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 A309507 Number of ways the n-th triangular number T(n) = A000217(n) can be written as the difference of two positive triangular numbers. 7
 0, 1, 1, 1, 3, 3, 1, 2, 5, 3, 3, 3, 3, 7, 3, 1, 5, 5, 3, 7, 7, 3, 3, 5, 5, 7, 7, 3, 7, 7, 1, 3, 7, 7, 11, 5, 3, 7, 7, 3, 7, 7, 3, 11, 11, 3, 3, 5, 8, 11, 7, 3, 7, 15, 7, 7, 7, 3, 7, 7, 3, 11, 5, 3, 15, 7, 3, 7, 15, 7, 5, 5, 3, 11, 11, 7, 15, 7, 3, 9, 9, 3, 7 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Equivalently, a(n) is the number of triples [n,k,m] with k>0 satisfying the Diophantine equation n*(n+1) + k*(k+1) - m*(m+1) = 0. Any such triple satisfies a triangle inequality, n+k > m. The n for which there is a triple [n,n,m] are listed in A053141. - Bradley Klee, Mar 01 2020; edited by N. J. A. Sloane, Mar 31 2020 LINKS Alois P. Heinz, Table of n, a(n) for n = 1..20000 J. S. Myers, R. Schroeppel, S. R. Shannon, N. J. A. Sloane, and P. Zimmermann, Three Cousins of Recaman's Sequence, arXiv:2004:14000 [math.NT], April 2020. M. A. Nyblom, On the representation of the integers as a difference of nonconsecutive triangular numbers, Fibonacci Quarterly 39:3 (2001), pp. 256-263. FORMULA a(n) = 1 <=> n in { A068194 } \ { 1 }. a(n) is even <=> n in { A001108 } \ { 0 }. a(n) = number of odd divisors of n*(n+1) (or, equally, of T(n)) that are greater than 1. - N. J. A. Sloane, Apr 03 2020 EXAMPLE a(5) = 3: T(5) = T(6)-T(3) = T(8)-T(6) = T(15)-T(14). a(7) = 1: T(7) = T(28)-T(27). a(8) = 2: T(8) = T(13)-T(10) = T(36)-T(35). a(9) = 5: T(9) = T(10)-T(4) = T(11)-T(6) = T(16)-T(13) = T(23)-T(21) = T(45)-T(44). a(49) = 8: T(49) = T(52)-T(17) = T(61)-T(36) = T(94)-T(80) = T(127)-T(117) = T(178)-T(171) = T(247)-T(242) = T(613)-T(611) = T(1225)-T(1224). The triples with n <= 16 are: 2, 2, 3 3, 5, 6 4, 9, 10 5, 3, 6 5, 6, 8 5, 14, 15 6, 5, 8 6, 9, 11 6, 20, 21 7, 27, 28 8, 10, 13 8, 35, 36 9, 4, 10 9, 6, 11 9, 13, 16 9, 21, 23 9, 44, 45 10, 8, 13 10, 26, 28 10, 54, 55 11, 14, 18 11, 20, 23 11, 65, 66 12, 17, 21 12, 24, 27 12, 77, 78 13, 9, 16 13, 44, 46 13, 90, 91 14, 5, 15 14, 11, 18 14, 14, 20 14, 18, 23 14, 33, 36 14, 51, 53 14, 104, 105 15, 21, 26 15, 38, 41 15, 119, 120 16, 135, 136. - N. J. A. Sloane, Mar 31 2020 MATHEMATICA (* Bradley Klee, Mar 01 2020 *) TriTriples[TNn_] := Sort[Select[{TNn, (TNn + TNn^2 - # - #^2)/(2 #), (TNn + TNn^2 - # + #^2)/(2 #)} & /@ Complement[Divisors[TNn (TNn + 1)], {TNn}], And[And @@ (IntegerQ /@ #), And @@ (# > 0 & /@ #)] &]] Length[TriTriples[#]] & /@ Range[100] CROSSREFS Cf. A000217, A001108, A046079 (the same for squares), A068194, A100821 (the same for primes for n>1), A309332. See also A053141. The monotonic triples [n,k,m] with n <= k <= m are counted in A333529. Sequence in context: A010264 A262816 A089680 * A306690 A160326 A213662 Adjacent sequences: A309504 A309505 A309506 * A309508 A309509 A309510 KEYWORD nonn AUTHOR Alois P. Heinz, Aug 05 2019 STATUS approved

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Last modified September 21 19:18 EDT 2023. Contains 365503 sequences. (Running on oeis4.)