A001614 -id:A001614 - cp's OEIS frontend

A305848 Solution b() of the complementary equation a(n) + b(n) = 5n, where a(1) = 1. See Comments.

4, 8, 12, 15, 19, 23, 26, 30, 34, 37, 41, 44, 48, 52, 55, 59, 63, 66, 70, 73, 77, 81, 84, 88, 92, 95, 99, 102, 106, 110, 113, 117, 120, 124, 128, 131, 135, 139, 142, 146, 149, 153, 157, 160, 164, 168, 171, 175, 178, 182, 186, 189, 193, 196, 200, 204, 207

Offset: 1

Clark Kimberling, Jun 11 2018

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial value. Let x = (5 - sqrt(5))/2 and y = (5 + sqrt(5))/2. Let r = y - 2 = golden ratio (A001622). It appears that

2 - r <= n*x - a(n) < r and 2 - r < b(n) - n*y < r for all n >= 1.

			a(1) = 1, so b(1) = 5 - a(1) = 4. In order for a() and b() to be increasing and complementary, we have a(2) = 2, a(3) = 3, a(4) = 5, etc.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A001622, A305848, A305849, A001614, A118011.

mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
u = 5; v = 5; z = 220;
c = {v}; a = {1}; b = {Last[c] - Last[a]};
Do[AppendTo[a, mex[Flatten[{a, b}], Last[a]]];
  AppendTo[c, u Length[c] + v];
  AppendTo[b, Last[c] - Last[a]], {z}];
c = Flatten[Position[Differences[a], 2]];
a  (* A305847 *)
b  (* A305848 *)
c  (* A305849 *)
(* Peter J. C. Moses, May 30 2018 *)

A036571 Binary packing of Connell sequence (shifted once right).

Original entry on oeis.org

0, 1, 3, 11, 27, 91, 347, 859, 2907, 11099, 43867, 109403, 371547, 1420123, 5614427, 22391643, 55946075, 190163803, 727034715, 2874518363, 11464452955, 45824191323, 114543668059, 389421575003

Offset: 0

Antti Karttunen

nonn

Binary representation of n has 1's at positions specified by Connell sequence (A001614).

			347=101011011 in binary, with 1's at positions 1,2,4,5,7,9.

Cf. A001614, A048721, A048722.

from itertools import count, islice
from math import isqrt
def A036571_gen(): # generator of terms
    c = 0
    for n in count(1):
        yield c
        c += 1<<(m:=n<<1)-(k:=isqrt(m))-int((m<<2)>(k<<2)*(k+1)+1)-1
A036571_list = list(islice(A036571_gen(),25)) # Chai Wah Wu, Jul 26 2022

a(0)=0, a(n) = a(n-1) + 2^((2*n - floor((1/2)*(1 + sqrt(8*n - 7)))) - 1).

A118012 a(n) = 4*A117384(n) - n; a self-inverse permutation of the natural numbers.

Original entry on oeis.org

3, 6, 1, 8, 11, 2, 13, 4, 15, 18, 5, 20, 7, 22, 9, 24, 27, 10, 29, 12, 31, 14, 33, 16, 35, 38, 17, 40, 19, 42, 21, 44, 23, 46, 25, 48, 51, 26, 53, 28, 55, 30, 57, 32, 59, 34, 61, 36, 63, 66, 37, 68, 39, 70, 41, 72, 43, 74, 45, 76, 47, 78, 49, 80, 83, 50, 85, 52, 87, 54, 89, 56

Offset: 1

Paul D. Hanna, Apr 10 2006

nonn

A117384 is defined by A117384(n) = A117384(k) when k = 4*A117384(n) - n. A001614 is the Connell sequence generated as: 1 odd, 2 even, 3 odd, .. and A118011 is the complement of A001614.

Cf. A117384, A001614, A118011.

a(a(n)) = n; a(A118011(n)) = a(4*n - A001614(n)) = A001614(n).

