1,1

Also numbers m such that binomial(m+2,m) mod 2 = 0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Oct 20 2007

Also numbers m such that floor(1+(m/2)) mod 2 = 0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Oct 20 2007

Partial sums of the sequence 2,1,3,1,3,1,3,1,3,1, ... which has period 2. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Oct 20 2007

In groups of four add and divide by two the odd and even numbers - George E. Antoniou (george.antoniou(AT)montclair.edu), Dec 12 2001.

Comments from Jeremy Gardiner (jeremy.gardiner(AT)btinternet.com) on the "mystery calculator". There are 6 cards.

Card 0 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, ... (A005408 sequence)

Card 1 2, 3, 6, 7, 10, 11, 14, 15, 18, 19, 22, 23, 26, 27, 30, 31, 34, 35, 38, 39, ... (this sequence)

Card 2 4, 5, 6, 7, 12, 13, 14, 15, 20, 21, 22, 23, 28, 29, 30, 31, 36, 37, 38, 39, ... ( A047566)

Card 3 8, 9, 10, 11, 12, 13, 14, 15, 24, 25, 26, 27, 28, 29, 30, 31, 40, 41, 42, ... (A115419)

Card 4 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 48, 49, 50, ... (A115420)

Card 5 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, ... (A115421)

The trick: You secretly select a number between 1 and 63 from one of the cards. You indicate to me the cards on which that number appears; I tell you the number you selected!

The solution: I add together the first term from each of the indicated cards. The total equals the selected number. The numbers in each sequence all have a "1" in the same position in their binary expansion. Example: You indicate cards 1, 3 and 5. Your selected number is 2+8+32 = 42.

Numbers having a 1 in position 1 of their binary expansion. One of the mystery calculator sequences: A005408, A042964, A047566, A115419, A115420, A115421. - Jeremy Gardiner (jeremy.gardiner(AT)btinternet.com), Jan 22 2006

Maths Magic, Mystery Calculator.

G.f.: (2+x+x^2)/((1-x)*(1-x^2)). a(n)=a(n-1)+2+(-1)^n - Michael Somos, Jan 12 2000.

a(n) = 2n if n is odd else n = 2n-1. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Oct 16 2003

a(n)=[2+(-1)^n+(-1)^(n+1)]*n-[1+(-1)^n]/2, n>=1 - Paolo P. Lava (ppl(AT)spl.at), Feb 15 2008

a(n)=(3+(-1)^n)/2 + 2n = 2n+2-(n mod 2). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Oct 20 2007

(PARI) a(n)=2*n+2-n%2

a(n) = A047406(n)/2.

Cf. A000040, A133620, A133621, A133622, A133630, A133635.

Cf. A133872, A133882, A133890, A133900, A133910.

Sequence in context: A073439 A107998 A053438 this_sequence A062837 A073170 A014689

Adjacent sequences: A042961 A042962 A042963 this_sequence A042965 A042966 A042967

nonn,new

njas

Edited by njas, Jun 30 2008 at the suggestion of R. J. Mathar