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 1-10 of 10 results.

A198385 Second of a triple of squares in arithmetic progression.

Original entry on oeis.org

25, 100, 169, 225, 289, 400, 625, 676, 625, 841, 900, 1156, 1369, 1225, 1681, 1521, 1600, 2500, 2025, 2704, 2601, 2500, 3721, 2809, 3025, 4225, 3364, 3600, 4225, 4225, 4225, 4624, 5625, 5476, 7225, 4900, 6724, 6084, 5329, 5625, 6400, 7225, 7225, 7225, 7921
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 24 2011

Keywords

Links

Programs

  • Haskell
    a198385 n = a198385_list !! (n-1)
a198385_list = map (^ 2) a198389_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]];
Flatten[DeleteCases[triples /@ Range[wmax], {}], 2][[All, 2]] (* Jean-François Alcover, Oct 19 2021 *)

Formula

a(n) = A198389(n)^2.
a(n) - A198384(n) = A198386(n) - a(n) = A198387(n).
A198436(n) = a(A198409(n)).

A198384 First of a triple of squares in arithmetic progression.

Original entry on oeis.org

1, 4, 49, 9, 49, 16, 289, 196, 25, 1, 36, 196, 529, 49, 961, 441, 64, 1156, 81, 784, 441, 100, 2401, 289, 121, 2209, 4, 144, 1225, 529, 169, 784, 2601, 2116, 5041, 196, 3844, 1764, 49, 225, 256, 1681, 1225, 289, 1681, 2401, 6241, 9, 4624, 324, 9409, 361
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 24 2011

Keywords

Links

Programs

  • Haskell
    a198384 n = a198384_list !! (n-1)
a198384_list = map (^ 2) a198388_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]];
Flatten[DeleteCases[triples /@ Range[wmax], {}], 2][[All, 1]] (* Jean-François Alcover, Oct 19 2021 *)

Formula

a(n) = A198388(n)^2.
A198385(n) - a(n) = A198386(n) - A198385(n) = A198387(n).
A198435(n) = a(A198409(n)).

Extensions

Thanks to Benoit Jubin, who had the idea for sequences A198384 .. A198390 and A198435 .. A198441.

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.

A198386 Third of a triple of squares in arithmetic progression.

Original entry on oeis.org

49, 196, 289, 441, 529, 784, 961, 1156, 1225, 1681, 1764, 2116, 2209, 2401, 2401, 2601, 3136, 3844, 3969, 4624, 4761, 4900, 5041, 5329, 5929, 6241, 6724, 7056, 7225, 7921, 8281, 8464, 8649, 8836, 9409, 9604, 9604, 10404, 10609, 11025, 12544, 12769, 13225
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 24 2011

Keywords

Links

Programs

  • Haskell
    a198386 n = a198386_list !! (n-1)
a198386_list = map (^ 2) a198390_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]];
Flatten[DeleteCases[triples /@ Range[wmax], {}], 2][[All, 3]] (* Jean-François Alcover, Oct 19 2021 *)

Formula

a(n) = A198390(n)^2.
a(n) - A198385(n) = A198385(n) - A198384(n) = A198387(n).
A198437(n) = a(A198409(n)).

A198438 Common differences in triples of squares in arithmetic progression, that are not a multiples of other triples in (A198384, A198385, A198386).

Original entry on oeis.org

24, 120, 240, 336, 840, 840, 720, 1320, 2520, 2016, 3696, 2184, 5280, 5544, 6240, 3960, 3360, 9360, 10920, 10296, 4896, 6864, 14280, 18480, 6840, 22440, 17160, 10920, 20064, 28560, 9240, 31824, 31920, 31416, 26520, 16320, 12144, 41496, 47880, 43680, 50160
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 25 2011

Keywords

Links

Programs

  • Haskell
    a198438 n = a198438_list !! (n-1)
a198438_list = map a198387 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];
#[[2]] - #[[1]]& /@ DeleteCases[tt, t_List /; GCD@@t > 1 && MemberQ[tt, t/GCD@@t]] (* Jean-François Alcover, Oct 22 2021 *)

Formula

a(n) = A198387(A198409(n)) = A198436(n) - A198435(n) = A198437(n) - A198436(n).

A222154 Numbers n such that there are three distinct triples (k, k+n, k+2n) of squares.

