 67322784388376, 269731897678032

Offset: 1

Antti Karttunen, Sep 14 2006

A. Karttunen, Python program for computing this sequence and the associated image.
A. Karttunen, Terms a(1)-a(512) drawn as binary strings, in Wolframesque fashion.

A122233 shows the same sequence in binary. Compare to similar Wolframesque plots given in A080070, A122229, A122235, A122239, A122242, A122245.

A122235 a(n) = A014486(A122234(n)).

Original entry on oeis.org

44, 216, 968, 3860, 16132, 62064, 247236, 1044612, 4073156, 16161828, 64513624, 253336008, 1046901060, 4267950372, 16347521428, 68075401492, 268150646664, 1086041921700, 4254535157576, 17346201751972, 66879000490408, 276319489325472

Offset: 1

Antti Karttunen, Sep 14 2006

A. Karttunen, Python program for computing this sequence and the associated image.
A. Karttunen, Terms a(1)-a(512) drawn as binary strings, in Wolframesque fashion.

A122236 shows the same sequence in binary. Compare to similar Wolframesque plots given in A080070, A122229, A122232, A122239, A122242, A122245.

A122239 a(n) = A014486(A122238(n)).

Original entry on oeis.org

52, 240, 964, 3972, 15556, 64532, 248288, 988964, 4164356, 15899248, 64719124, 257019652, 1070118936, 4197239188, 16299415152, 65592597568, 259741591312, 1093901323332, 4233842104068, 16616683414632, 70137217092164

Offset: 1

Antti Karttunen, Sep 14 2006

A122240 shows the same sequence in binary.

A. Karttunen, Python program for computing this sequence and the associated image.
A. Karttunen, Terms a(1)-a(512) drawn as binary strings, in Wolframesque fashion.

Compare to similar Wolframesque plots given in A080070, A122229, A122232, A122235, A122242, A122245.

A122244 Iterates of A122237, starting from 5.

Original entry on oeis.org

5, 21, 55, 183, 512, 1724, 6085, 20899, 66106, 231841, 888275, 3188220

Offset: 1

Antti Karttunen, Sep 14 2006

nonn

A122245.

(define (A122244 n) (if (= 1 n) 5 (A122237 (A122244 (- n 1)))))

A179757 a(n) = A014486(A179756(n)).

Original entry on oeis.org

56, 228, 932, 3736, 15512, 62040, 242264, 969136, 3985840, 15943080, 62096808, 248388496, 1021909904, 4087635248, 15883066672, 63532285248, 261545377088, 1046181434696, 4065904153928, 16263616920064, 66960253254144

Offset: 1

Antti Karttunen, Aug 03 2010

nonn

A. Karttunen, Table of n, a(n) for n = 1..1024
A. Karttunen, Python program for computing this sequence and the associated images.
A. Karttunen, Terms a(1)-a(768) drawn as binary strings, one pixel per bit.
A. Karttunen, Terms a(1)-a(700) drawn as binary strings, 3x3 pixels per bit.

A122242, A122245, A179755. Cf. also A179763, A179828, A179829.

A179777 Position of ones in A179776.

Original entry on oeis.org

1, 2, 6, 10, 15, 17, 22, 25, 31, 33, 34, 35, 39, 41, 42, 43, 47, 49, 50, 51, 55, 57, 62, 64, 65, 71, 79, 80, 84, 89, 92, 94, 99, 101, 103, 108, 110, 111, 115, 119, 124, 126, 127, 128, 132, 134, 135, 136, 140, 142, 144, 149, 151, 156, 160, 164, 165, 166, 167, 168

Offset: 1

Antti Karttunen, Aug 03 2010

nonn

This seems to give the positions where "L"'s occur in the central column of A122245 (please zoom into the illustration given here). Conjecture, from a(56)=164 onward, all integers >= 164 present.

A. Karttunen, Table of n, a(n) for n = 1..150
A. Karttunen, Terms a(1)-a(700) of A122245 drawn as binary strings, 3x3 pixels per bit.

Cf. A179778-A179779, and also A179772, A179832.

A154473 a(n) = A014486(A154472(n)).

Original entry on oeis.org

842, 11090, 13202, 46882, 60994, 231272004, 198873570440, 266349291297936, 64442911458703648, 3667589230123774528, 3336154829743802737792, 17601566387699271821281536, 1023499990310357893964861952

Offset: 0

Antti Karttunen, with terms a(0)-a(100) also independently computed by Wouter Meeussen, with the given Mathematica program, Jan 11 2009

This sequence gives the parenthesis expressions shown at the upper right corner image of the page 103 of NKS, with the left brackets (black squares) converted to 1's and the right brackets (white squares) converted to 0's and then interpreting each such number as a binary number and converted to decimal. A154474 shows the corresponding binary representations. Compare to A080070, A122242, A122245.

