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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A008344, A047398, A047461.

Programs

Magma

[n : n in [0..150] | n mod 8 in [1, 3, 4, 6]]; // Wesley Ivan Hurt, May 31 2016

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 *)

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

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