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.

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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)).

A198390 Square root of third term of a triple of squares in arithmetic progression.

Original entry on oeis.org

7, 14, 17, 21, 23, 28, 31, 34, 35, 41, 42, 46, 47, 49, 49, 51, 56, 62, 63, 68, 69, 70, 71, 73, 77, 79, 82, 84, 85, 89, 91, 92, 93, 94, 97, 98, 98, 102, 103, 105, 112, 113, 115, 119, 119, 119, 119, 123, 124, 126, 127, 133, 136, 137, 138, 140, 141, 142, 146
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 24 2011

Keywords

Comments

There is a connection to the leg sums of Pythagorean triangles.
See a comment on the primitive case under A198439, which applies mutatis mutandis. - Wolfdieter Lang, May 23 2013
Are these just the positive multiples of A001132? - Charles R Greathouse IV, May 28 2013
n appears A331671(n) times. - Ray Chandler, Feb 26 2020

Examples

			Connection to leg sums of Pythagorean triangles: a(2) = 14 because (in the notation of the Zumkeller link) (u,v,w)= (2,10,14) = 2*(1,5,7), and this corresponds to the non-primitive Pythagorean triangle 2*(x=(7-1)/1,y=(1+7)/2,z=5) = 2*(3,4,5) with leg sum 2*(3+4) = 14. - _Wolfdieter Lang_, May 23 2013

Links

Crossrefs

Cf. A198386, A198409, A198439, A198441, A331671.

Programs

  • Haskell
    a198390 n = a198390_list !! (n-1)
a198390_list = map (\(,,x) -> x) ts where
   ts = [(u,v,w) | w <- [1..], v <- [1..w-1], u <- [1..v-1],
                   w^2 - v^2 == v^2 - u^2]
  • 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, v, w}]]]]][[2]];
Flatten[DeleteCases[triples /@ Range[wmax], {}], 2][[All, 3]] (* Jean-François Alcover, Oct 20 2021 *)
  • PARI
    is(n)=my(t=n^2);forstep(i=2-n%2,n-2,2, if(issquare((t+i^2)/2), return(1))); 0 \\ Charles R Greathouse IV, May 28 2013

Formula

A198386(n) = a(n)^2.
A198441(n) = a(A198409(n)).

A198388 Square root of first term of a triple of squares in arithmetic progression.

Original entry on oeis.org

1, 2, 7, 3, 7, 4, 17, 14, 5, 1, 6, 14, 23, 7, 31, 21, 8, 34, 9, 28, 21, 10, 49, 17, 11, 47, 2, 12, 35, 23, 13, 28, 51, 46, 71, 14, 62, 42, 7, 15, 16, 41, 35, 17, 41, 49, 79, 3, 68, 18, 97, 19, 56, 7, 42, 20, 69, 98, 34, 21, 93, 31, 63, 22, 85, 94, 23, 49, 73
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 24 2011

Keywords

Comments

There is a connection to |x-y| of Pythagorean triangles (x,y,z). See a comment on the primitive Pythagorean triangle case under A198441 which applies mutatis mutandis. - Wolfdieter Lang, May 23 2013

Examples

			Connection to Pythagorean triangles: a(2) = 2 because (in the notation of the Zumkeller link) (u,v,w) = 2*(1,5,7) and the corresponding Pythagorean triangle is 2*((7-1)/2,(1+7)/2,5) = 2*(3,4,5) with |x-y| = 2*(4-3) = 2. - _Wolfdieter Lang_, May 23 2013

Links

Crossrefs

Cf. A198384, A198409, A198439.

Programs

  • Haskell
    a198388 n = a198388_list !! (n-1)
a198388_list = map (\(x,,) -> x) ts where
   ts = [(u,v,w) | w <- [1..], v <- [1..w-1], u <- [1..v-1],
                   w^2 - v^2 == v^2 - u^2]
  • 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, v, w}]]]]][[2]];
Flatten[DeleteCases[triples /@ Range[wmax], {}], 2][[All, 1]] (* Jean-François Alcover, Oct 20 2021 *)

Formula

A198384(n) = a(n)^2.
A198439(n) = a(A198409(n)).

A198389 Square root of second term of a triple of squares in arithmetic progression.

Original entry on oeis.org

5, 10, 13, 15, 17, 20, 25, 26, 25, 29, 30, 34, 37, 35, 41, 39, 40, 50, 45, 52, 51, 50, 61, 53, 55, 65, 58, 60, 65, 65, 65, 68, 75, 74, 85, 70, 82, 78, 73, 75, 80, 85, 85, 85, 89, 91, 101, 87, 100, 90, 113, 95, 104, 97, 102, 100, 111, 122, 106, 105, 123, 109
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 24 2011

Keywords

Comments

Apart from its initial 1, A001653 is a subsequence: for all n>1 exists an m such that A198388(m)=1 and a(m)=A001653(n). [observed by Zak Seidov, Reinhard Zumkeller, Oct 25 2011]
There is a connection to hypotenuses of Pythagorean triangles. See a comment for the primitive case on A198441 which applies here mutatis mutandis. - Wolfdieter Lang, May 23 2013

Examples

			Connection to Pythagorean triangle hypotenuses: a(20) = 10 because (in the notation of the Zumkeller link) (u,v,w) = 2*(1,5,7) and the Pythagorean triangle is 2*(x=(7-1)/2,y=(1+7)/2,5) = 2*(3,4,5) with hypotenuse 2*5 = 10. - _Wolfdieter Lang_, May 23 2013

Links

Crossrefs

Cf. A198385, A198409, A198440.

Programs

  • Haskell
    a198389 n = a198389_list !! (n-1)
a198389_list = map (\(,x,) -> x) ts where
   ts = [(u,v,w) | w <- [1..], v <- [1..w-1], u <- [1..v-1],
                   w^2 - v^2 == v^2 - u^2]
  • 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, v, w}]]]]][[2]];
Flatten[DeleteCases[triples /@ Range[wmax], {}], 2][[All, 2]] (* Jean-François Alcover, Oct 20 2021 *)

Formula

A198385(n) = a(n)^2.
A198440(n) = a(A198409(n)).
Previous Showing 11-14 of 14 results.

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