

A184218


a(n) = largest k such that A000217(n+1) = A000217(n) + (A000217(n) mod k), or 0 if no such k exists.


2



0, 0, 0, 0, 9, 14, 20, 27, 35, 44, 54, 65, 77, 90, 104, 119, 135, 152, 170, 189, 209, 230, 252, 275, 299, 324, 350, 377, 405, 434, 464, 495, 527, 560, 594, 629, 665, 702, 740, 779, 819, 860, 902, 945, 989, 1034, 1080, 1127, 1175, 1224, 1274, 1325, 1377
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OFFSET

1,5


COMMENTS

From the definition, a(n) = A000217(n)  (n + 1) if A000217(n)  (n + 1) > (n + 1), or 0 otherwise, where A000217 are the triangular numbers.


LINKS

G. C. Greubel and Vincenzo Librandi, Table of n, a(n) for n = 1..10000 [Originally computed by Remi Eismann]
Index entries for linear recurrences with constant coefficients, signature (3,3,1).


FORMULA

a(n) = (n+1)*(n2)/2 = A000096(n2) for n >= 5 and a(n) = 0 for n <= 4.  M. F. Hasler, Jan 10 2011
From Chai Wah Wu, Jun 21 2016: (Start)
a(n) = 3*a(n1)  3*a(n2) + a(n3) for n > 7.
G.f.: x^5*(5*x^2  13*x + 9)/(1  x)^3. (End)


EXAMPLE

For n = 3 we have A000217(3) = 6, A000217(4) = 10; there is no k such that 10  6 = 4 = (6 mod k), hence a(3) = 0.
For n = 5 we have A000217(5) = 15, A000217(6) = 21; 9 is the largest k such that 21  15 = 6 = (15 mod k), hence a(5) = 9; a(5) = A000217(5)  (5 + 1) = 15  6 = 9.
For n = 24 we have A000217(24) = 300, A000217(25) = 325; 275 is the largest k such that 325  300 = 25 = (300 mod k), hence a(24) = 275; a(24) = A000217(24)  (24 + 1) = 275.


MATHEMATICA

Join[{0, 0, 0, 0}, LinearRecurrence[{3, 3, 1}, {9, 14, 20}, 100]] (* G. C. Greubel, Jun 22 2016 *)
lim = 10^4; Table[SelectFirst[Reverse@ Range@ lim, Function[k, PolygonalNumber[n + 1] == # + Mod[#, k] &@ PolygonalNumber@ n]], {n, 53}] /. {k_ /; MissingQ@ k > 0, k_ /; k == lim > 0} (* Michael De Vlieger, Jun 30 2016, Version 10.4 *)


PROG

(Magma) [0, 0, 0, 0] cat [(n+1)*(n2)/2: n in [5..60]]; // Vincenzo Librandi, Jun 22 2016


CROSSREFS

Cf. essentially the same as A000096, A000217, A000027, A130703, A184219, A118534, A117078, A117563, A001223.
Sequence in context: A332588 A034703 A006624 * A272527 A272308 A186778
Adjacent sequences: A184215 A184216 A184217 * A184219 A184220 A184221


KEYWORD

nonn,easy


AUTHOR

Rémi Eismann, Jan 10 2011


STATUS

approved



