A092092 Back and Forth Summant S(n, 3): a(n) = Sum{i=0..floor(2n/3)} (n-3i).

1, 1, 0, 3, 2, 0, 5, 3, 0, 7, 4, 0, 9, 5, 0, 11, 6, 0, 13, 7, 0, 15, 8, 0, 17, 9, 0, 19, 10, 0, 21, 11, 0, 23, 12, 0, 25, 13, 0, 27, 14, 0, 29, 15, 0, 31, 16, 0, 33, 17, 0, 35, 18, 0, 37, 19, 0, 39, 20, 0, 41, 21, 0, 43, 22, 0, 45, 23, 0, 47, 24, 0, 49, 25, 0, 51, 26, 0, 53, 27, 0, 55, 28

Offset: 1

Author

Jahan Tuten (jahant(AT)indiainfo.com), Mar 29 2004

Keywords

• nonn
• easy

Comments

The terms for n>1 can also be defined by: a(n)=0 if n==0 (mod 3), and otherwise a(n) equals the inverse of 3 in Z/nZ*. - José María Grau Ribas, Jun 18 2013

The subsequence of nonzero terms is essentially the same as A026741. - Giovanni Resta, Jun 18 2013

References

• F. Smarandache, Back and Forth Summants, Arizona State Univ., Special Collections, 1972.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• J. Dezert, editor, Smarandacheials, Mathematics Magazine, Aurora, Canada
• Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Crossrefs

Cf. A092094, A092397.

Other values of k: A000004 (k = 1, 2), A027656 (k = 4), A092093 (k = 5).

Cf. A226782 - A226787 for inverses of 4,5,6,.. in Z/nZ*.

Programs

Maple

f:= proc(n) local t;
t:= n mod 3;
if t = 0 then 0 elif t = 1 then 2/3*(n+1/2) else (n+1)/3 fi
end proc:
map(f, [$1..100]); # Robert Israel, May 19 2016

Mathematica

LinearRecurrence[{0, 0, 2, 0, 0, -1}, {1, 1, 0, 3, 2, 0}, 100] (* Jean-François Alcover, Jun 04 2020 *)

PARI

S(n, k=3) = local(s, x); s = n; x = n - k; while (x >= -n, s = s + x; x = x - k); s;

Formula

a(3n) = 0; a(3n+1) = 2n+1; a(3n+2) = n+1.

G.f.: x*(1+x+x^3) / ( (x-1)^2*(1+x+x^2)^2 ). - R. J. Mathar, Jun 26 2013

a(n) = Sum_{k=1..n} k*( floor((3k-1)/n)-floor((3k-2)/n) ). - Anthony Browne, May 17 2016

Extensions

Edited and extended by David Wasserman, Dec 19 2005