A047415 Numbers that are congruent to {1, 3, 4, 6} mod 8.

1, 3, 4, 6, 9, 11, 12, 14, 17, 19, 20, 22, 25, 27, 28, 30, 33, 35, 36, 38, 41, 43, 44, 46, 49, 51, 52, 54, 57, 59, 60, 62, 65, 67, 68, 70, 73, 75, 76, 78, 81, 83, 84, 86, 89, 91, 92, 94, 97, 99, 100, 102, 105, 107, 108, 110, 113, 115, 116, 118, 121, 123, 124

Offset

1,2

Comments

Consider an operation SS(n) defined for a specific sequence b where b(n) is the n-th term of b. This operation is defined as follows: SS(1) = b(1); if b(n+1) > SS(n), SS(n+1) = SS(n) + b(n+1), otherwise SS(n+1) = SS(n) - b(n+1) (If b(n) = A000027(n), then SS(n) = A008344(n+1)). If the sequence b can represent any permutation of the first n natural numbers, then a(n) is the maximum possible value of SS(n). - Iain Fox, Sep 15 2020 (see link by Math StackExchage)

Links

Iain Fox, Table of n, a(n) for n = 1..10000

Math StackExchange, Strange Sum of Numbers 1 to 100, September 2020.

Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Formula

G.f.: x*(1+x+2*x^3) / ( (1+x^2)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011

a(n) = 2*(n-1)-(i^(n*(n+1))-1)/2, where i=sqrt(-1). - Bruno Berselli, Dec 06 2011

From Wesley Ivan Hurt, May 31 2016: (Start)

a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4.

a(n) = (1+i)*(4*n-4*n*i+3*i-3+i^(1-n)-i^n)/4 where i=sqrt(-1).

a(2*k) = A047398(k), a(2*k-1) = A047461(k). (End)

E.g.f.: (4 + sin(x) - cos(x) + (4*x - 3)*exp(x))/2. - Ilya Gutkovskiy, May 31 2016

Sum_{n>=1} (-1)^(n+1)/a(n) = 3*Pi/16 - (sqrt(2)+1)*log(2)/8 + sqrt(2)*log(sqrt(2)+2)/4. - Amiram Eldar, Dec 24 2021

Maple

A047415:=n->2*(n-1)-(I^(n*(n+1))-1)/2: seq(A047415(n), n=1..100); # Wesley Ivan Hurt, May 31 2016

Mathematica

Select[Range[108], MemberQ[{1, 3, 4, 6}, Mod[#, 8]]&] (* Bruno Berselli, Dec 06 2011 *)

Prog

(Maxima) makelist(2*(n-1)-(%i^(n*(n+1))-1)/2, n, 1, 55); /* Bruno Berselli, Dec 06 2011 */
(PARI) a(n)=2*(n-1)-(I^(n*(n+1))-1)/2 \\ Charles R Greathouse IV, Dec 06 2011
(Magma) [n : n in [0..150] | n mod 8 in [1, 3, 4, 6]]; // Wesley Ivan Hurt, May 31 2016

Crossrefs

Cf. A008344, A047398, A047461.

Sequence in context: A329862 A153236 A158705 * A087805 A213040 A173318

Adjacent sequences: A047412 A047413 A047414 * A047416 A047417 A047418

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved