A102370 -id:A102370 - cp's OEIS frontend

A103205 Write numbers in decimal under each other, then read diagonals in upward direction.

0, 1, 2, 3, 4, 5, 6, 7, 8, 19, 10, 11, 12, 13, 14, 15, 16, 17, 18, 29, 20, 21, 22, 23, 24, 25, 26, 27, 28, 39, 30, 31, 32, 33, 34, 35, 36, 37, 38, 49, 40, 41, 42, 43, 44, 45, 46, 47, 48, 59, 50, 51, 52, 53, 54, 55, 56, 57, 58, 69, 60, 61, 62, 63, 64, 65, 66, 67, 68, 79, 70, 71, 72, 73, 74, 75, 76, 77, 78, 89, 80, 81, 82, 83, 84, 85, 86, 87, 88

Offset: 0

N. J. A. Sloane, Mar 27 2005

Decimal analog of A102370.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp. Preprint versions: [pdf, ps].

Cf. A102370 (base 2), A109681 (base 3), A325644 (base 4), A325645 (base 5), A325692 (base 6), A325693 (base 7), A325805 (base 8), A325829 (base 9), this sequence (base 10).

A037093 "Sloping binary representation" of Fibonacci numbers, slope = +1.

Original entry on oeis.org

0, 1, 3, 14, 57, 229, 916, 7761, 29567, 117474, 469113, 3973641, 15138352, 60146777, 240187355, 2070207870, 7733090689, 30791909229, 260408711716, 991495872825, 3942106110215, 15739612088946, 133333733918417

Offset: 0

Antti Karttunen, Jan 28 1999

			When Fibonacci numbers are written in binary (see A004685), under each other as:
0000000 (0)
0000001 (1)
0000001 (1)
0000010 (2)
0000011 (3)
0000101 (5)
0001000 (8)
0001101 (13)
0010101 (21)
0100010 (34)
0110111 (55)
1011001 (89)
and one starts collecting their bits from column-0 to SW-direction (from the least to the most significant end), one gets 000... (0), ...00001 (1), ...00011 (3), ...001110 (14), etc. (See A102370 for similar transformation done on nonnegative integers).

Same sequence in octal: A037098. Cf. also: A102370, A000045, A037094-A037095, A036284.

a(n) := Sum(bit_n(A000045(n+i), i)*(2^i), i=0..inf) [ bit_n := (x, n) -> `mod`(floor(x/(2^n)), 2); ]

In practice, n can be used as an upper limit instead of infinity.

Entry revised Dec 29 2007

A325644 "Sloping quaternary numbers": write numbers in quaternary under each other (right-justified), read diagonals in upward direction, convert to decimal.

Original entry on oeis.org

0, 1, 2, 7, 4, 5, 6, 11, 8, 9, 10, 15, 12, 13, 30, 19, 16, 17, 18, 23, 20, 21, 22, 27, 24, 25, 26, 31, 28, 29, 46, 35, 32, 33, 34, 39, 36, 37, 38, 43, 40, 41, 42, 47, 44, 45, 62, 51, 48, 49, 50, 55, 52, 53, 54, 59, 56, 57, 58, 63, 60, 125, 78, 67, 64, 65, 66, 71, 68, 69, 70

Offset: 0

Seiichi Manyama, Sep 07 2019

			    0
    1
    2
    3
   10
   11
   12
   13
   20
   21
   22
   23
   30
   31
   32
   33
  100
...
The upward-sloping diagonals are:
0
1
2
13
10
11
12
23
20
21
22
33
30
31
132
103
100
...
giving 0, 1, 2, "7", 4, 5, 6, "11", 8, 9, 10, "15", 12, 13, "30", "19", 16, ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Wikipedia, Quaternary numeral system

Cf. A102370 (base 2), A109681 (base3), this sequence (base 4), A325645 (base 5), A325692 (base 6), A325693 (base 7), A325805 (base 8), A325829 (base 9), A103205 (base 10).

def A(m, n)
  ary = [0]
  n.times{|i|
    (m ** i - i..m ** (i + 1) - i - 2).each{|j|
      ary << (0..i).inject(0){|s, k| s + (j + k).to_s(m)[-1 - k].to_i * m ** k}
    }
  }
  ary
end
p A(4, 4)

A325645 "Sloping quinary numbers": write numbers in quinary under each other (right-justified), read diagonals in upward direction, convert to decimal.

