cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-15 of 15 results.

A231551 Position of n in A231550.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 5, 8, 15, 14, 9, 12, 11, 10, 13, 16, 31, 30, 17, 28, 19, 18, 29, 24, 23, 22, 25, 20, 27, 26, 21, 32, 63, 62, 33, 60, 35, 34, 61, 56, 39, 38, 57, 36, 59, 58, 37, 48, 47, 46, 49, 44, 51, 50, 45, 40, 55, 54, 41, 52, 43, 42, 53, 64, 127, 126, 65
Offset: 0

Views

Author

Alex Ratushnyak, Nov 10 2013

Keywords

Comments

This permutation transforms the enumeration system of positive irreducible fractions A002487/A002487' (Calkin-Wilf) into the enumeration system A020651/A020650, and A162911/A162912 (Drib) the enumeration system into A245327/A245326. - Yosu Yurramendi, Jun 16 2015

Links

Crossrefs

Cf. A003188, A006068, A231550.

Programs

  • Mathematica
    Join[{0, 1}, Table[d = Reverse@IntegerDigits[n, 2]; FromDigits[Reverse@Append[FoldList[BitXor, d[[1]], Most@Rest@d], d[[-1]]], 2], {n, 2, 67}]] (* Ivan Neretin, Dec 28 2016 *)
  • Python
    for n in range(99):
  bits = [0]*64
  orig = [0]*64
  l = int.bit_length(int(n))
  t = n
  for i in range(l):
    bits[i] = orig[i] = t&1
    t>>=1
  #for i in range(1, l-1):  bits[i] ^= orig[i-1]   # A231550
  for i in range(1, l-1):  bits[i] ^= bits[i-1]   # A231551
  #for i in range(l-1):  bits[i] ^= orig[i+1]      # A003188
  #for i in range(1, l):  bits[l-1-i] ^= bits[l-i]  # A006068
  t = 0
  for i in range(l):  t += bits[i]<
  • R
    maxrow <- 8 # by choice
b01 <- 0 # b01 is going to be A010059
a <- 1
for(m in 0:maxrow) for(k in 0:(2^m-1)){
   b01[2^(m+1)+    k] <-     b01[2^m+k]
     a[2^(m+1)+    k] <-       a[2^m+k]  + 2^(m+b01[2^(m+1)+    k])
   b01[2^(m+1)+2^m+k] <- 1 - b01[2^m+k]
     a[2^(m+1)+2^m+k] <-       a[2^m+k]  + 2^(m+b01[2^(m+1)+2^m+k])
}
(a <- c(0,a))
# Yosu Yurramendi, Apr 10 2017
  • R
    maxblock <- 8 # by choice
a <- 1:3
for(n in 4:2^maxblock){
ones <- which(as.integer(intToBits(n)) == 1)
nbit <- as.integer(intToBits(n))[1:tail(ones, n = 1)]
anbit <- nbit
for(i in 2:(length(anbit) - 1))
   anbit[i] <- bitwXor(anbit[i], anbit[i-1])  # ?bitwXor
a <- c(a, sum(anbit*2^(0:(length(anbit) - 1))))
}
(a <- c(0,a))
# Yosu Yurramendi, Apr 25 2021

Formula

A231550(a(n)) = a(A231550(n)) = n.
a(n) = A258996(A284460(n)) = A284459(A092569(n)), n > 0. - Yosu Yurramendi, Apr 10 2017
a(n) = A054429(A153154(n)), n > 0. - Yosu Yurramendi, Oct 04 2021

A245328 Denominators in recursive bijection from positive integers to positive rationals (the bijection is f(1) = 1, f(2n) = 1/(f(n)+1), f(2n+1) = f(n)+1).

