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.

A185128 a(n) = larger member of n-th pair of distinct, positive, triangular numbers whose sum and difference are also triangular numbers.

Original entry on oeis.org

21, 171, 703, 990, 3741, 4186, 6786, 8778, 30628, 38781, 77028, 188191, 203203, 219453, 318801, 359128, 416328, 678030, 763230, 928203, 1023165, 1342341, 1505980, 1983036, 2114596, 2185095, 2349028, 2795430, 3219453, 3744216, 4928230, 6049981, 7036876, 7478778
Offset: 1

Views

Author

Martin Renner, Jan 20 2012

Keywords

Comments

Comments from N. J. A. Sloane, Dec 28 2024 (Start):
Beiler's Table 81 (see link) lists eight sequences based on finding pairs of distinct, positive, triangular numbers whose sum and difference are also triangular numbers. The sequences are A185128 (the present sequence), A185129, A185223, A185233, A185243, A185253, A185257, and A185258.
The order of the pairs is the same in each sequence, and is determined by the terms of the present sequence.
It would be nice to have an analytic solution to the corresponding Diophantine equations. Beiler does not discuss this, but it is probably in Volume 2 of Dickson's "History of the Theory of Numbers".
(End)

Examples

			a(2) = 171, since the pair of triangular numbers 171 = 18*(18+1)/2 and 105 = 14*(14+1)/2 produce the sum 276 = 23*(23+1)/2 and the difference 66 = 11*(11+1)/2 which are both 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, A185129, A185223, A185233, A185243, A185253, A185257, A185258.

Programs

  • Mathematica
    Module[{trs=Accumulate[Range[3900]]},Union[Select[Sort/@Subsets[trs,{2}],AllTrue[{Sqrt[ 8Total[#]+ 1],Sqrt[8Abs[#[[1]]-#[[2]]]+1]},OddQ]&]]][[All,2]]//Sort (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 02 2018 *)
  • 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], ", "));););} \\ Michel Marcus, Jan 08 2015

Formula

a(n) = A000217(A185223(n)). - R. J. Mathar, Feb 11 2018. (This is a restatement of the definition.)

Extensions

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

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)

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

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)

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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