A. Karttunen and W. Meeussen, Table of n, a(n) for n = 0..100
S. Wolfram, A New Kind of Science, Wolfram Media Inc., (2002), p. 102, p. 103 and pages 104, 896-898.

Cf. A154475, A154476.

init=e[e[e][e]][e][e]
toDeca[ w_ ]:=FromDigits[ ToExpression[ Characters[ ToString[ w ] ]/.{"e"->Sequence[], "["->"1","]"->"0"} ],2 ]
toDeca /@ NestList[ #/.e[x_ ][y_ ]->x[x[y]]&, init, 100]

A179755 a(n) = A014486(A179754(n)).

Original entry on oeis.org

50, 216, 868, 3492, 13976, 56472, 225880, 897624, 3590576, 14471600, 57886120, 229868968, 919477136, 3706264464, 14825053488, 58832739632, 235330977088, 948740144448, 3794960504136, 15061020431688, 60244082031104

Offset: 1

Antti Karttunen, Aug 03 2010

nonn

A. Karttunen, Table of n, a(n) for n = 1..1024
A. Karttunen, Python program for computing this sequence and the associated images.
A. Karttunen, Terms a(1)-a(768) drawn as binary strings, one pixel per bit.
A. Karttunen, Terms a(1)-a(700) drawn as binary strings, 3x3 pixels per bit.

a(n+1) = A004758(A179757(n)). Cf. also A122242, A122245.

A179417 a(n) is the binary number (shown here in decimal) constructed from quadratic residues of 65537 in range [(n^2)+1,(n+1)^2] in such a way that quadratic residues are mapped to 1-bits, and non-quadratic residues (as well as the multiples of 65537) to 0-bits, with the lower end of range mapped to less significant, and the higher end of range to more significant bits.

Original entry on oeis.org

1, 5, 24, 104, 279, 2001, 4131, 17453, 88826, 362532, 1655660, 6120642, 25376649, 128526482, 301370205, 1756488602, 8046359747, 30854867177, 73845140753, 488906501177, 2106640948770, 6573967883049, 29711211505300

Offset: 0

Antti Karttunen, Jul 27 2010

The binary width of terms are 1, 3, 5, 7, 9, ... i.e., the successive odd numbers, as their partial sums give the squares, 1, 4, 9, 16, ... at which points there certainly is always a quadratic residue, which thus gives the most significant bit for each number.

			In the range [(2^2)+1, (2+1)^2] (i.e., [5,9]) we have A165471(5)=A165471(6)=A165471(7)=-1 and A165471(8)=A165471(9)=+1, i.e., there are quadratic non-residues at points 5, 6 and 7, and quadratic residues at 8 and 9, so we construct a binary number 11000, which is 24 in decimal, thus a(2)=24.

Antti Karttunen, Table of n, a(n) for n = 0..256
Antti Karttunen, Terms a(0)-a(255) drawn as a bit triangle, 1 pixel per bit.
Antti Karttunen, Terms a(0)-a(255) drawn as a bit triangle, 2x2 pixels per bit.
Antti Karttunen, Terms a(0)-a(255) drawn as a bit triangle, 3x3 pixels per bit.

Cf. A179418.

Compare to similar bit triangle illustrations given in A080070, A122229, A122232, A122235, A122239, A122242, A122245.

cp's OEIS Frontend

A122229 a(n) = A014486(A122228(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

A122232 a(n) = A014486(A122231(n)).

Views

Author

Keywords

Links

Crossrefs

A122235 a(n) = A014486(A122234(n)).

Views

Author

Keywords

Links

Crossrefs

A122239 a(n) = A014486(A122238(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

A122244 Iterates of A122237, starting from 5.

Views

Author

Keywords

Crossrefs

Programs

Scheme

A179757 a(n) = A014486(A179756(n)).

Views

Author

Keywords

Links

Crossrefs

A179777 Position of ones in A179776.

Views

Author

Keywords

Comments

Links

Crossrefs

A154473 a(n) = A014486(A154472(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A179755 a(n) = A014486(A179754(n)).

Views

Author

Keywords

Links

Crossrefs

Views

Author

Keywords

Comments

Examples

Links

Crossrefs