Original entry on oeis.org

1, 2, 1, 3, 2, 3, 1, 5, 3, 5, 2, 4, 3, 4, 1, 8, 5, 8, 3, 7, 5, 7, 2, 7, 4, 7, 3, 5, 4, 5, 1, 13, 8, 13, 5, 11, 8, 11, 3, 12, 7, 12, 5, 9, 7, 9, 2, 11, 7, 11, 4, 10, 7, 10, 3, 9, 5, 9, 4, 6, 5, 6, 1, 21, 13, 21, 8, 18, 13, 18, 5, 19, 11, 19, 8, 14, 11, 14, 3, 19, 12, 19, 7, 17, 12, 17, 5, 16, 9, 16, 7, 11, 9, 11, 2, 18, 11, 18, 7, 15
Offset: 1

Views

Author

Yosu Yurramendi, Jul 18 2014

Keywords

Comments

A245327(n)/a(n) enumerates all the reduced nonnegative rational numbers exactly once.
If the terms (n>0) are written as an array (left-aligned fashion) with rows of length 2^m, m = 0,1,2,3,...
1,
2,1,
3,2, 3,1,
5,3, 5,2, 4,3, 4,1,
8,5, 8,3, 7,5, 7,2, 7,4, 7,3,5,4,5,1,
13,8,13,5,11,8,11,3,12,7,12,5,9,7,9,2,11,7,11,4,10,7,10,3,9,5,9,4,6,5,6,1,
then the sum of the m-th row is 3^m (m = 0,1,2,), and each column k is a Fibonacci sequence.
If the rows are written in a right-aligned fashion:
1,
2,1,
3,2,3,1,
5,3,5,2,4,3,4,1,
8,5, 8,3, 7,5, 7,2,7,4,7,3,5,4,5,1,
13,8,13,5,11,8,11,3,12,7,12,5,9,7,9,2,11,7,11,4,10,7,10,3,9,5,9,4,6,5,6,1,
then each column is an arithmetic sequence. The differences of the arithmetic sequences, except the first on the right, give the sequence A093873 (Numerators in Kepler's tree of harmonic fractions) (a(2^(m+1)-1-k) - a(2^m-1-k) = A093873(k), m = 0,1,2,..., k = 0,1,2,...,2^m-1).
If the sequence is considered by blocks of length 2^m, m = 0,1,2,..., the blocks of this sequence are permutations of terms of blocks from A002487 (Stern's diatomic series or Stern-Brocot sequence), and, more precisely, the reverses of blocks of A020651 ( a(2^m+k) = A020651(2^(m+1)-1-k), m = 0,1,2,..., k = 0,1,2,...,2^m-1).
Moreover, each block is the bit-reversed permutation of the corresponding block of A245326.

Links

Crossrefs

Cf. A002487, A020651, A093873, A245326, A245327, A273493.

Programs

  • Mathematica
    f[n_] := Which[n == 1, 1, EvenQ@ n, 1/(f[n/2] + 1), True, f[(n - 1)/2] + 1]; Table[Denominator@ f@ k, {n, 7}, {k, 2^(n - 1), 2^n - 1}] // Flatten (* Michael De Vlieger, Mar 02 2017 *)
  • PARI
    a(n) = my(A=0); forstep(i=logint(n, 2), 0, -1, if(bittest(n, i), A++, A=1/(A+1))); denominator(A) \\ Mikhail Kurkov, Mar 12 2023
  • R
    N <- 25 # arbitrary
a <- c(1,2,1)
for(n in 1:N){
  a[4*n]   <- a[2*n] + a[2*n+1]
  a[4*n+1] <- a[2*n]
  a[4*n+2] <- a[2*n] + a[2*n+1]
  a[4*n+3] <-          a[2*n+1]
}
a

Formula

a(2n) = A245327(2n+1) , a(2n+1) = A245328(2n) , n=1,2,3,...
a((2*n+1)*2^m - 1) = A273493(n), n > 0, m >= 0. For n = 0 A273493(0) = 1 is needed. - Yosu Yurramendi, Mar 02 2017
a(n) = A002487(1+A284459(n)). - Yosu Yurramendi, Aug 23 2021

A052913 a(n+2) = 5*a(n+1) - 2*a(n), with a(0) = 1, a(1) = 4.

Original entry on oeis.org

1, 4, 18, 82, 374, 1706, 7782, 35498, 161926, 738634, 3369318, 15369322, 70107974, 319801226, 1458790182, 6654348458, 30354161926, 138462112714, 631602239718, 2881086973162, 13142230386374, 59948977985546, 273460429154982, 1247404189803818, 5690100090709126
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Comments

