A089781 -id:A089781 - cp's OEIS frontend

A116966 a(n) = n + {1,2,0,1} according as n == {0,1,2,3} mod 4.

1, 3, 2, 4, 5, 7, 6, 8, 9, 11, 10, 12, 13, 15, 14, 16, 17, 19, 18, 20, 21, 23, 22, 24, 25, 27, 26, 28, 29, 31, 30, 32, 33, 35, 34, 36, 37, 39, 38, 40, 41, 43, 42, 44, 45, 47, 46, 48, 49, 51, 50, 52, 53, 55, 54, 56, 57, 59, 58, 60, 61, 63, 62, 64, 65, 67, 66, 68

Offset: 0

N. J. A. Sloane, Mar 31 2006

In each group of four consecutive numbers, swap 2nd and 3rd terms. - Zak Seidov, Mar 31 2006
First differences of A089781. - Reinhard Zumkeller, Aug 15 2015
From Guenther Schrack, May 31 2017: (Start)
Permutation of the positive integers partitioned into quadruples [4k+1,4k+3,4k+2,4k+4].
Partial sums: A116996. (End)

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

Cf. A115391, A116996.

Cf. A140081, A089781.

a116966 n = a116966_list !! n
a116966_list = zipWith (+) [0..] $ drop 2 a140081_list
-- Reinhard Zumkeller, Aug 15 2015

/* By definition: */ [ n + [1,2,0,1][1+(n mod 4)]: n in [0..70] ]; // Bruno Berselli, Nov 25 2012

f:=proc(i) if i mod 4 = 0 then i+1 elif i mod 4 = 1 then i+2 elif i mod 4 = 2 then i else i+1; fi; end;

b := {1, 2, 0, 1}; a[n_] := n + b[[1 + Mod[n, 4]]]; Table[a[n], {n, 0, 60}] (* Stefan Steinerberger, Mar 31 2006 *)
CoefficientList[Series[(2 x^3 - x^2 + 2 x + 1) / ((x - 1)^2 (x + 1) (x^2 + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 19 2013 *)

makelist(n+1+(%i^(n*(n-1))-(-1)^n)/2, n, 0, 70); /* Bruno Berselli, Nov 25 2012 */

Vec((2*x^3-x^2+2*x+1) / ((x-1)^2*(x+1)*(x^2+1)) + O(x^66) ) \\ Joerg Arndt, Apr 30 2013

a(n) = n+1+(i^(n(n-1))-(-1)^n)/2, where i=sqrt(-1). - Bruno Berselli, Nov 25 2012
G.f.: (2*x^3-x^2+2*x+1) / ((x-1)^2*(x+1)*(x^2+1)). - Colin Barker, Apr 30 2013
a(n) = A140081(n+2) + n. - Reinhard Zumkeller, Aug 15 2015
From Guenther Schrack, May 31 2017: (Start)
a(n) = n + 1 + ((-1)^(n*(n-1)/2) - (-1)^n)/2.
a(n) = a(n-4) + 4, n > 3.
a(n) = a(n-1) + a(n-4) - a(n-5), n > 4. (End)
Sum_{n>=0} (-1)^n/a(n) = Pi/4 + log(2)/2. - Amiram Eldar, Jan 31 2023

A111244 Partial sums of A084385.

Original entry on oeis.org

1, 3, 7, 10, 17, 22, 31, 37, 45, 56, 69, 79, 91, 106, 123, 137, 153, 172, 193, 211, 231, 254, 279, 301, 325, 352, 381, 407, 435, 466, 499, 529, 561, 596, 633, 667, 703, 742, 783, 821, 861, 904, 949, 991, 1035, 1082, 1131, 1177, 1225, 1276, 1329, 1379, 1431, 1486, 1543

Offset: 1

N. J. A. Sloane, Oct 31 2005

nonn

a(n+1) = a(n) + (smallest number coprime with a(n) and not already added). - Reinhard Zumkeller, Aug 15 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A084385, A089781.

a111244 n = a111244_list !! (n-1)
a111244_list = scanl1 (+) a084385_list
-- Reinhard Zumkeller, Aug 15 2015

A132425 a(1)=1. a(2)=2. For n >= 3, a(n) is the smallest integer which is > a(n-1), is not coprime to a(n-1) and is such that a(n)-a(n-1) does not equal a(m)-a(m-1) for any m < n.

Original entry on oeis.org

1, 2, 4, 8, 14, 21, 24, 32, 42, 51, 63, 77, 88, 104, 117, 132, 150, 155, 175, 196, 218, 242, 268, 296, 326, 358, 392, 427, 469, 518, 554, 592, 629, 646, 665, 690, 713, 744, 771, 804, 843, 888, 928, 957, 1001, 1053, 1101, 1152, 1198, 1248, 1302, 1358, 1416, 1473

Offset: 1

Leroy Quet, Aug 20 2007

nonn

Is {a(n+1)-a(n)} a permutation of the positive integers?

Cf. A089781, A132426.

a = {1, 2}; For[n = 1, n < 1500, n++, n = a[[ -1]]; c = 0; While[c == 0, n++; If[GCD[n, a[[ -1]]] > 1, b = 0; For[j = 1, j < Length[a], j++, If[a[[j + 1]] - a[[j]] == n - a[[ -1]], b = 1]]; If[b == 0, AppendTo[a, n]; c = 1]]]]; a (* Stefan Steinerberger, Oct 21 2007 *)

More terms from Stefan Steinerberger, Oct 21 2007

cp's OEIS Frontend

A116966 a(n) = n + {1,2,0,1} according as n == {0,1,2,3} mod 4.

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A111244 Partial sums of A084385.

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A132425 a(1)=1. a(2)=2. For n >= 3, a(n) is the smallest integer which is > a(n-1), is not coprime to a(n-1) and is such that a(n)-a(n-1) does not equal a(m)-a(m-1) for any m < n.

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