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.

A053631 Pythagorean spiral: a(n-1)+1, a(n) and a(n)+1 are the sides of a right triangle (a primitive Pythagorean triangle).

Original entry on oeis.org

2, 4, 12, 84, 3612, 6526884, 21300113901612, 226847426110843688722000884, 25729877366557343481074291996721923093306518970391612, 331013294649039928396936390888878360035026305412754995683702777533071737279144813617823976263475290370884
Offset: 1

Views

Author

Henry Bottomley, Mar 21 2000

Keywords

Comments

To derive a list of Pythagorean triples from this sequence, we note that the difference between the second and the third terms in the Pythagorean triple is 1 and that the last term of the previous triple gives us the first term in the next triple. Therefore the sequence is completely determined by the initial triple.
A053631 gives us a list of Pythagorean triples beginning with (3,4,5), since a(1)=2. Using any initial value h>1, (2h-1,2h^2-2h,2h^2-2h+1) forms a Pythagorean triple; we can use b(1)=2h-1 and the recursive formula b(n)=b(n-1)^2-b(n-1)+1 for n>1, we can create infinitely many of spirals of this type. - Haoqi Chen, Teena Carroll

Examples

			For n=3, a(n-1) = 4, so we want a right triangle with sides 4 + 1 = 5, a(n), and a(n)+1.  Solving (x+1)^2 = x^2 + 5^2 gives x = 12, so a(3) = 12. - _Michael B. Porter_, Jul 19 2016

Links

Crossrefs

Apart from the initial term, the sequence is the same as A127690.
Cf. A046092

Programs

  • Maple
    a[1]:= 2:
for n from 2 to 10 do a[n]:= a[n-1] + a[n-1]^2/2 od:
seq(a[i],i=1..10); # Robert Israel, Jul 08 2015
  • Mathematica
    NestList[# + #^2/2 &, 2, 9] (* Robert G. Wilson v, Dec 12 2012 *)
  • Maxima
    a[1]:2$
a[n]:=a[n-1] + (a[n-1]^2)/2$
A053631(n):=a[n]$
makelist(A053631(n),n,1,10); /* Martin Ettl, Nov 08 2012 */
  • PARI
    main(size)={v=vector(size); v[1]=2;for(n=2,size,v[n]=v[n-1]+v[n-1]^2/2);return(v)} /* Anders Hellström, Jul 08 2015 */

Formula

a(1)=2; for n >= 2: a(n) = a(n-1) + a(n-1)^2/2 = A046092(a(n-1)/2).
a(n) = A053630(n) - 1. - Robert G. Wilson v, Jul 29 2014
a(n) = 2*A007018(n-1). - Ivan Neretin, Jul 26 2015

Extensions

Corrected and extended by James Sellers, Mar 22 2000
a(1) = 2 added by Zak Seidov, Apr 10 2007

A127689 a(1)=3; for n>1, a(n) is least number such that a(n) > a(n-1) and a(1)^2+...+a(n)^2 is a square.

Original entry on oeis.org

3, 4, 12, 84, 132, 12324, 15960, 26280, 27300, 66660, 115188, 9777193284, 23465263884, 48701491080, 40900397690640, 680008604512020, 127049882801497788, 247290967245178188, 335580091290976716, 1045885075937364972, 1607091702050097396, 3419793695168900508, 5138020847719969956, 10059508412964112740
Offset: 1

Views

Author

Artur Jasinski, Jan 23 2007

Keywords

Comments

Without the a(n) > a(n-1) constraint, the sequence would be A018930.

Examples

			a(2)=4 because 3^2+4^2=5^2, a(3)=12 because 3^2+4^2+12^2=13^2 etc.

Crossrefs

Cf. A018930, A127690, A127691.

Programs

  • Mathematica
    a = {3}; For[k = 1 + a[[Length[a]]], Length[a] < 11, While[ ! IntegerQ[Sqrt[(k)^2 + Sum[(a[[t]])^2, {t, 1, Length[a]}]]], k++ ]; AppendTo[a, k]]; a
  • PARI
    q=3; s=9; for(n=1,30, b=0; fordiv(s,d, if(d*d>=s,break); t=(s\d-d)/2; if(t>q,b=t); ); q=b; print1(q,", "); s+=q^2); \\ Max Alekseyev, Nov 23 2012

Extensions

More terms from Max Alekseyev, Nov 23 2012

A118017 Largest denominators in even Egyptian fraction representation of 1.

Original entry on oeis.org

12, 84, 3612, 6526884, 21300113901612
Offset: 1

Views

Author

Teena Carroll, Jul 06 2011

Keywords

Comments

An Egyptian Fraction representation of a rational number a/b is a list of distinct unit fractions with sum a/b. We will call it an even Egyptian Fraction representation if only even integers are used as denominators. The n-th term of this sequence gives the largest denominators that appear in an Egyptian Fraction Representation of one with length n+3. For instance, 1/2 + 1/4 + 1/6 + 1/12 gives a 4-term even representation of one, which is the shortest possible Egyptian even fraction representation of one.
This sequence can be derived from the Sylvester sequence (A000058). If s(n) represents the Sylvester sequence, s(n)-1 is the largest denominator appearing in an n term Egyptian fraction representation of 1. There is a one-to-one correspondence between k-term representations and (k+1)-term even representation for k<12. An even representation has to have at least 4 terms, thus a(1) is related to s(3). a(1) = 2*(s(3) - 1), etc.
Excluding the first two terms of the Pythagorean spiral sequence (A053631) yields this sequence.

Examples

			a(2) = 1/2*12^2 + 12 so a(2) = 84.

References

  • Mohammad K. Azarian, Sylvester's Sequence and the Infinite Egyptian Fraction Decomposition of 1, Problem 958, College Mathematics Journal, Vol. 42, No. 4, September 2011, p. 330. Solution published in Vol. 43, No. 4, September 2012, pp. 340-342

Crossrefs

Cf. A000058, A007018, A053631.
A053631 and A127690 are very similar to this sequence.

Formula

a(n+1) = 1/2*a(n)^2 + a(n).
Showing 1-3 of 3 results.

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