Main diagonal of the array: m(1,j)=3^(j-1), m(i,1)=1; m(i,j) = m(i-1,j) + m(i,j-1): 1 3 9 27 81 ... / 1 4 13 40 ... / 1 5 18 58 ... / 1 6 24 82 ... - Benoit Cloitre, Aug 05 2002
a(n) is also the number of 3 X n matrices of integers for which the upper-left hand corner is a 1, the rows and columns are weakly increasing, and two adjacent entries differ by at most 1. - Richard Stanley, Jun 06 2010
a(n) is the number of compositions of n when there are 4 types of 1 and 2 types of other natural numbers. - Milan Janjic, Aug 13 2010
If a Stern's sequence based enumeration system of positive irreducible fractions is considered (for example, A007305/A047679, or A162909/A162910, or A071766/A229742, or A245325/A245326, ...), and if it is organized by blocks or levels (n) with 2^n terms (n>=0), and the products numerator*denominator, term by term, are summed at each level n, then the resulting sequence of integers is a(n). - Yosu Yurramendi, May 23 2015
Number of 1’s in the substitution system {0 -> 110, 1 -> 11110} at step n from initial string "1" (1 -> 11110 -> 11110111101111011110110 -> ...) . - Ilya Gutkovskiy, Apr 10 2017

Links

Crossrefs

Cf. A007482 (inverse binomial transform).

Programs

  • GAP
    a:=[1,4];; for n in [3..30] do a[n]:=5*a[n-1]-2*a[n-2]; od; a; # G. C. Greubel, Oct 16 2019
  • Magma
    I:=[1,4]; [n le 2 select I[n] else 5*Self(n-1)-2*Self(n-2): n in [1..35]]; // Vincenzo Librandi, May 24 2015
  • Magma
    R:=PowerSeriesRing(Integers(), 25); Coefficients(R!((1-x)/(1-5*x+2*x^2))); // Marius A. Burtea, Oct 16 2019
  • Maple
    spec := [S,{S=Sequence(Union(Prod(Sequence(Z),Union(Z,Z)),Z,Z))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
seq(coeff(series((1-x)/(1-5*x+2*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 16 2019
  • Mathematica
    Transpose[NestList[{Last[#],5Last[#]-2First[#]}&, {1,4},20]][[1]] (* Harvey P. Dale, Mar 12 2011 *)
LinearRecurrence[{5, -2}, {1, 4}, 25] (* Jean-François Alcover, Jan 08 2019 *)
  • PARI
    Vec((1-x)/(1-5*x+2*x^2) + O(x^30)) \\ Michel Marcus, Mar 05 2015
  • Sage
    def A052913_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-x)/(1-5*x+2*x^2)).list()
A052913_list(30) # G. C. Greubel, Oct 16 2019

Formula

G.f.: (1-x)/(1-5*x+2*x^2).
a(n) = Sum_{alpha=RootOf(1 - 5*z + 2*z^2)} (1/17)*(3+alpha)*alpha^(-1-n).
a(n) = ((17+3*sqrt(17))/34)*((5+sqrt(17))/2)^n + ((17-3*sqrt(17))/34)*((5-sqrt(17))/2)^n. - N. J. A. Sloane, Jun 03 2002
a(n) = A107839(n) - A107839(n-1). - R. J. Mathar, May 21 2015
a(n) = 2*A020698(n-1), n>1. - R. J. Mathar, Nov 23 2015
E.g.f.: (1/17)*exp(5*x/2)*(17*cosh(sqrt(17)*x/2) + 3*sqrt(17)*sinh(sqrt(17)*x/2)). - Stefano Spezia, Oct 16 2019
a(n) = 3*A107839(n-1) + (-1)^n*A152594(n) with A107839(-1) = 0. - Klaus Purath, Jul 29 2020

Extensions

Typo in definition corrected by Bruno Berselli, Jun 07 2010

A180201 Inverse permutation to A180200.

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 6, 7, 11, 10, 8, 9, 13, 12, 14, 15, 23, 22, 20, 21, 17, 16, 18, 19, 27, 26, 24, 25, 29, 28, 30, 31, 47, 46, 44, 45, 41, 40, 42, 43, 35, 34, 32, 33, 37, 36, 38, 39, 55, 54, 52, 53, 49, 48, 50, 51, 59, 58, 56, 57, 61, 60, 62, 63, 95, 94, 92, 93, 89, 88, 90, 91, 83
Offset: 0

