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 1-7 of 7 results.

A086909 Middle side of the first primitive d-arithmetic triangle, where d=A072330(n).

Original entry on oeis.org

4, 26, 28, 52, 76, 98, 124, 134, 158, 148, 172, 206, 218, 266, 244, 316, 292, 362, 388, 388, 364, 364, 386, 398, 518, 556, 494, 532, 556, 508, 532, 602, 602, 628, 724, 676, 758, 746, 734, 854, 916, 806, 868, 916, 844, 892, 866, 868, 1036, 1022, 988, 964, 974
Offset: 1

Views

Author

Lekraj Beedassy, Sep 19 2003

Keywords

References

  • J. A. MacDougall, "Heron Triangles With Sides In Arithmetic Progression", Journal of Recreational Mathematics 31(3) 2002-2003, pp. 192-194.

Links

Crossrefs

Cf. A072330, A072360, A089019, A089020, A096672, A096673, A096674.

Programs

  • Mathematica
    terms = 1000;
nmax = 12 terms;
okQ[n_] := AllTrue[FactorInteger[n][[All, 1]], MatchQ[Mod[#, 12], 1|11]&];
A072330 = Select[Range[nmax], okQ];
a[n_] := Module[{a, b, c, d, p}, d = If[n <= Length[A072330], A072330[[n]], Print["nmax = ", nmax, " insufficient"]; Exit[]]; If[n==1, 4, For[b = 2d, True, b++, a = b-d; c = b+d; p = (a+b+c)/2; If[IntegerQ[p] && IntegerQ[ Sqrt[p(p-a)(p-b)(p-c)]] && GCD[a, b, c] == 1, Return[b]]]]];
a /@ Range[terms] (* Jean-François Alcover, Mar 06 2020 *)

Extensions

Extended by Ray Chandler, Jul 03 2004

A072360 One-sixth the area of the smallest primitive d-arithmetic triangle, where d=A072330(n).

Original entry on oeis.org

1, 26, 21, 91, 95, 196, 341, 536, 790, 259, 559, 1030, 654, 2926, 549, 4029, 1241, 4706, 5529, 5335, 1729, 1001, 1544, 2786, 9324, 12649, 4446, 8645, 9591, 1651, 3059, 10234, 3010, 3925, 19005, 2535, 16676, 14174, 8074, 25620, 33205, 8060
Offset: 1

Views

Author

Lekraj Beedassy, Jul 18 2002

Keywords

Comments

Such a triangle has middle side 2*x'.

Links

Crossrefs

Cf. A072330, A086909, A089019, A089020, A096672, A096673, A096674.

Programs

  • Mathematica
    terms = 1000;
nmax = 12 terms;
okQ[n_] := AllTrue[FactorInteger[n][[All, 1]], MatchQ[Mod[#, 12], 1|11]&];
A072330 = Select[Range[nmax], okQ];
a[n_] := Module[{a, b, c, d, p, area}, d = If[n <= Length[A072330], A072330[[n]], Print["nmax = ", nmax, " insufficient"]; Exit[]]; If[n == 1, 1, For[b = 2 d, True, b++, a = b - d; c = b + d; p = (a + b + c)/2; If[IntegerQ[p] && IntegerQ[area = Sqrt[p (p - a) (p - b) (p - c)]] && GCD[a, b, c] == 1, Return[area/6]]]]];
a /@ Range[terms] (* Jean-François Alcover, Mar 07 2020 *)

Formula

a(n) = x'*y'/2, where (x', y') is the fundamental solution to x^2 - 3*y^2 = d^2, where d=A072330(n).

Extensions

Edited, corrected and extended by Ray Chandler, Jul 03 2004

A089019 First nontrivial x satisfying x^2 - 3*y^2 = d^2, where d=A072330(n).

Original entry on oeis.org

2, 13, 14, 26, 38, 49, 62, 67, 79, 74, 86, 103, 109, 133, 122, 158, 146, 181, 194, 194, 182, 182, 193, 199, 259, 278, 247, 266, 278, 254, 266, 301, 301, 314, 362, 338, 379, 373, 367, 427, 458, 403, 434, 458, 422, 446, 433, 434, 518, 511, 494, 482, 487, 523
Offset: 1

Views

Author

Lekraj Beedassy, Nov 04 2003

Keywords

Links

Crossrefs

For corresponding y see A089020.
Cf. A072330, A072360, A086909, A089020, A096672, A096673, A096674.

