A042964 - OEIS

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A042964

Numbers congruent to 2 or 3 mod 4.

2, 3, 6, 7, 10, 11, 14, 15, 18, 19, 22, 23, 26, 27, 30, 31, 34, 35, 38, 39, 42, 43, 46, 47, 50, 51, 54, 55, 58, 59, 62, 63, 66, 67, 70, 71, 74, 75, 78, 79, 82, 83, 86, 87, 90, 91, 94, 95, 98, 99, 102, 103, 106, 107, 110, 111, 114, 115, 118, 119, 122, 123, 126, 127 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,1`
	COMMENTS	`Also numbers m such that binomial(m+2, m) mod 2 = 0. - Hieronymus Fischer, Oct 20 2007` `Also numbers m such that floor(1+(m/2)) mod 2 = 0. - Hieronymus Fischer, Oct 20 2007` `Partial sums of the sequence 2, 1, 3, 1, 3, 1, 3, 1, 3, 1, ... which has period 2. - Hieronymus Fischer, Oct 20 2007` `In groups of four add and divide by two the odd and even numbers. - George E. Antoniou, Dec 12 2001` `Comments from Jeremy Gardiner 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, Jan 22 2006` `A133872(a(n)) = 0; complement of A042948. - Reinhard Zumkeller, Oct 03 2008` `Also the 2nd Witt transform of A040000 [Moree]. - R. J. Mathar, Nov 08 2008` `In general, sequences of numbers congruent to {a,a+i} mod k will have a closed form of (k-2i)(2n-1+(-1)^n)/4+in+a, from offset 0. - Gary Detlefs, Oct 29 2013` `Union of A004767 and A016825; Fixed points of A098180. - Wesley Ivan Hurt, Jan 14 2014, Oct 13 2015`
	LINKS	`Table of n, a(n) for n=1..64.` `Maths Magic, Mystery Calculator.` `Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160.` `Index entries for linear recurrences with constant coefficients, signature (1,1,-1).`
	FORMULA	`a(n) = A047406(n)/2.` `From Michael Somos, Jan 12 2000: (Start)` `G.f.: (2+x+x^2)/((1-x)(1-x^2)).` `a(n) = a(n-1) + 2 + (-1)^n. (End)` `a(n) = 2n if n is odd, otherwise n = 2n - 1. - Amarnath Murthy, Oct 16 2003` `a(n) = (2 + (-1)^n + (-1)^(n+1))n - (1 + (-1)^n)/2, n >= 1. - Paolo P. Lava, Feb 15 2008` `a(n) = (3 + (-1)^(n-1))/2 + 2(n-1) = 2n + 2 - (n mod 2). - Hieronymus Fischer, Oct 20 2007` `a(n) = 4n - a(n-1) - 3 (with a(1) = 2). - Vincenzo Librandi, Nov 17 2010` `a(n) = 2*n + ((-1)^(n-1) - 1)/2. - Gary Detlefs, Oct 29 2013`
	MAPLE	`A042964:=n->2*n+((-1)^(n-1)-1)/2; seq(A042964(n), n=1..100); # Wesley Ivan Hurt, Jan 07 2014`
	MATHEMATICA	`Flatten[Table[4n + {2, 3}, {n, 0, 31}]] (* Alonso del Arte, Feb 07 2013 )` `Select[Range[200], MemberQ[{2, 3}, Mod[#, 4]]&] ( or ) LinearRecurrence[ {1, 1, -1}, {2, 3, 6}, 90] ( Harvey P. Dale, Nov 28 2018 *)`
	PROG	`(PARI) a(n)=2n+2-n%2` `(MAGMA) [2n+((-1)^(n-1)-1)/2 : n in [1..100]]; // Wesley Ivan Hurt, Oct 13 2015` `(MAGMA) [n: n in [1..150] \| n mod 4 in [2, 3]]; // Vincenzo Librandi, Oct 13 2015` `(PARI) Vec((2+x+x^2)/((1-x)*(1-x^2)) + O(x^100)) \\ Altug Alkan, Oct 13 2015`
	CROSSREFS	`Cf. A000040, A133620, A133621, A133622, A133630, A133635.` `Cf. A133872, A133882, A133890, A133900, A133910.` `Card trick: A005408, A047566, A115419, A115420, A115421.` `Cf. A004767, A016825 A040000, A042948, A047406, A098180.` `Sequence in context: A107998 A276884 A053438 * A338064 A230375 A062837` `Adjacent sequences: A042961 A042962 A042963 * A042965 A042966 A042967`
	KEYWORD	`nonn,easy`
	AUTHOR	`N. J. A. Sloane`
	EXTENSIONS	`Edited by N. J. A. Sloane, Jun 30 2008 at the suggestion of R. J. Mathar` `Corrected by Jaroslav Krizek, Dec 18 2009`
	STATUS	`approved`

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Last modified February 27 20:01 EST 2021. Contains 341658 sequences. (Running on oeis4.)