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

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

1,1

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)

References

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

Links

Table of n, a(n) for n=1..34.

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.

Formula

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

Example

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.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A125358 A126516 A241219 * A007261 A119105 A015880

Adjacent sequences: A185125 A185126 A185127 * A185129 A185130 A185131

Keyword

nonn

Author

Martin Renner, Jan 20 2012

Extensions

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

Status

approved