Views

Author

Reinhard Zumkeller, Aug 15 2010

Keywords

Comments

A180199(n) = a(a(n));
a(A180198(n)) = A180198(a(n)) = A180200(n);
a(A075427(n)) = A075427(n).
This permutation transforms the enumeration system of positive irreducible fractions A245325/A245326 into the enumeration system A007305/A047679 (Stern-Brocot), and enumeration system A071766/A229742 (HCS) into A162909/A162910 (Bird). - Yosu Yurramendi, Jun 09 2015

Links

Programs

  • R
    #
maxn <- 63 # by choice
a <- 1
for(n in 1:maxn){
a[2*n  ] <- 2*a[n] + (n%%2 == 0)
a[2*n+1] <- 2*a[n] + (n%%2 != 0)}
a <- c(0, a)
# Yosu Yurramendi, May 23 2020

Formula

a(n) = A233280(A258746(n)) = A117120(A233280(n)), n > 0. - Yosu Yurramendi, Apr 10 2017 [Corrected by Yosu Yurramendi, Mar 14 2025]
a(0) = 0, a(1) = 1, for n > 0 a(2*n) = 2*a(n) + [n even], a(2*n + 1) = 2*a(n) + [n odd]. - Yosu Yurramendi, May 23 2020
From Alan Michael Gómez Calderón, Mar 04 2025: (Start)
a(n) = A054429(n) XOR floor(n/2) for n > 0.
a(n) = A054429(A003188(n)) for n > 0. (End)
a(n) = A154436(A054429(n)), n > 0. - Yosu Yurramendi, Mar 11 2025

A231550 Permutation of nonnegative integers: for each bit[i] in the binary representation, except the most and the least significant bits, set bit[i] = bit[i] XOR bit[i-1], where bit[i-1] is the less significant bit, XOR is the binary logical exclusive or operator.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 5, 8, 11, 14, 13, 12, 15, 10, 9, 16, 19, 22, 21, 28, 31, 26, 25, 24, 27, 30, 29, 20, 23, 18, 17, 32, 35, 38, 37, 44, 47, 42, 41, 56, 59, 62, 61, 52, 55, 50, 49, 48, 51, 54, 53, 60, 63, 58, 57, 40, 43, 46, 45, 36, 39, 34, 33, 64, 67, 70, 69, 76
Offset: 0

Views

Author

Alex Ratushnyak, Nov 10 2013

Keywords

Comments

This permutation transforms the enumeration system of positive irreducible fractions A020651/A020650 into the enumeration system A002487/A002487' (Calkin-Wilf), and enumeration system A245327/A245326 into A162911/A162912 (Drib). - Yosu Yurramendi, Jun 16 2015

Links

Crossrefs

Cf. A003188, A006068, A231551.

Programs

  • Mathematica
    Join[{0, 1}, Table[d = IntegerDigits[n, 2]; FromDigits[Join[{d[[1]]}, BitXor[Most@Rest@d, Rest@Rest@d], {d[[-1]]}], 2], {n, 2, 68}]] (* Ivan Neretin, Dec 28 2016 *)
  • PARI
    a(n) = bitxor(n, if(n>3, bitand(n<<1, bitneg(0,logint(n,2))))); \\ Kevin Ryde, Jul 17 2021
  • Python
    for n in range(99):
  bits = [0]*64
  orig = [0]*64
  l = int.bit_length(int(n))
  t = n
  for i in range(l):
    bits[i] = orig[i] = t&1
    t>>=1
  for i in range(1, l-1):  bits[i] ^= orig[i-1]   # A231550
  #for i in range(1, l-1):  bits[i] ^= bits[i-1]   # A231551
  #for i in range(l-1):  bits[i] ^= orig[i+1]      # A003188
  #for i in range(1,l):  bits[l-1-i] ^= bits[l-i]  # A006068
  t = 0
  for i in range(l):  t += bits[i]<
  • R
    a <- 1
maxlevel <- 8 # by choice
#
for(m in 0:maxlevel) for(k in 0:(2^m-1)){
  a[2^(m+1)  +2*k] <- 2*a[2^m+k]
  a[2^(m+2)-1-2*k] <- 2*a[2^m+k] + 1
}
(a <- c(0,a))
# Yosu Yurramendi, Apr 10 2017

Formula

a(A231551(n)) = A231551(a(n)) = n.
a(n) = A284460(A258996(n)) = A092569(A284460(n)), n > 0. - Yosu Yurramendi, Apr 10 2017
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