A047596 Numbers that are congruent to {2, 3, 4, 5} mod 8.

2, 3, 4, 5, 10, 11, 12, 13, 18, 19, 20, 21, 26, 27, 28, 29, 34, 35, 36, 37, 42, 43, 44, 45, 50, 51, 52, 53, 58, 59, 60, 61, 66, 67, 68, 69, 74, 75, 76, 77, 82, 83, 84, 85, 90, 91, 92, 93, 98, 99, 100, 101, 106, 107, 108, 109, 114, 115, 116, 117, 122, 123

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

For n > 2, a(n) is the decimal value that results from the conversion of n-1 to binary whose last two bits are altered by either of the following rules: 00->010, 01->011, 10->100, 11->101. For example a(10) = 19 because 10 - 1 = 9 = '10'01'->'10'011' = 19. - Franck Maminirina Ramaharo, Jul 25 2018

Links

• Bruno Berselli, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Crossrefs

Cf. A010873, A047463, A047621.

Programs

GAP

Filtered([1..130],n->n mod 8=2 or n mod 8=3 or n mod 8=4 or n mod 8=5); # Muniru A Asiru, Jul 27 2018

Magma

[n: n in [1..120] | n mod 8 in [2..5]]; // Bruno Berselli, Jul 17 2012

Maple

A047596:=n->2*n-1-I^(n*(n+1))-(1+I^(2*n))/2: seq(A047596(n), n=1..100); # Wesley Ivan Hurt, Jun 01 2016

Mathematica

Select[Range[120], MemberQ[{2, 3, 4, 5}, Mod[#, 8]] &] (* or *) LinearRecurrence[{1, 0, 0, 1, -1}, {2, 3, 4, 5, 10}, 60] (* Bruno Berselli, Jul 17 2012 *)

Maxima

makelist(2*n-1-%i^(n*(n+1))-(1+(-1)^n)/2,n,1,60); /* Bruno Berselli, Jul 17 2012 */

PARI

Vec((2+x+x^2+x^3+3*x^4)/((1+x)*(1-x)^2*(1+x^2))+O(x^60)) \\ Bruno Berselli, Jul 17 2012

Formula

G.f.: x*(2+x+x^2+x^3+3*x^4)/((1+x)*(1-x)^2*(1+x^2)). [Bruno Berselli, Jul 17 2012]

a(n) = 2*n-1-i^(n*(n+1))-(1+(-1)^n)/2, where i=sqrt(-1). [Bruno Berselli, Jul 17 2012]

a(n) = 2n - A010873(n-1). - Wesley Ivan Hurt, Jul 07 2013

From Wesley Ivan Hurt, Jun 01 2016: (Start)

a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.

a(2k) = A047621(k), a(2k-1) = A047463(k). (End)

E.g.f.: 3 + sin(x) - cos(x) + (2*x - 1)*sinh(x) + 2*(x - 1)*cosh(x). - Ilya Gutkovskiy, Jun 02 2016

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/16 - (sqrt(2)+3)*log(2)/8 + sqrt(2)*log(sqrt(2)+2)/4. - Amiram Eldar, Dec 25 2021