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.

A185223 A185128(n) is the a(n)-th triangular number.

Original entry on oeis.org

6, 18, 37, 44, 86, 91, 116, 132, 247, 278, 392, 613, 637, 662, 798, 847, 912, 1164, 1235, 1362, 1430, 1638, 1735, 1991, 2056, 2090, 2167, 2364, 2537, 2736, 3139, 3478, 3751, 3867, 4298, 4422, 4553, 5202, 6068, 6391, 6500, 7241, 7859, 7957, 8378, 9309, 9793
Offset: 1

Views

Author

Martin Renner, Jan 20 2012

Keywords

Comments

Side lengths where both triangular numbers are the same (A053141) are not included. - R. J. Mathar, Feb 11 2018
See A185128 for further information.

Examples

			A185128(2) = 171 which is the 18th triangular number, so a(2) = 18.

References

  • Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966, p. 197, no. 8.

Links

Crossrefs

Cf. A000217, A185128, A185129, A185233, A185243, A185253, A185257, A185258.

Programs

  • PARI
    lista(nn) = {v = vector(nn, n, n*(n+1)/2); for (n=2, nn, for (k=1, n-1, if (ispolygonal(v[n]+v[k], 3) && ispolygonal(v[n]-v[k], 3), print1(n, ", "));););} \\ Michel Marcus, Jan 08 2015

Extensions

Edited (with a simpler definition) by N. J. A. Sloane, Dec 28 2024

A185253 a(n) = A185128(n) - A185129(n).

Original entry on oeis.org

6, 66, 325, 210, 1596, 2701, 1326, 903, 1225, 16836, 6903, 82621, 141778, 181503, 63546, 52975, 354903, 10440, 13530, 405450, 7140, 989121, 1329265, 511566, 668746, 437580, 2102275, 2001000, 2469753, 3229611, 1428895, 3096316, 1963171, 6843150, 856086, 4276350
Offset: 1

Views

Author

Martin Renner, Jan 20 2012

Keywords

References

  • Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966, p. 197, no. 8.

Links

  • N. J. A. Sloane, Annotated scan of Beiler's Table 81, based on page 197 of Beiler's "Recreations in the Theory of Numbers: The Queen of Mathematics Entertains", New York, Dover, First ed., 1964.

Crossrefs

Cf. A000217, A185128, A185129, A185223, A185233, A185243, A185257, A185258.

Programs

  • Mathematica
    kmax=2000; TriangularQ[n_]:=IntegerQ[(Sqrt[1+8n]-1)/2]; A000217[n_]:=n(n+1)/2; a={}; For[k=1, k<=kmax, k++, For[h=1, A000217[h]<A000217[k], h++, If[TriangularQ[d=A000217[k] - A000217[h]] && TriangularQ[A000217[k]+A000217[h]], AppendTo[a,d]]]]; a (* Stefano Spezia, Sep 02 2024 *)
  • PARI
    lista(nn) = {v = vector(nn, n, n*(n+1)/2); for (n=2, nn, for (k=1, n-1, if (ispolygonal(v[n]+v[k], 3) && ispolygonal(v[n]-v[k], 3), print1(v[n]-v[k], ", "));););} \\ Michel Marcus, Jan 08 2015

Extensions

Edited by N. J. A. Sloane, Dec 28 2024 (replaced definition with simpler and more explicit formula from Michel Marcus, Jan 08 2015)

A185243 a(n) = A185128(n) + A185129(n).

Original entry on oeis.org

36, 276, 1081, 1770, 5886, 5671, 12246, 16653, 60031, 60726, 147153, 293761, 264628, 257403, 574056, 665281, 477753, 1345620, 1512930, 1450956, 2039190, 1695561, 1682695, 3454506, 3560446, 3932610, 2595781, 3589860, 3969153, 4258821, 8427565, 9003646, 12110581
Offset: 1

Views

Author

Martin Renner, Jan 20 2012

Keywords

Comments

See A185128 for further information.

References

  • Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966, p. 197, no. 8.

Links

  • N. J. A. Sloane, Annotated scan of Beiler's Table 81, based on page 197 of Beiler's "Recreations in the Theory of Numbers: The Queen of Mathematics Entertains", New York, Dover, First ed., 1964.

Crossrefs

Cf. A000217, A185128, A185129, A185223, A185233, A185253, A185257, A185258.

Programs

  • PARI
    lista(nn) = {v = vector(nn, n, n*(n+1)/2); for (n=2, nn, for (k=1, n-1, if (ispolygonal(v[n]+v[k], 3) && ispolygonal(v[n]-v[k], 3), print1(v[n]+v[k], ", "));););} \\ Michel Marcus, Jan 08 2015

Extensions

Edited by N. J. A. Sloane, Dec 28 2024 (replaced definition with simpler and more explicit formula from Michel Marcus, Jan 08 2015)

A185129 a(n) = smaller member of n-th pair of distinct, positive, triangular numbers whose sum and difference are also triangular numbers.

Original entry on oeis.org

15, 105, 378, 780, 2145, 1485, 5460, 7875, 29403, 21945, 70125, 105570, 61425, 37950, 255255, 306153, 61425, 667590, 749700, 522753, 1016025, 353220, 176715, 1471470, 1445850, 1747515, 246753, 794430, 749700, 514605, 3499335, 2953665, 5073705, 635628, 8382465
Offset: 1

Views

Author

Martin Renner, Jan 20 2012

Keywords

Comments

See A185128 for further information.

