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.

Showing 1-3 of 3 results.

A079000 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is odd".

Original entry on oeis.org

1, 4, 6, 7, 8, 9, 11, 13, 15, 16, 17, 18, 19, 20, 21, 23, 25, 27, 29, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 95, 97
Offset: 1

Views

Author

Matthew Vandermast, Feb 01 2003

Keywords

Comments

a(a(n)) = 2n + 3 for n>1.

Examples

			a(2) cannot be 2 because 2 is even; it cannot be 3 because that would require 2 to be a member of the sequence. Hence a(2)=4 and the next odd member of the sequence is the fourth member.

References

  • Hsien-Kuei Hwang, S Janson, TH Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint, 2016; http://140.109.74.92/hk/wp-content/files/2016/12/aat-hhrr-1.pdf. Also Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585
  • N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.

Links

Crossrefs

Cf. A079250-A079259, A079313, A079325, A064437, A003605, A079352, A079358.
Cf. also A080596, A080731, A080752, A121053, A080032.
Partial sums give A080566. Differences give A079948.

Programs

  • Maple
    Digits := 50; A079000 := proc(n) local k,j; if n<=2 then n^2; else k := floor(evalf(log( (n+3)/6 )/log(2)) ); j := n-(9*2^k-3); 12*2^k-3+3*j/2 +abs(j)/2; fi; end;
A002264 := n->floor(n/3): A079944 := n->floor(log[2](4*(n+2)/3))-floor(log[2](n+2)): A000523 := n->floor(log[2](n)): f := n->A079944(A002264(n-4)): g := n->A000523(A002264(n+2)/2): A079000 := proc(n) if n>3 then RETURN(simplify(3*n+3-3*2^g(n)+(-1)^f(n)*(9*2^g(n)-n-3))/2) else if n>0 then RETURN([1,4,6][n]) else RETURN(0) fi fi: end;
  • Mathematica
    a[1] = 1; a[n_] := (k = Floor[Log[2, (n+3)/6]]; j = n-(9*2^k - 3); 12*2^k-3 + 3*j/2 + Abs[j]/2); Table[a[n], {n, 1, 71}] (* Jean-François Alcover, May 21 2012, after Maple *)

Formula

a(1) = 1, a(2) = 4, then a(9*2^k-3+j) = 12*2^k-3+3*j/2+|j|/2 for k>=0, -3*2^k <= j <= 3*2^k. Also a(3n) = 3*b(n/3), a(3n+1) = 2*b(n)+b(n+1), a(3n+2) = b(n)+2*b(n+1) for n>=2, where b = A079905. - N. J. A. Sloane and Benoit Cloitre, Feb 20 2003
a(n+1) - 2*a(n) + a(n-1) = 1 for n = 9*2^k - 3, k>=0, = -1 for n = 2 and 3*2^k-3, k>=1 and = 0 otherwise.
a(n) = (3*n + 3 - 3*2^g(n) + (-1)^f(n)*(9*2^g(n) - n - 3))/2 for n>3, f(n) = A079944(A002264(n-4)) and g(n) = A000523(A002264(n+2)/2). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 23 2003
Also a(n) = n + 3*2^A000523(A002264(n+2)/2)*(1 - 3*A080584(n-4)) + A080584(n-4)*(n+3) for n>3, where A080584(n)=A079944(A002264(n)). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 24 2003

A079313 a(n) is taken to be the smallest positive integer not already present which is consistent with the condition "n is a member of the sequence if and only if a(n) is odd".

Original entry on oeis.org

1, 3, 5, 2, 7, 8, 9, 11, 13, 12, 15, 17, 19, 16, 21, 23, 25, 20, 27, 29, 31, 24, 33, 35, 37, 28, 39, 41, 43, 32, 45, 47, 49, 36, 51, 53, 55, 40, 57, 59, 61, 44, 63, 65, 67, 48, 69, 71, 73, 52, 75, 77, 79, 56, 81, 83, 85, 60, 87, 89, 91, 64, 93, 95, 97, 68, 99, 101, 103, 72, 105
Offset: 1

Views

Author

J. C. Lagarias and N. J. A. Sloane, Feb 11 2003

Keywords

Comments

The sequence obeys the rule: "The concatenation of a(n) and a(a(n)) is odd". Example: "1" and the 1st term, concatenated, is 11; "3" and the 3rd term, concatenated, is 35; "5" and the 5th term, concatenated, is 57; "2" and the 2nd term, concatenated, is 23; etc.

Links

Crossrefs

Equals A080032 + 1. Cf. A079000, A079250-A079259, A080029-A080031.

Programs

  • Mathematica
    Rest@ CoefficientList[Series[x*(-3*x^11 + 2*x^10 - x^9 + 7*x^7 - x^6 + 2*x^5 + 5*x^4 + 2*x^3 + 5*x^2 + 3*x + 1)/(x^8 - 2*x^4 + 1), {x, 0, 120}], x] (* Michael De Vlieger, Dec 17 2024 *)

Formula

For n >= 5 a(n) is given by: a(4t-2) = 4t, a(4t-1) = 6t-3, a(4t) = 6t-1, a(4t+1) = 6t+1.
All odd numbers occur; the only even numbers which occur are 2 and the multiples of 4 excluding 4 itself.
From Chai Wah Wu, Apr 13 2024: (Start)
a(n) = 2*a(n-4) - a(n-8) for n > 12.
G.f.: x*(-3*x^11 + 2*x^10 - x^9 + 7*x^7 - x^6 + 2*x^5 + 5*x^4 + 2*x^3 + 5*x^2 + 3*x + 1)/(x^8 - 2*x^4 + 1). (End)

Extensions

More terms from Matthew Vandermast, Mar 20 2003

A079252 Even numbers not in A079000.

Original entry on oeis.org

2, 10, 12, 14, 22, 24, 26, 28, 30, 32, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132, 134, 136, 138, 140, 190, 192, 194, 196, 198, 200, 202, 204, 206
Offset: 1

Views

Author

N. J. A. Sloane, Feb 04 2003

Keywords

Links

Programs

  • Mathematica
    nmax = 150;
b[1] = 1; b[n_] := (k = Floor[Log[2, (n + 3)/6]]; j = n - (9 2^k - 3); 12 2^k - 3 + 3 j/2 + Abs[j]/2);
Complement[Range[2, b[nmax], 2], Table[b[n], {n, 1, nmax}]] (* Jean-François Alcover, Feb 13 2019 *)

Formula

See A079250 for formula.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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