Programs

Formula

a(n) = A086909(n)/2.

Extensions

Extended by Ray Chandler, Jul 03 2004

A089020 First value y satisfying x^2 - 3*y^2 = d^2, where d=A072330(n).

Original entry on oeis.org

1, 4, 3, 7, 5, 8, 11, 16, 20, 7, 13, 20, 12, 44, 9, 51, 17, 52, 57, 55, 19, 11, 16, 28, 72, 91, 36, 65, 69, 13, 23, 68, 20, 25, 105, 15, 88, 76, 44, 120, 145, 40, 87, 119, 29, 85, 24, 17, 155, 132, 93, 31, 44, 104, 52, 92, 95, 19, 140, 200, 28, 105, 231, 35, 100, 185, 105, 120
Offset: 1

Views

Author

Lekraj Beedassy, Nov 04 2003

Keywords

Links

Crossrefs

For corresponding x see A089019.
Cf. A072330, A072360, A086909, A089019, A096672, A096673, A096674.

Programs

  • Mathematica
    terms = 1000;
nmax = 12 terms;
okQ[n_] := AllTrue[FactorInteger[n][[All, 1]], MatchQ[Mod[#, 12], 1|11]&];
A072330 = Select[Range[nmax], okQ];
a[n_] := Module[{a, b, c, d, p}, d = If[n <= Length[A072330], A072330[[n]], Print["nmax = ", nmax, " insufficient"]; Exit[]]; If[n == 1, 1, For[b = 2 d, True, b++, a = b - d; c = b + d; p = (a + b + c)/2; If[IntegerQ[p] && IntegerQ[Sqrt[p (p - a) (p - b) (p - c)]] && GCD[a, b, c] == 1, Return[ Sqrt[b^2 - 4 d^2]/(2 Sqrt[3])]]]]];
a /@ Range[terms] (* Jean-François Alcover, Mar 07 2020 *)

Extensions

Extended by Ray Chandler, Jul 03 2004

A096672 Smallest side of the first primitive d-arithmetic triangle, where d=A072330(n).

Original entry on oeis.org

3, 15, 15, 29, 39, 51, 65, 73, 87, 75, 89, 109, 111, 157, 123, 185, 149, 205, 221, 219, 185, 183, 195, 205, 291, 327, 255, 291, 305, 255, 269, 325, 303, 317, 411, 339, 411, 397, 375, 481, 533, 409, 461, 507, 425, 471, 435, 435, 593, 565, 521, 485, 493, 555
Offset: 1

Views

Author

Ray Chandler, Jul 03 2004

Keywords

Crossrefs

Cf. A072330, A072360, A086909, A089019, A089020, A096673, A096674.

A096673 Largest side of the first primitive d-arithmetic triangle, where d=A072330(n).

Original entry on oeis.org

5, 37, 41, 75, 113, 145, 183, 195, 229, 221, 255, 303, 325, 375, 365, 447, 435, 519, 555, 557, 543, 545, 577, 591, 745, 785, 733, 773, 807, 761, 795, 879, 901, 939, 1037, 1013, 1105, 1095, 1093, 1227, 1299, 1203, 1275, 1325, 1263, 1313, 1297, 1301, 1479, 1479
Offset: 1

Views

Author

Ray Chandler, Jul 03 2004

Keywords

Crossrefs

Cf. A072330, A072360, A086909, A089019, A089020, A096672, A096674.

A096674 Semiperimeter of the first primitive d-arithmetic triangle, where d=A072330(n).

Original entry on oeis.org

6, 39, 42, 78, 114, 147, 186, 201, 237, 222, 258, 309, 327, 399, 366, 474, 438, 543, 582, 582, 546, 546, 579, 597, 777, 834, 741, 798, 834, 762, 798, 903, 903, 942, 1086, 1014, 1137, 1119, 1101, 1281, 1374, 1209, 1302, 1374, 1266, 1338, 1299, 1302, 1554, 1533
Offset: 1

Views

Author

Ray Chandler, Jul 03 2004

Keywords

Crossrefs

Cf. A072330, A072360, A086909, A089019, A089020, A096672, A096673.
Showing 1-7 of 7 results.

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