Examples

			a(2) = 105, corresponding to the second pair of triangular numbers 171 = 18*(18+1)/2 and 105 = 14*(14+1)/2, which produce the sum 276 = 23*(23+1)/2 and the difference 66 = 11*(11+1)/2, both of which are triangular numbers.

References

  • Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966, p. 197, no. 8.

Links

  • N. J. A. Sloane, Annotated scan of Beiler's Table 81, based on page 197 of Beiler's "Recreations in the Theory of Numbers: The Queen of Mathematics Entertains", New York, Dover, First ed., 1964.

Crossrefs

Cf. A000217, A185128, A185223, A185233, A185243, A185253, A185257, A185258.

Programs

  • Mathematica
    kmax=2000; TriangularQ[n_]:=IntegerQ[(Sqrt[1+8n]-1)/2]; A000217[n_]:=n(n+1)/2; a={}; For[k=1, k<=kmax, k++, For[h=1, A000217[h]<A000217[k], h++, If[TriangularQ[A000217[k] - A000217[h]] && TriangularQ[A000217[k]+A000217[h]], AppendTo[a, A000217[h]]]]]; a (* Stefano Spezia, Sep 02 2024 *)
  • PARI
    lista(n) = {v = vector(nn, n, n*(n+1)/2); for (n=2, nn, for (k=1, n-1, if (ispolygonal(v[n]+v[k], 3) && ispolygonal(v[n]-v[k], 3), print1(v[k], ", "));););} \\ Michel Marcus, Jan 08 2015

Extensions

Edited by N. J. A. Sloane, Dec 28 2024

A185233 A185129(n) is the a(n)-th triangular number.

Original entry on oeis.org

5, 14, 27, 39, 65, 54, 104, 125, 242, 209, 374, 459, 350, 275, 714, 782, 350, 1155, 1224, 1022, 1425, 840, 594, 1715, 1700, 1869, 702, 1260, 1224, 1014, 2645, 2430, 3185, 1127, 4094, 3317, 1274, 5124, 6060, 3834, 3059, 6174, 5565, 7749, 8349, 7395, 7344
Offset: 1

Views

Author

Martin Renner, Jan 20 2012

Keywords

Comments

See A185128 for further information.

Examples

			A185122(2) = 105 which is the 14th triangular number, so a(2) = 14.

References

  • Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966, p. 197, no. 8.

Links

  • N. J. A. Sloane, Annotated scan of Beiler's Table 81, based on page 197 of Beiler's "Recreations in the Theory of Numbers: The Queen of Mathematics Entertains", New York, Dover, First ed., 1964.

Crossrefs

Cf. A000217, A185128, A185129, A185223, A185243, A185253, A185257, A185258.

Programs

  • PARI
    lista(nn) = {v = vector(nn, n, n*(n+1)/2); for (n=2, nn, for (k=1, n-1, if (ispolygonal(v[n]+v[k], 3) && ispolygonal(v[n]-v[k], 3), print1(k, ", "));););} \\ Michel Marcus, Jan 08 2015

Extensions

Edited (with a simpller definition) by N. J. A. Sloane, Dec 28 2024

A185257 A185243(n) is the a(n)-th triangular number.

Original entry on oeis.org

8, 23, 46, 59, 108, 106, 156, 182, 346, 348, 542, 766, 727, 717, 1071, 1153, 977, 1640, 1739, 1703, 2019, 1841, 1834, 2628, 2668, 2804, 2278, 2679, 2817, 2918, 4105, 4243, 4921, 4028, 5936, 5528, 4728, 7302, 8576, 7453, 7184, 9516, 9630, 11107, 11828, 11889, 12241
Offset: 1

Views

Author

Martin Renner, Jan 20 2012

Keywords

Comments

See A185128 for further information.

References

  • Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966, p. 197, no. 8.

Links

  • N. J. A. Sloane, Annotated scan of Beiler's Table 81, based on page 197 of Beiler's "Recreations in the Theory of Numbers: The Queen of Mathematics Entertains", New York, Dover, First ed., 1964.

Crossrefs

Cf. A000217, A185128, A185129, A185223, A185233, A185243, A185253, A185258.

Extensions

Edited (with a simpler definition) by N. J. A. Sloane, Dec 28 2024

A185258 A185253(n) is the a(n)-th triangular number.

Original entry on oeis.org

3, 11, 25, 20, 56, 73, 51, 42, 49, 183, 117, 406, 532, 602, 356, 325, 842, 144, 164, 900, 119, 1406, 1630, 1011, 1156, 935, 2050, 2000, 2222, 2541, 1690, 2488, 1981, 3699, 1308, 2924, 4371, 897, 311, 5113, 5735, 3783, 5549, 1807, 696, 5654, 6478
Offset: 1

Views

Author

Martin Renner, Jan 20 2012

Keywords

Comments

See A185128 for further information.

References

  • Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966, p. 197, no. 8.

Links

  • N. J. A. Sloane, Annotated scan of Beiler's Table 81, based on page 197 of Beiler's "Recreations in the Theory of Numbers: The Queen of Mathematics Entertains", New York, Dover, First ed., 1964.

Crossrefs

Cf. A000217, A185128, A185129, A185223, A185233, A185243, A185253, A185257.

Extensions

Edited (with a simpler definition) by N. J. A. Sloane, Dec 28 2024
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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