A073794 -id:A073794 - cp's OEIS frontend

A053985 Replace 2^k with (-2)^k in binary expansion of n.

0, 1, -2, -1, 4, 5, 2, 3, -8, -7, -10, -9, -4, -3, -6, -5, 16, 17, 14, 15, 20, 21, 18, 19, 8, 9, 6, 7, 12, 13, 10, 11, -32, -31, -34, -33, -28, -27, -30, -29, -40, -39, -42, -41, -36, -35, -38, -37, -16, -15, -18, -17, -12, -11, -14, -13, -24, -23, -26, -25, -20, -19

Offset: 0

Henry Bottomley, Apr 03 2000

Base 2 representation for n (in lexicographic order) converted from base -2 to base 10.
Maps natural numbers uniquely onto integers; within each group of positive values, maximum is in A002450; a(n)=n iff n can be written only with 1's and 0's in base 4 (A000695).
a(n) = A004514(n) - n. - Reinhard Zumkeller, Dec 27 2003
Schroeppel gives formula n = (a(n) + b) XOR b where b = binary ...101010, and notes this formula is reversible.  The reverse a(n) = (n XOR b) - b is a bit twiddle to transform 1 bits to -1.  Odd position 0 or 1 in n is flipped by "XOR b" to 1 or 0, then "- b" gives 0 or -1.  Only odd position 1's are changed, so b can be any length sure to cover those. - Kevin Ryde, Jun 26 2020

			a(9)=-7 because 9 is written 1001 base 2 and (-2)^3 + (-2)^0 = -8 + 1 = -7.
Or by Schroeppel's formula, b = binary 1010 then a(9) = (1001 XOR 1010) - 1010 = decimal -7. - _Kevin Ryde_, Jun 26 2020

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Michael Beeler, R. William Gosper, and Richard Schroeppel, HAKMEM, MIT Artificial Intelligence Laboratory report AIM-239, February 1972, item 128 by Schroeppel, page 62. Also HTML transcription. Figure 7 path drawn 0, 1, -2, -1, 4, ... is the present sequence.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 45, 49.
Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625 [math.CO], 2020-2021.
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions

Inverse of A005351. Cf. A039724, A007088, A065369, A073791, A073792, A073793, A073794, A073795, A073796 and A073835.

f[n_Integer, b_Integer] := Block[{l = IntegerDigits[n]}, Sum[l[[ -i]]*(-b)^(i - 1), {i, 1, Length[l]}]]; a = Table[ FromDigits[ IntegerDigits[n, 2]], {n, 0, 80}]; b = {}; Do[b = Append[b, f[a[[n]], 2]], {n, 1, 80}]; b
(* Second program: *)
Array[FromDigits[IntegerDigits[#, 2], -2] &, 62, 0] (* Michael De Vlieger, Jun 27 2020 *)

a(n) = fromdigits(binary(n), -2) \\ Rémy Sigrist, Sep 01 2018

def A053985(n): return  -(b:=int('10'*(n.bit_length()+1>>1),2)) + (n^b) if n else 0 # Chai Wah Wu, Nov 18 2022

From Ralf Stephan, Jun 13 2003: (Start)
G.f.: (1/(1-x)) * Sum_{k>=0} (-2)^k*x^2^k/(1+x^2^k).
a(0) = 0, a(2*n) = -2*a(n), a(2*n+1) = -2*a(n)+1. (End)
a(n) = Sum_{k>=0} A030308(n,k)*A122803(k). - Philippe Deléham, Oct 15 2011
a(n) = (n XOR b) - b where b = binary ..101010 [Schroeppel].  Any b of this form (A020988) with bitlength(b) >= bitlength(n) suits. - Kevin Ryde, Jun 26 2020

A065369 Replace 3^k with (-3)^k in ternary expansion of n.

