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 1-6 of 6 results.

A076093 Squares arising in A076991.

Original entry on oeis.org

9, 16, 25, 36, 49, 64, 81, 64, 81, 100, 121, 81, 64, 49, 64, 100, 121, 144, 81, 64, 36, 49, 64, 81, 64, 81, 100, 169, 196, 225, 196, 225, 256, 324, 289, 196, 121, 81, 100, 81, 144, 121, 144, 169, 196, 169, 196, 225, 289, 324, 289, 324, 361, 484, 441, 289
Offset: 1

Views

Author

David Garber, Nov 02 2002

Keywords

Links

Crossrefs

Cf. A076991, A077396.

A077396 Square roots of squares arising in A076991.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 9, 8, 9, 10, 11, 9, 8, 7, 8, 10, 11, 12, 9, 8, 6, 7, 8, 9, 8, 9, 10, 13, 14, 15, 14, 15, 16, 18, 17, 14, 11, 9, 10, 9, 12, 11, 12, 13, 14, 13, 14, 15, 17, 18, 17, 18, 19, 22, 21, 17, 14, 13, 12, 11, 10, 11, 13, 14, 15, 12, 13, 15, 18, 19, 20, 21, 19, 16, 13, 14, 16
Offset: 1

Views

Author

David Garber, Nov 03 2002

Keywords

Links

Crossrefs

Cf. A076093, A076991.

A158787 a(n) = position of n in A076991.

Original entry on oeis.org

1, 2, 14, 10, 27, 3, 68, 4, 21, 16, 5, 44, 23, 22, 48, 38, 6, 17, 42, 170, 7, 24, 63, 193, 13, 8, 40, 87, 25, 26, 19, 75, 53, 9, 41, 15, 61, 39, 47, 62, 20, 45, 11, 60, 33, 28, 118, 114, 29, 66, 167, 120, 12, 86, 137, 92, 175, 64, 78, 77, 84, 58, 59, 148, 90, 267, 37, 50, 82
Offset: 1

Views

Author

Zak Seidov, Mar 26 2009

Keywords

Comments

A076991 is conjectured to be a permutation of the positive integers.

Crossrefs

Cf. A076991 a(1) = 1, a(2) = 2, a(n) = smallest positive integer not occurring earlier such that the sum of three successive terms is a square.

A075989 Number of k satisfying 1<=k<=n and {n/k} >= 1/2, where {n/k} is the fractional part of n/k, i.e., {n/k} = n/k - floor(n/k).

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 2, 2, 3, 2, 5, 2, 4, 5, 6, 3, 6, 6, 7, 6, 7, 6, 11, 6, 8, 9, 10, 9, 12, 9, 10, 10, 13, 12, 15, 10, 11, 14, 17, 12, 16, 13, 16, 15, 16, 17, 20, 15, 16, 18, 19, 16, 23, 20, 21, 18, 19, 20, 25, 20, 22, 23, 26, 21, 24, 21, 24, 27, 28, 25, 28, 22, 25, 28, 29, 26, 31, 30
Offset: 1

Views

Author

Clark Kimberling, Sep 28 2002

Keywords

Examples

			For n = 5, the fractional parts of k/n are 0, 1/2, 2/3, 1/4, 0; a(5) = 2 counts 1/2 and 2/3. A075988(5) = 1 counts 1/4 and A000005(5) = 2 counts the 0's.

Links

Crossrefs

Cf. A000005, A075988, A076991.

Programs

  • Maple
    seq(nops(select(k -> frac(n/k) >= 1/2, [$1..n])), n=1..100); # Robert Israel, Sep 25 2016
  • Mathematica
    Table[Count[Range@ n, k_ /; n/k - Floor[n/k] >= 1/2], {n, 78}] (* Michael De Vlieger, Sep 25 2016 *)
  • PARI
    a(n)=n-sum(i=1,n,frac(n/i)>=1/2)
  • PARI
    a(n)=sum(k=1,n,floor(2*n/(2*k+1))-floor(2*n/(2*k+2))) \\ Benoit Cloitre, Oct 21 2012
  • PARI
    A075989(n)=sum(k=1,n,2*n\(2*k+1)-n\(k+1)) \\ M. F. Hasler, Oct 21 2012

Formula

a(n) + A075988(n) = n-d(n), where d(n) = A000005(n) is the number of divisors of n.
a(n) = Sum_{k=1..n} (floor(2*n/(2*k+1)) - floor(2*n/(2*k+2))); a(n) = (2*(log 2)-1)*n + O(n^(1/2)). Conjecture: a(n) = (2*(log 2)-1)*n + O(n^(1/4 + epsilon)) like for the divisor and Circle problems. - Benoit Cloitre, Oct 21 2012
Conjecture: Let f(a,b)=1, if (a+b) mod |a-b| != (a mod |a-b|)+(b mod |a-b|), and 0 otherwise. a(n) = Sum_{k=1..n-1} f(n,k). - Benedict W. J. Irwin, Sep 22 2016

A130205 a(n) = n^2 - a(n-1) - a(n-2), with a(1) = 1 and a(2) = 2.