Original entry on oeis.org

3360, 13440, 30240, 43680, 53760, 84000, 120960, 127680, 164640, 174720, 215040, 272160, 336000, 393120, 406560, 483840, 510720, 567840, 658560, 665280, 698880, 756000, 860160, 971040, 1088640, 1092000, 1145760, 1149120, 1212960, 1344000, 1367520, 1481760
Offset: 1

Views

Author

Zdenek Cervenka, Feb 10 2013

Keywords

Examples

			The following 3 triples have a common difference of 3360: (2^2, 58^2, 82^2), (46^2, 74^2, 94^2), and (97^2, 113^2, 127^2).

Links

Crossrefs

Cf. A198387, A222155.

A222155 Numbers n such that there are four distinct triples (k, k+n, k+2n) of squares.

Original entry on oeis.org

1367520, 5470080, 12307680, 21880320, 32672640, 34188000, 49230720, 58544640, 67008480, 71831760, 73513440, 87521280, 110769120, 130690560, 136752000, 165469920, 196922880, 209969760, 221205600, 231110880, 234178560, 268033920, 287327040, 294053760, 307692000
Offset: 1

Views

Author

Zdenek Cervenka, Feb 10 2013

Keywords

Examples

			The following 4 triples have a common difference of 1367520: (802^2, 1418^2, 1838^2), (2162^2, 2458^2, 2722^2), (2729^2, 2969^2, 3191^2), and (6049^2, 6161^2, 6271^2).

Crossrefs

Cf. A198387, A222154.

A214155 Numbers n such that there are five distinct triples (k, k+n, k+2n) of squares.

Original entry on oeis.org

287327040, 294053760, 1149308160, 2585943360, 4597232640, 7183176000, 10343773440, 14079024960, 18388930560, 23273490240, 28732704000, 34766571840, 41375093760, 48558269760, 64648584000, 73555722240, 83037514560, 93093960960, 103725061440
Offset: 1

Views

Author

Zdenek Cervenka, Feb 16 2013

Keywords

Examples

			The following 5 triples all have a common difference of 287327040:
(4342^2, 17498^2, 24362^2), (12454^2, 21034^2, 27014^2), (23266^2, 28786^2, 33406^2), (51778^2, 54482^2, 57058^2), and (57073^2, 59537^2, 61903^2).

Crossrefs

Cf. A198387, A222154, A222155.

A226858 Numbers n such that there are six distinct triples (k, k+n, k+2n) of squares.

Original entry on oeis.org

258594336000, 1034377344000, 2327349024000, 4137509376000, 6464858400000, 9309396096000, 12671122464000, 16550037504000, 20946141216000, 25859433600000
Offset: 1

Views

Author

Zdenek Cervenka, Jun 20 2013

Keywords

Comments

For the first 10 terms we have a(n) = n^2 * a(1). Are there any other primitive terms other than a(1)?

Examples

			These 6 triples have a common difference of 9309396096000: (579774^2, 3105726^2, 4353726^2), (781560^2, 3149640^2, 4385160^2), (2241720^2, 3786120^2, 4862520^2), (4187880^2, 5181480^2, 6013080^2), (9320040^2, 9806760^2, 10270440^2), and (10273140^2, 10716660^2, 11142540^2).

Crossrefs

Cf. A198387, A222154, A222155, A214155, A226954.

A226954 Numbers n such that there are seven distinct triples (k, k+n, k+2n) of squares.

Original entry on oeis.org

12671122464000, 50684489856000, 114040102176000, 202737959424000, 316778061600000, 456160408704000, 620885000736000, 810951837696000, 1026360919584000, 1267112246400000, 1533205818144000
Offset: 1

Views

Author

Zdenek Cervenka, Jun 26 2013

Keywords

Comments

For the first 11 terms we have a(n) = n^2 * a(1). Are there any other primitive terms other than a(1)?

Examples

			These 7 triples of squares have a common difference of 12671122464000: (676403^2, 3623347^2, 5079347^2), (911820^2, 3674580^2, 5116020^2), (2615340^2, 4417140^2, 5672940^2), (4885860^2, 6045060^2, 7015260^2), (5664815^2, 6690385^2, 7578415^2), (10873380^2, 11441220^2, 11982180^2) and (11985330^2, 12502770^2, 12999630^2).

Crossrefs

Cf. A198387, A222154, A222155, A214155, A226858.
Showing 1-10 of 10 results.

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