Original entry on oeis.org

0, 1, 2, -3, -2, -1, -6, -5, -4, 9, 10, 11, 6, 7, 8, 3, 4, 5, 18, 19, 20, 15, 16, 17, 12, 13, 14, -27, -26, -25, -30, -29, -28, -33, -32, -31, -18, -17, -16, -21, -20, -19, -24, -23, -22, -9, -8, -7, -12, -11, -10, -15, -14, -13, -54, -53, -52, -57, -56, -55, -60, -59, -58, -45, -44, -43, -48, -47, -46

Offset: 0

Marc LeBrun, Oct 31 2001

Base 3 representation for n (in lexicographic order) converted from base -3 to base 10.
Notation: (3)[n](-3)
Fixed point of the morphism 0-> 0,1,2 ; 1-> -3,-2,-1 ; 2-> -6,-5,-4 ; ...; n-> -3n,-3n+1,-3n+2. - Philippe Deléham, Oct 22 2011

			15 = +1(9)+2(3)+0(1) -> +1(+9)+2(-3)+0(+1) = +3 = a(15)

Rémy Sigrist, Table of n, a(n) for n = 0..19682
Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625 [math.CO], 2020-2021.

Cf. A007089, A053985, A065367, A073785, A073791, A073792, A073793, A073794, A073795, A073796, A073835.

f[n_Integer, b_Integer] := Block[{l = IntegerDigits[n]}, Sum[l[[ -i]]*(-b)^(i - 1), {i, 1, Length[l]}]]; a = Table[ FromDigits[ IntegerDigits[n, 3]], {n, 0, 80}]; b = {}; Do[b = Append[b, f[a[[n]], 3]], {n, 1, 80}]; b

a(n) = fromdigits(digits(n, 3), -3) \\ Rémy Sigrist, Feb 06 2020

a(n) = Sum_{k>=0} A030341(n,k)*(-3)^k. - Philippe Deléham, Oct 22 2011

a(3*k+m) = -3*a(k)+m for 0 <= m < 3. - Chai Wah Wu, Jan 16 2020

A073835 Replace 10^k with (-10)^k in decimal expansion of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -30, -29, -28, -27, -26, -25, -24, -23, -22, -21, -40, -39, -38, -37, -36, -35, -34, -33, -32, -31, -50, -49, -48, -47, -46, -45, -44, -43, -42, -41, -60

Offset: 0

Robert G. Wilson v, Aug 12 2002

Base 10 representation for n (in lexicographic order) converted from base -10 to base 10.

A bijection from N = [0..oo) to Z = (-oo..+oo), or enumeration of the integers. - M. F. Hasler, Oct 17 2018

Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625, Dec 08, 2020

Cf. A001477, A039723, A053985, A065369, A073791, A073792, A073793, A073794, A073795 & A073796.

f[n_Integer, b_Integer] := Block[{l = IntegerDigits[n]}, Sum[l[[ -i]]*(-b)^(i - 1), {i, 1, Length[l]}]]; a = Table[FromDigits[ IntegerDigits[n, 10]], {n, 0, 80}]; b = {}; Do[ b = Append[b, f[a[[n]], 10]], {n, 1, 80}]; b (* Typo fixed by Harvey P. Dale, Oct 03 2013 *)

a(n)=fromdigits(digits(n),-10) \\ M. F. Hasler, Oct 17 2018

a(10*k+m) = -10*a(k)+m for 0 <= m < 10. - Chai Wah Wu, Jan 16 2020

A073791 Replace 4^k with (-4)^k in base 4 expansion of n.

Original entry on oeis.org

0, 1, 2, 3, -4, -3, -2, -1, -8, -7, -6, -5, -12, -11, -10, -9, 16, 17, 18, 19, 12, 13, 14, 15, 8, 9, 10, 11, 4, 5, 6, 7, 32, 33, 34, 35, 28, 29, 30, 31, 24, 25, 26, 27, 20, 21, 22, 23, 48, 49, 50, 51, 44, 45, 46, 47, 40, 41, 42, 43, 36, 37, 38, 39, -64, -63, -62, -61, -68, -67, -66, -65, -72, -71, -70, -69

Offset: 0

Robert G. Wilson v, Aug 12 2002

Base 4 representation for n converted from base -4 to base 10.

Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625 [math.CO], 2020-2021.

Cf. A007090, A007608, A053985, A065369, A073792, A073793, A073794, A073795, A073796 & A073835.

Cf. A066321, A320283.

f[n_Integer, b_Integer] := Block[{l = IntegerDigits[n]}, Sum[l[[ -i]]*(-b)^(i - 1), {i, 1, Length[l]}]]; a = Table[ FromDigits[ IntegerDigits[n, 4]], {n, 0, 80}]; b = {}; Do[b = Append[b, f[a[[n]], 4]], {n, 1, 80}]; b

a(n) = subst(Pol(digits(n,4)), x, -4); \\ Michel Marcus, Jan 30 2019

a(4*k+m) = -4*a(k)+m for 0 <= m < 4. - Chai Wah Wu, Jan 16 2020

A073792 Replace 5^k with (-5)^k in base 5 expansion of n.

Original entry on oeis.org

0, 1, 2, 3, 4, -5, -4, -3, -2, -1, -10, -9, -8, -7, -6, -15, -14, -13, -12, -11, -20, -19, -18, -17, -16, 25, 26, 27, 28, 29, 20, 21, 22, 23, 24, 15, 16, 17, 18, 19, 10, 11, 12, 13, 14, 5, 6, 7, 8, 9, 50, 51, 52, 53, 54, 45, 46, 47, 48, 49, 40, 41, 42, 43, 44, 35, 36, 37, 38, 39, 30, 31, 32, 33, 34

Offset: 0

Robert G. Wilson v, Aug 12 2002

Base 5 representation for n converted from base -5 to base 10.

Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625 [math.CO], 2020-2021.

Cf. A007091, A073786, A053985, A065369, A073791, A073793, A073794, A073795, A073796 & A073835.

f[n_Integer, b_Integer] := Block[{l = IntegerDigits[n]}, Sum[l[[ -i]]*(-b)^(i - 1), {i, 1, Length[l]}]]; a = Table[ FromDigits[ IntegerDigits[n, 5]], {n, 0, 80}]; b = {}; Do[b = Append[b, f[a[[n]], 5]], {n, 1, 80}]; b

a(5*k+m) = -5*a(k)+m for 0 <= m < 5. - Chai Wah Wu, Jan 16 2020

A073793 Replace 6^k with (-6)^k in base 6 expansion of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, -6, -5, -4, -3, -2, -1, -12, -11, -10, -9, -8, -7, -18, -17, -16, -15, -14, -13, -24, -23, -22, -21, -20, -19, -30, -29, -28, -27, -26, -25, 36, 37, 38, 39, 40, 41, 30, 31, 32, 33, 34, 35, 24, 25, 26, 27, 28, 29, 18, 19, 20, 21, 22, 23, 12, 13, 14, 15, 16, 17, 6, 7, 8, 9, 10, 11

Offset: 0

Robert G. Wilson v, Aug 12 2002

Base 6 representation for n converted from base -6 to base 10.

Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625 [math.CO], 2020-2021.

Cf. A007092, A073787, A053985, A065369, A073791, A073792, A073794, A073795, A073796 & A073835.

f[n_Integer, b_Integer] := Block[{l = IntegerDigits[n]}, Sum[l[[ -i]]*(-b)^(i - 1), {i, 1, Length[l]}]]; a = Table[ FromDigits[ IntegerDigits[n, 6]], {n, 0, 80}]; b = {}; Do[b = Append[b, f[a[[n]], 6]], {n, 1, 80}]; b

a(6*k+m) = -6*a(k)+m for 0 <= m < 6. - Chai Wah Wu, Jan 16 2020

A073795 Replace 8^k with (-8)^k in base 8 expansion of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, -8, -7, -6, -5, -4, -3, -2, -1, -16, -15, -14, -13, -12, -11, -10, -9, -24, -23, -22, -21, -20, -19, -18, -17, -32, -31, -30, -29, -28, -27, -26, -25, -40, -39, -38, -37, -36, -35, -34, -33, -48, -47, -46, -45, -44, -43, -42, -41, -56, -55, -54, -53, -52, -51, -50, -49

Offset: 0

Robert G. Wilson v, Aug 12 2002

Base 8 representation for n (in lexicographic order) converted from base -8 to base 10.

Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625, Dec 08, 2020

Cf. A007094, A073789, A053985, A065369, A073791, A073792, A073793, A073794, A073796 & A073835.

f[n_Integer, b_Integer] := Block[{l = IntegerDigits[n]}, Sum[l[[ -i]]*(-b)^(i - 1), {i, 1, Length[l]}]]; a = Table[ FromDigits[ IntegerDigits[n, 8]], {n, 0, 80}]; b = {}; Do[b = Append[b, f[a[[n]], 8]], {n, 1, 80}]; b

a(8*k+m) = -8*a(k)+m for 0 <= m < 8. - Chai Wah Wu, Jan 16 2020

A073796 Replace 9^k with (-9)^k in base 9 expansion of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, -9, -8, -7, -6, -5, -4, -3, -2, -1, -18, -17, -16, -15, -14, -13, -12, -11, -10, -27, -26, -25, -24, -23, -22, -21, -20, -19, -36, -35, -34, -33, -32, -31, -30, -29, -28, -45, -44, -43, -42, -41, -40, -39, -38, -37, -54, -53, -52, -51, -50, -49, -48, -47, -46

Offset: 0

Robert G. Wilson v, Aug 12 2002

Base 9 representation for n (in lexicographic order) converted from base -9 to base 10.

Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625, Dec 08, 2020

Cf. A007095, A073790, A053985, A065369, A073791, A073792, A073793, A073794, A073795 & A073835.

f[n_Integer, b_Integer] := Block[{l = IntegerDigits[n]}, Sum[l[[ -i]]*(-b)^(i - 1), {i, 1, Length[l]}]]; a = Table[ FromDigits[ IntegerDigits[n, 9]], {n, 0, 80}]; b = {}; Do[b = Append[b, f[a[[n]], 9]], {n, 1, 80}]; b

a(9*k+m) = -9*a(k)+m for 0 <= m < 9. - Chai Wah Wu, Jan 16 2020

A180352 A permutation of Motzkin numbers by reversal of indices in blocks of length 7.

Original entry on oeis.org

127, 51, 21, 9, 4, 2, 1, 113634, 41835, 15511, 5798, 2188, 835, 323, 142547559, 50852019, 18199284, 6536382, 2356779, 853467, 310572, 208023278209, 73007772802, 25669818476, 9043402501, 3192727797, 1129760415, 400763223

Offset: 1

Matt Insall (insall(AT)mst.edu), Aug 29 2010

The sequence shows A001006(7) down to A001006(1), then A001006(14) down to A001006(8), etc., varying the index in a sawtooth pattern.
When I initially logged in to use the OEIS, it 'pre-loaded' the search box with a sequence. The seven terms 127,51,21,9,4,2,1 are obtained from that original one by reversing the terms.
"Periodic" term reversals of this sort are routinely encountered when writing out convolution sums.

Cf. A001006, A073794

A180352 = lambda n: A001006(7*((n-1)//7+1)-((n-1) % 7)) # D. S. McNeil, Dec 06 2010

Let r(n) = 7, 6, 5 ,4, 3, 2, 1 (n>=1), extended with r(n+7)=7+r(n), then a(n) = A001006(r(n)).

cp's OEIS Frontend

A053985 Replace 2^k with (-2)^k in binary expansion of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A065369 Replace 3^k with (-3)^k in ternary expansion of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A073835 Replace 10^k with (-10)^k in decimal expansion of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A073791 Replace 4^k with (-4)^k in base 4 expansion of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A073792 Replace 5^k with (-5)^k in base 5 expansion of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A073793 Replace 6^k with (-6)^k in base 6 expansion of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A073795 Replace 8^k with (-8)^k in base 8 expansion of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A073796 Replace 9^k with (-9)^k in base 9 expansion of n.

Views