A022856 a(n) = n-2 + Sum_{i = 1..n-2} (a(i+1) mod a(i)) for n >= 3 with a(1) = a(2) = 1.

1, 1, 1, 2, 3, 5, 8, 12, 17, 23, 30, 38, 47, 57, 68, 80, 93, 107, 122, 138, 155, 173, 192, 212, 233, 255, 278, 302, 327, 353, 380, 408, 437, 467, 498, 530, 563, 597, 632, 668, 705, 743, 782, 822, 863, 905, 948, 992, 1037, 1083, 1130, 1178, 1227

Offset

1,4

Comments

Essentially triangular numbers + 2, but with three extra initial terms.

Links

G. C. Greubel, Table of n, a(n) for n = 1..5000

Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]

Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.

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

Formula

For n > 3, a(n) = (n^2 - 7*n + 16)/2 = A027689(n-4)/2 = A000217(n-4) + 2 = A000124(n-4) + 1. - Henry Bottomley, Jun 27 2000

a(n) = Sum_{k=0..2} A007318(n-k-2, k) for n > 3. - Johannes W. Meijer, Aug 11 2013

Sum_{n>=1} 1/a(n) = 3 + 2*Pi*tanh(sqrt(15)*Pi/2)/sqrt(15). - Amiram Eldar, Dec 13 2022

Mathematica

a[n_] := If[n<4, 1, (n^2-7n+16)/2]; Array[a, 60] (* Jean-François Alcover, Mar 08 2017 *)

Prog

(PARI) for(n=1, 100, print1(if(n<4, 1, (n^2 - 7*n +16)/2), ", ")) \\ G. C. Greubel, Jul 13 2017

Crossrefs

Cf. A000124, A000217, A007318, A027689.

Sequence in context: A104664 A338921 A333343 * A089071 A177205 A275580

Adjacent sequences: A022853 A022854 A022855 * A022857 A022858 A022859

Keyword

nonn,easy

Author

Clark Kimberling

Status

approved