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.

A018930 a(1)=3; for n>1, a(n) is smallest positive integer such that a(1)^2+...+a(n)^2 = m^2 for some m.

Original entry on oeis.org

3, 4, 12, 84, 132, 12324, 1836, 105552, 255084, 197580, 10358340, 13775220, 1936434780, 51299286012, 123205977516, 862441842612, 1310543298204, 667510076211780, 207181940072172, 110912831751840, 1698410314006284
Offset: 1

Views

Author

Charles Reed (charles.reed(AT)bbs.ewgateway.org)

Keywords

Links

Crossrefs

Cf. A018928, A018929, A127689.

Programs

  • Mathematica
    f[n_]:=Module[{a={3}}, Do[AppendTo[a,First[y/. {ToRules[Reduce[{y^2+a.a == x^2,x>0,y>0}, {y,x},Integers]]}]], {n-1}]; a]; f[21]//Timing (* Jean-François Alcover, Jan 26 2007 *)
  • PARI
    print1("3, "); s=9; for(n=1,30, d=divisors(s); t=d[#d\2]; q=(s\t-t)/2; print1(q,", "); s+=q^2); \\ Max Alekseyev, Nov 23 2012

Extensions

More terms from David W. Wilson.

A127690 a(1)=3; for n>1, a(n) is such that a(1)^2+...+a(n)^2 = (1+a(n))^2.

Original entry on oeis.org

3, 4, 12, 84, 3612, 6526884, 21300113901612, 226847426110843688722000884, 25729877366557343481074291996721923093306518970391612, 331013294649039928396936390888878360035026305412754995683702777533071737279144813617823976263475290370884
Offset: 1

Views

Author

Artur Jasinski, Jan 23 2007, Jan 29 2007

Keywords

Examples

			a(2)=4 because (3^2+4^2=5^2) and (4+1=5), a(3)=12 because (3^2+4^2+12^2=13^2) and (12+1=13) a(5)= 3612 because (3^2+4^2+12^2+84^2+3612^2=3613^2) and (3612+1=3613) etc.

Links

Crossrefs

Cf. A018930, A127689, A127691.
Apart from the initial term, the sequence is the same as A053631.

Programs

  • Mathematica
    a = {3}; For[k = 1 + a[[Length[a]]], Length[a] < 5, While[ ! ((IntegerQ[Sqrt[(k)^2 + Sum[(a[[t]])^2, {t, 1, Length[a]}]]]) && (Sqrt[(k)^2 + Sum[(a[[t]])^2, {t, 1, Length[a]}]] == k + 1)), k++ ]; AppendTo[a, k]]; a
a = {3}; For[k = 1 + a[[Length[a]]], Length[a] < 12, s2 = Plus @@ (a^2); t = Reduce[{y^2 + s2 == (y + 1)^2}, y, Integers]; t = t /. {Equal -> Rule}; k = y /. t; AppendTo[a, k]]; a (* Daniel Huber *)

Formula

For n>2, a(n) = (a(1)^2 + a(2)^2 + ... + a(n-1)^2 - 1)/2 = ((a(n-1) + 1)^2 - 1)/2. - Max Alekseyev, Nov 23 2012
a(n) = A053630(n-1)-1 for n>=2. - R. J. Mathar, Apr 23 2007

A307077 Let a(1)=3; for n > 1, let a(n) be the least positive integer k such that k > a(n-1), a(1)^2 + ... + a(n-1)^2 + k^2 is a square and the Pythagorean triple sqrt(a(1)^2 + ... + a(n-1)^2), a(n), sqrt(a(1)^2 + ... + a(n)^2) is primitive.

Original entry on oeis.org

3, 4, 12, 84, 132, 12324, 89892, 2447844, 28350372, 295742791596, 171480834409712412, 633511848768467916, 1616599508725767821225590810932, 4158520496012961741299012805876, 115366949386695884000892071516523067413910188
Offset: 1

Views

Author

Rohan Hemasingha, May 30 2019

Keywords

Comments

For n > 1, a(n) is the even value of a primitive Pythagorean triple where the larger odd value of the triple equals the smaller odd value of a primitive Pythagorean triple with even value a(n+1) (see A239381). - Torlach Rush, Jan 27 2023

Crossrefs

Cf. A018930, A127689, A239381.

Programs

  • PARI
    lista(NN) = s=9;k=3;print1(k);for(n=1,NN-1,v=divisors(s);j=#v;while(v[j]*(v[j]+2*k)>s,j--);while(gcd((s-v[j]^2)/(2*v[j]), s)!=1, j--);print1(", ", k=(s-v[j]^2)/(2*v[j]));s+=k^2); \\ Jinyuan Wang, May 31 2019

Formula

The numbers are generated by using the well-known characterization of primitive Pythagorean triples, namely (a,b,c) is a PPT iff there are positive integers j,k of opposite parity with j > k, and gcd(j,k)=1 such that a = j^2 - k^2, b = 2jk, c = j^2 + k^2.

Extensions

a(14)-a(15) from Jinyuan Wang, Jun 01 2019
Showing 1-3 of 3 results.

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