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 1-5 of 5 results.

A198409 Positions in sequences A198384, A198385 and A198386 to indicate triples of squares in arithmetic progression, that are not multiples of earlier triples.

Original entry on oeis.org

1, 3, 5, 7, 10, 13, 15, 23, 24, 26, 30, 35, 39, 42, 45, 47, 51, 54, 62, 69, 70, 72, 83, 84, 88, 97, 98, 102, 107, 114, 115, 124, 126, 129, 136, 141, 142, 143, 156, 157, 167, 169, 172, 177, 181, 188, 191, 201, 205, 208, 214, 218, 229, 230, 237, 244, 249, 253
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 25 2011

Keywords

Comments

A198435(n) = A198384(a(n)); A198439(n) = A198388(a(n));
A198436(n) = A198385(a(n)); A198440(n) = A198389(a(n));
A198437(n) = A198386(a(n)); A198441(n) = A198390(a(n));
A198438(n) = A198387(a(n)).

Links

Programs

  • Haskell
    import Data.List (elemIndices)
a198409 n = a198409_list !! (n-1)
a198409_list = map (+ 1) $ elemIndices 1 $ map a008966 $
   zipWith gcd a198384_list $ zipWith gcd a198385_list a198386_list
  • Mathematica
    wmax = 1000;
triples[w_] := Reap[Module[{u, v}, For[u = 1, u < w, u++, If[IntegerQ[v = Sqrt[(u^2 + w^2)/2]], Sow[{u^2, v^2, w^2}]]]]][[2]];
tt = Flatten[DeleteCases[triples /@ Range[wmax], {}], 2];
Position[tt, t_List /; SquareFreeQ[GCD@@t]] // Flatten (* Jean-François Alcover, Oct 24 2021 *)

Formula

A008966(A050873(A198439(n),A050873(A198440(n),A050873(A198441(n))))) = 1.

A198387 Common differences in triples of squares in arithmetic progression.

Original entry on oeis.org

24, 96, 120, 216, 240, 384, 336, 480, 600, 840, 864, 960, 840, 1176, 720, 1080, 1536, 1344, 1944, 1920, 2160, 2400, 1320, 2520, 2904, 2016, 3360, 3456, 3000, 3696, 4056, 3840, 3024, 3360, 2184, 4704, 2880, 4320, 5280, 5400, 6144, 5544, 6000, 6936, 6240, 5880
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 24 2011

Keywords

Links

Programs

  • Haskell
    a198387 n = a198387_list !! (n-1)
a198387_list = zipWith (-) a198385_list a198384_list
  • Mathematica
    wmax = 1000;
triples[w_] := Reap[Module[{u, v}, For[u = 1, u < w, u++, If[IntegerQ[v = Sqrt[(u^2 + w^2)/2]], Sow[{u^2, v^2, w^2}]]]]][[2]];
#[[2]] - #[[1]]& /@ Flatten[DeleteCases[triples /@ Range[wmax], {}] , 2] (* Jean-François Alcover, Oct 21 2021 *)

Formula

a(n) = A198385(n) - A198384(n) = A198386(n) - A198385(n).
A198438(n) = a(A198409(n)).

A198435 First term of a triple of squares in arithmetic progression, which is not a multiple of another triple in (A198384,A198385,A198386).

Original entry on oeis.org

1, 49, 49, 289, 1, 529, 961, 2401, 289, 2209, 529, 5041, 49, 1681, 1681, 6241, 9409, 49, 961, 5329, 16129, 14161, 7921, 289, 25921, 2209, 12769, 27889, 14161, 1, 39601, 2401, 5329, 10609, 25921, 49729, 58081, 529, 961, 10609, 7921, 36481, 82369, 22801, 47089
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 25 2011

Keywords

Links

Crossrefs

Cf. A198384, A198409, A198437, A198438, A198439

Programs

  • Haskell
    a198435 n = a198435_list !! (n-1)
a198435_list = map a198384 a198409_list
  • Mathematica
    wmax = 1000;
triples[w_] := Reap[Module[{u, v}, For[u = 1, u < w, u++, If[IntegerQ[v = Sqrt[(u^2 + w^2)/2]], Sow[{u^2, v^2, w^2}]]]]][[2]];
tt = Flatten[DeleteCases[triples /@ Range[wmax], {}], 2];
DeleteCases[tt, t_List /; GCD @@ t>1 && MemberQ[tt, t/GCD @@ t]][[All, 1]] (* Jean-François Alcover, Oct 20 2021 *)

Formula

a(n) = A198439(n)^2 = A198384(A198409(n));
A198436(n) - a(n) = A198437(n) - A198436(n) = A198438(n).

A198436 Second term of a triple of squares in arithmetic progression, which is not a multiple of another triple in (A198384, A198385, A198386).

Original entry on oeis.org

25, 169, 289, 625, 841, 1369, 1681, 3721, 2809, 4225, 4225, 7225, 5329, 7225, 7921, 10201, 12769, 9409, 11881, 15625, 21025, 21025, 22201, 18769, 32761, 24649, 29929, 38809, 34225, 28561, 48841, 34225, 37249, 42025, 52441, 66049, 70225, 42025, 48841, 54289
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 25 2011

Keywords

Links

Crossrefs

Cf. A198385, A198409, A198435, A198437, A198438, A198440.

Programs

  • Haskell
    a198436 n = a198436_list !! (n-1)
a198436_list = map a198385 a198409_list
  • Mathematica
    wmax = 1000;
triples[w_] := Reap[Module[{u, v}, For[u = 1, u < w, u++, If[IntegerQ[v = Sqrt[(u^2 + w^2)/2]], Sow[{u^2, v^2, w^2}]]]]][[2]];
tt = Flatten[DeleteCases[triples /@ Range[wmax], {}], 2];
DeleteCases[tt, t_List /; GCD @@ t>1 && MemberQ[tt, t/GCD @@ t]][[All, 2]] (* Jean-François Alcover, Oct 20 2021 *)

Formula

a(n) = A198440(n)^2 = A198385(A198409(n)).
a(n) - A198435(n) = A198437(n) - a(n) = A198438(n).

A198437 Third term of a triple of squares in arithmetic progression, which is not a multiple of another triple in (A198384,A198385,A198386).

Original entry on oeis.org

49, 289, 529, 961, 1681, 2209, 2401, 5041, 5329, 6241, 7921, 9409, 10609, 12769, 14161, 14161, 16129, 18769, 22801, 25921, 25921, 27889, 36481, 37249, 39601, 47089, 47089, 49729, 54289, 57121, 58081, 66049, 69169, 73441, 78961, 82369, 82369, 83521, 96721
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 25 2011

Keywords

Links

Crossrefs

Cf. A198435, A198436, A198438, A198409, A198441.

Programs

  • Haskell
    a198437 n = a198437_list !! (n-1)
a198437_list = map a198386 a198409_list
  • Mathematica
    wmax = 1000;
triples[w_] := Reap[Module[{u, v}, For[u = 1, u < w, u++, If[IntegerQ[v = Sqrt[(u^2 + w^2)/2]], Sow[{u^2, v^2, w^2}]]]]][[2]];
tt = Flatten[DeleteCases[triples /@ Range[wmax], {}], 2];
DeleteCases[tt, t_List /; GCD @@ t>1 && MemberQ[tt, t/GCD @@ t]][[All, 3]] (* Jean-François Alcover, Oct 20 2021 *)

Formula

a(n) = A198441(n)^2 = A198386(A198409(n));
a(n) - A198436(n) = A198436(n) - A198435(n) = A198438(n).
Showing 1-5 of 5 results.

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