Original entry on oeis.org

1, 2, 6, 8, 11, 17, 21, 26, 34, 40, 47, 57, 65, 74, 86, 96, 107, 121, 133, 146, 162, 176, 191, 209, 225, 242, 262, 280, 299, 321, 341, 362, 386, 408, 431, 457, 481, 506, 534, 560, 587, 617, 645, 674, 706, 736, 767, 801, 833, 866, 902, 936, 971, 1009, 1045, 1082
Offset: 1

Views

Author

Zak Seidov, May 16 2007

Keywords

Comments

Any three consecutive terms sum up to a perfect square. First 9 terms coincide with A076991.
Changing a(1) leaves a(5+3m) constant for m >= 0. Changing a(2) leaves a(4+3m) constant for m >= 0. - Richard R. Forberg, Jun 05 2013

Examples

			1+2+6=3^2, 2+6+8=4^2, 6+8+11=5^2.
G.f. = x + 2*x^2 + 6*x^3 + 8*x^4 + 11*x^5 + 17*x^6 + 21*x^7 + 26*x^8 + ...

Links

Crossrefs

Cf. A076991, A001840.

Programs

  • Maple
    A130205 := proc(n)
    option remember;
    if n <= 2 then
        n;
    else
        n^2-procname(n-1)-procname(n-2) ;
    end if;
end proc:
seq(A130205(n),n=1..50) ; # R. J. Mathar, Aug 06 2016
  • Mathematica
    a[1]=1;a[2]=2;a[n_]:=a[n]=n^2-a[n-1]-a[n-2]; Table[a[n],{n,100}]
a[ n_] := Quotient[ (n + 1)^2, 3] + 1 - Mod[n, 3]; (* Michael Somos, Aug 04 2016 *)
  • PARI
    a(n)=(n^2+2*n+4)\3 - n%3 \\ Charles R Greathouse IV, Aug 03 2016

Formula

a(1)=1, a(2)=2; n>2: a(n)=n^2-a(n-1)-a(n-2).
G.f.: x*(1+3*x^2-3*x^3+x^4)/(1+x+x^2)/(1-x)^3. - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009; checked and corrected by R. J. Mathar, Sep 16 2009
a(n) = floor((n^2+2*n+1)/3) + 1 - (n mod 3). - Ivan Neretin, May 25 2015
For n>6, a(n)=2*a(n-3)-a(n-6)+6. - Zak Seidov, Aug 05 2016
a(n) = (3*n^2+6*n+1 +8*A049347(n)+7*A049347(n-1))/9.. - R. J. Mathar, Aug 06 2016

A275708 a(1)=1, a(2)=2; thereafter a(n) is the smallest positive integer not yet used such that a(n)+a(n-1)-a(n-2) is a square.

Original entry on oeis.org

1, 2, 3, 8, 4, 5, 15, 6, 10, 12, 7, 9, 14, 11, 19, 17, 18, 24, 30, 43, 23, 21, 27, 58, 33, 26, 16, 35, 45, 39, 22, 42, 29, 38, 40, 34, 31, 28, 52, 25, 36, 53, 32, 37, 20, 66, 54, 13, 50, 44, 55, 70, 49, 46, 67
Offset: 1

Views

Author

Zak Seidov, Aug 06 2016

Keywords

Comments

Apparently this is a permutation of positive numbers. Out of the first 10000 terms the missing numbers are:
8974, 9298, 9342, 9380, 9386, 9425, 9429, 9454, 9495, 9497, 9525,...,
while the maximal term is a(9919)=10802.
Corresponding squares:
4, 9, 9, 1, 16, 16, 1, 16, 9, 4, 16, 16, 16, 25, 16, 25, 36, 49, 36, 1, 25, 64, 64, 1, ...

Examples

			-1+2+3=4, -2+3+8=9, -3+8+4=9.

Crossrefs

Cf. A076991.

Programs

  • Mathematica
    s={1,2};Do[a = s[[-1]] - s[[-2]]; k = 1; While[(b=k^2-a)<=0 || MemberQ[s,b], k++]; AppendTo[s, k^2 - a], {100}]; s
Showing 1-6 of 6 results.

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