A175631 a(n) = (n-th pentagonal number) modulo (n-th triangular number).

0, 2, 0, 2, 5, 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, 1430

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

For n>=3, a(n) = A000096(n-2).

From Chai Wah Wu, Oct 12 2018: (Start)

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 5.

G.f.: x^2*(2 - 6*x + 8*x^2 - 3*x^3)/(1 - x)^3. (End)

E.g.f.: (x/2)*(2 + 3*x - (2 - x)*exp(x)). - G. C. Greubel, Jan 30 2022

Example

a(1)=0 because (1(3-1)/2) mod (1(1+1)/2) = 1 mod 1 = 0,
a(2)=2 because (2(6-1)/2) mod (2(2+1)/2) = 5 mod 3 = 2.

Mathematica

Table[Mod[n(3n-1)/2, n(n+1)/2], {n, 100}]
Module[{nn=60}, Mod[#[[1]], #[[2]]]&/@Thread[{PolygonalNumber[ 5, Range[ nn]], Accumulate[ Range[nn]]}]] (* Harvey P. Dale, Nov 19 2022 *)

Prog

(Magma) [n lt 4 select 1+(-1)^n else n*(n-3)/2: n in [1..60]]; // G. C. Greubel, Jan 30 2022
(Sage)
def A175631(n): return 1+(-1)^n if (n<4) else 9*binomial(n/3, 2)
[A175631(n) for n in (1..60)] # G. C. Greubel, Jan 30 2022

Crossrefs

Cf. A000096 (n(n+3)/2), A000217 (triangular numbers), A000326 (pentagonal numbers), A175630 (n-th pentagonal number mod (n+2)).

Cf. A080956, A132315, A132316, A132317.

Sequence in context: A024308 A059432 A256488 * A243816 A243159 A339327

Adjacent sequences: A175628 A175629 A175630 * A175632 A175633 A175634

Keyword

nonn

Author

Zak Seidov, Jul 29 2010

Status

approved