A185253 -id:A185253 - cp's OEIS frontend

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

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

Martin Renner, Jan 20 2012

nonn

See A185128 for further information.

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

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.

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

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

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

Martin Renner, Jan 20 2012

nonn

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)

			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.

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

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.

Cf. A000217, A185129, A185223, A185233, A185243, A185253, A185257, A185258.

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 *)

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

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

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

Martin Renner, Jan 20 2012

nonn

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

See A185128 for further information.

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

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

R. J. Mathar, Table of n, a(n) for n = 1..58
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.

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

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

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

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

Martin Renner, Jan 20 2012

nonn

See A185128 for further information.

			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.

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

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.

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

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 *)

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

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

Martin Renner, Jan 20 2012

nonn

See A185128 for further information.

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

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

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.

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

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

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

Martin Renner, Jan 20 2012

nonn

See A185128 for further information.

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

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.

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

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

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

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

Martin Renner, Jan 20 2012

nonn

See A185128 for further information.

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

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.

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

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

A129965 Triangular numbers that are the difference of nonnegative cubes.

Original entry on oeis.org

0, 1, 91, 4095, 5886, 7875, 8128, 8911, 9045, 17955, 21736, 23653, 47278, 93961, 115921, 130816, 184528, 259560, 379756, 488566, 575128, 658378, 758296, 810901, 873181, 885115, 1060696, 1155960, 1358776, 1385280, 1997001, 2616328, 2685403

Offset: 1

Peter Pein (petsie(AT)dordos.net), Jun 13 2007, Jun 14 2007

nonn

			A000217(13) = Sum_{k=1..13} k = 91 = 216 - 125 = 6^3 - 5^3, so 91 is in the sequence. - _Peter Munn_, Dec 05 2022

Robert Israel, Table of n, a(n) for n = 1..439 (terms <= 10^12)

Intersection of A000217 and A152043.

Cf. A038596, A129966, A185253.

M:= 10^7: # for terms <= M
S:= {0}:
for x from 1 while 3*x^2 - 3*x + 1 < M do
   if x^3 < M then Y:= 0 else Y:= ceil(x^3-M) fi;
  S:= S union select(t -> issqr(1+8*t),{seq(x^3 - y^3, y = Y .. x-1)});
od:
sort(convert(S,list)); # Robert Israel, Dec 05 2023

With[{n = 5000}, Intersection[(#1*((#1 + 1)/2) & ) /@ Range[0, n], Flatten[Outer[ #1^3 - #2^3 &, Range[n], Range[0, n - 1]]]]]

cp's OEIS Frontend

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

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

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

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

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

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

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

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