Original entry on oeis.org

0, 1, 2, 3, 9, 5, 6, 7, 8, 14, 10, 11, 12, 13, 19, 15, 16, 17, 18, 24, 20, 21, 22, 48, 29, 25, 26, 27, 28, 34, 30, 31, 32, 33, 39, 35, 36, 37, 38, 44, 40, 41, 42, 43, 49, 45, 46, 47, 73, 54, 50, 51, 52, 53, 59, 55, 56, 57, 58, 64, 60, 61, 62, 63, 69, 65, 66, 67, 68, 74, 70, 71

Offset: 0

Seiichi Manyama, Sep 07 2019

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Wikipedia, Quinary

Cf. A102370 (base 2), A109681 (base3), A325644 (base 4), this sequence (base 5), A325692 (base 6), A325693 (base 7), A325805 (base 8), A325829 (base 9), A103205 (base 10).

def A(m, n)
  ary = [0]
  n.times{|i|
    (m ** i - i..m ** (i + 1) - i - 2).each{|j|
      ary << (0..i).inject(0){|s, k| s + (j + k).to_s(m)[-1 - k].to_i * m ** k}
    }
  }
  ary
end
p A(5, 3)

A325692 "Sloping senary numbers": write numbers in senary (that is, base 6) under each other (right-justified), read diagonals in upward direction, convert to decimal.

Original entry on oeis.org

0, 1, 2, 3, 4, 11, 6, 7, 8, 9, 10, 17, 12, 13, 14, 15, 16, 23, 18, 19, 20, 21, 22, 29, 24, 25, 26, 27, 28, 35, 30, 31, 32, 33, 70, 41, 36, 37, 38, 39, 40, 47, 42, 43, 44, 45, 46, 53, 48, 49, 50, 51, 52, 59, 54, 55, 56, 57, 58, 65, 60, 61, 62, 63, 64, 71, 66, 67, 68, 69, 106, 77, 72

Offset: 0

Seiichi Manyama, Sep 07 2019

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Wikipedia, Senary

Cf. A102370 (base 2), A109681 (base 3), A325644 (base 4), A325645 (base 5), this sequence (base 6), A325693 (base 7), A325805 (base 8), A325829 (base 9), A103205 (base 10).

A325693 "Sloping septenary numbers": write numbers in septenary under each other (right-justified), read diagonals in upward direction, convert to decimal.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 13, 7, 8, 9, 10, 11, 12, 20, 14, 15, 16, 17, 18, 19, 27, 21, 22, 23, 24, 25, 26, 34, 28, 29, 30, 31, 32, 33, 41, 35, 36, 37, 38, 39, 40, 48, 42, 43, 44, 45, 46, 96, 55, 49, 50, 51, 52, 53, 54, 62, 56, 57, 58, 59, 60, 61, 69, 63, 64, 65, 66, 67, 68, 76, 70, 71

Offset: 0

Seiichi Manyama, Sep 07 2019

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A102370 (base 2), A109681 (base 3), A325644 (base 4), A325645 (base 5), A325692 (base 6), this sequence (base 7), A325805 (base 8), A325829 (base 9), A103205 (base 10).

A325805 "Sloping octal numbers": write numbers in octal under each other (right-justified), read diagonals in upward direction, convert to decimal.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 15, 8, 9, 10, 11, 12, 13, 14, 23, 16, 17, 18, 19, 20, 21, 22, 31, 24, 25, 26, 27, 28, 29, 30, 39, 32, 33, 34, 35, 36, 37, 38, 47, 40, 41, 42, 43, 44, 45, 46, 55, 48, 49, 50, 51, 52, 53, 54, 63, 56, 57, 58, 59, 60, 61, 126, 71, 64, 65, 66, 67, 68, 69, 70, 79

Offset: 0

Seiichi Manyama, Sep 07 2019

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Wikipedia, Octal

Cf. A102370 (base 2), A109681 (base 3), A325644 (base 4), A325645 (base 5), A325692 (base 6), A325693 (base 7), this sequence (base 8), A325829 (base 9), A103205 (base 10).

A325829 "Sloping nonary numbers": write numbers in nonary under each other (right-justified), read diagonals in upward direction, convert to decimal.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 17, 9, 10, 11, 12, 13, 14, 15, 16, 26, 18, 19, 20, 21, 22, 23, 24, 25, 35, 27, 28, 29, 30, 31, 32, 33, 34, 44, 36, 37, 38, 39, 40, 41, 42, 43, 53, 45, 46, 47, 48, 49, 50, 51, 52, 62, 54, 55, 56, 57, 58, 59, 60, 61, 71, 63, 64, 65, 66, 67, 68, 69, 70

Offset: 0

Seiichi Manyama, Sep 07 2019

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A102370 (base 2), A109681 (base 3), A325644 (base 4), A325645 (base 5), A325692 (base 6), A325693 (base 7), A325805 (base 8), this sequence (base 9), A103205 (base 10).

A103586 a(0)=1, for n > 0: n-th run consists of 2^n-1 copies of n+1.

Original entry on oeis.org

1, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 0

Benoit Cloitre, Mar 24 2005

a(A214489(n)) = A070939(A214489(n)).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.
O. Kullmann and X. Zhao, Bounds for variables with few occurrences in conjunctive normal forms, arXiv preprint arXiv:1408.0629 [math.CO], 2014-2017.
Ana Luzón, Manuel A. Morón, and Luis Felipe Prieto-Martínez, Commutators and commutator subgroups of the Riordan group, (2021).
Index entries for sequences related to binary expansion of n

Number of bits in binary representation of A102370(n).

Cf. A000225.

a103586 n = a070939 (n + a070939 n)
a103586_list = 1 : concat
   (zipWith (replicate . fromInteger) (tail a000225_list) [2..])
-- Reinhard Zumkeller, Jul 21 2012

Join[{1},Flatten[Table[PadRight[{},2^n-1,n+1],{n,6}]]] (* Harvey P. Dale, Aug 22 2021 *)

def A103586(n): return (m:=n.bit_length())+(n>=(1<Chai Wah Wu, Jun 30 2024

a(n) = A070939(n + A070939(n)) for n > 0. - Reinhard Zumkeller, Jul 21 2012

a(0) = 1 added, definition and offset adjusted by Reinhard Zumkeller, Jul 21 2012

A103185 a(n) = Sum_{ k >= 0 such that n + k == 0 mod 2^k } 2^(k-1).

Original entry on oeis.org

0, 1, 2, 1, 0, 5, 2, 1, 0, 1, 2, 1, 8, 5, 2, 1, 0, 1, 2, 1, 0, 5, 2, 1, 0, 1, 2, 17, 8, 5, 2, 1, 0, 1, 2, 1, 0, 5, 2, 1, 0, 1, 2, 1, 8, 5, 2, 1, 0, 1, 2, 1, 0, 5, 2, 1, 0, 1, 34, 17, 8, 5, 2, 1, 0, 1, 2, 1, 0, 5, 2, 1, 0, 1, 2, 1, 8, 5, 2, 1, 0, 1, 2, 1, 0, 5, 2, 1, 0, 1, 2, 17, 8, 5, 2, 1, 0, 1, 2, 1, 0, 5, 2, 1

Offset: 0

N. J. A. Sloane, Mar 18 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.

Cf. A102370(n).

f[n_] := Block[{k = 1, s = 0, l = Max[2, Floor[ Log[2, n + 1] + 2]]}, While[k < l, If[ Mod[n + k, 2^k] == 0, s = s + 2^(k - 1)]; k++ ]; s]; Table[ f[n], {n, 0, 103}] (* Robert G. Wilson v, Mar 18 2005 *)

A103185(n)=(sum(k=0,ceil(log(n+1)/log(2)),if((n+k)%2^k,0,2^k))-1)/2 \\ Benoit Cloitre, Mar 20 2005

def a102370(n): return 0 if n<1 else sum([(n + k)&(2**k) for k in range(len(bin(n)[2:]) + 1)])
def a(n): return (a102370(n) - n)/2 # Indranil Ghosh, May 03 2017

a(n) = (A102370(n) - n)/2.

More terms from Robert G. Wilson v, Mar 18 2005

cp's OEIS Frontend

A103205 Write numbers in decimal under each other, then read diagonals in upward direction.

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A037093 "Sloping binary representation" of Fibonacci numbers, slope = +1.

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Extensions

A325644 "Sloping quaternary numbers": write numbers in quaternary under each other (right-justified), read diagonals in upward direction, convert to decimal.

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Ruby

A325645 "Sloping quinary numbers": write numbers in quinary under each other (right-justified), read diagonals in upward direction, convert to decimal.

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Author

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Programs

Ruby

A325692 "Sloping senary numbers": write numbers in senary (that is, base 6) under each other (right-justified), read diagonals in upward direction, convert to decimal.

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Author

Keywords

Links

Crossrefs

A325693 "Sloping septenary numbers": write numbers in septenary under each other (right-justified), read diagonals in upward direction, convert to decimal.

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Author

Keywords

Links

Crossrefs

A325805 "Sloping octal numbers": write numbers in octal under each other (right-justified), read diagonals in upward direction, convert to decimal.

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Author

Keywords

Links

Crossrefs

A325829 "Sloping nonary numbers": write numbers in nonary under each other (right-justified), read diagonals in upward direction, convert to decimal.

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Author

Keywords

Links

Crossrefs

A103586 a(0)=1, for n > 0: n-th run consists of 2^n-1 copies of n+1.

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Haskell

Mathematica

Python

Formula

Extensions

A103185 a(n) = Sum_{ k >= 0 such that n + k == 0 mod 2^k } 2^(k-1).

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Python