A190716 a(2n) = 2n and a(2*n-1) = A054569(n).

Original entry on oeis.org

1, 2, 7, 4, 21, 6, 43, 8, 73, 10, 111, 12, 157, 14, 211, 16, 273, 18, 343, 20, 421, 22, 507, 24, 601, 26, 703, 28, 813, 30, 931, 32, 1057, 34, 1191, 36, 1333, 38, 1483, 40, 1641, 42, 1807, 44, 1981, 46, 2163, 48, 2353, 50, 2551, 52, 2757, 54, 2971, 56, 3193

Offset: 1

Johannes W. Meijer, May 18 2011

Equals the Row2 triangle sums of the Connell sequence A001614 as a triangle. The Row2(n) triangle sums are defined by Row2(n) = sum((-1)^(n+k)*T(n,k), k=1..n), see A180662.

Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A001614, A054569, A180662.

A190716:= n-> coeff (series (x*(1+2*x+4*x^2-2*x^3+3*x^4)/(1-x^2)^3, x, n+1), x, n): seq(A190716(n), n=1..49);

a[n_]:=(1-(-1)^n(n-1)^2+n^2)/2; Array[a,57] (* Stefano Spezia, Aug 19 2025 *)

a(2*n) = 2*n and a(2*n-1) = 4*n^2 - 6*n + 3.
G.f.: x*(1+2*x+4*x^2-2*x^3+3*x^4)/(1-x^2)^3.
From Stefano Spezia, Aug 19 2025: (Start)
a(n) = (1 - (-1)^n*(n - 1)^2 + n^2)/2.
E.g.f.: (1 + x + x^2)*sinh(x). (End)

A138585 The sequence is formed by concatenating subsequences S1, S2, ... each of finite length. S1 consists of the element 1. The n-th subsequence consist of numbers {(n/2)(n/2 - 1)+ 1, ..., (n/2)(n/2 + 1)} for n even, {((n-1)/2)^2, ..., (n-1)/2 * ((n-1)/2 + 2)} for n odd.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 3, 4, 5, 6, 4, 5, 6, 7, 8, 7, 8, 9, 10, 11, 12, 9, 10, 11, 12, 13, 14, 15, 13, 14, 15, 16, 17, 18, 19, 20, 16, 17, 18, 19, 20, 21, 22, 23, 24, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 31, 32, 33, 34, 35

Offset: 1

Ctibor O. Zizka, May 13 2008

A generalized Connell sequence.

Except for the first term the first element of each subsequence Sn (equivalently, each row of the triangle) gives A004652 (offset by 1), and the last element is A035106.

			S1: {1}
S2: {1,2}
S3: {1,2,3,}
S4: {3,4,5,6}
S5: {4,5,6,7,8}
S6: {7,8,9,10,11,12}, etc.
so concatenation of S1/S2/S3/S4/S5/S6/... gives:
1,1,2,1,2,3,3,4,5,6,4,5,6,7,8,7,8,9,10,11,12,...

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
Douglas E. Iannucci and Donna Mills-Taylor. On Generalizing the Connell Sequence. Journal of Integer Sequences 2 (1999), Article 99.1.7.

Cf. A001614.

S := proc(n) local s: if(n=1)then s:=1: elif(n mod 2 = 0)then s:=(n/2)*(n/2 -1)+1: else s:=((n-1)/2)^2: fi: seq(k,k=s..s+n-1): end: seq(S(n),n=1..12); # Nathaniel Johnston, Oct 01 2011

Corrected and edited by D. S. McNeil, Dec 12 2010

A193732 Connell-like sequence.

Original entry on oeis.org

1, 3, 4, 6, 8, 9, 11, 13, 15, 17, 18, 20, 22, 24, 26, 28, 30, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 110, 112, 114, 116, 118, 120

Offset: 1

Ctibor O. Zizka, Aug 08 2011

			Let prime(n) denote the n-th prime.
Because prime(1)=2, take first 2 odd numbers giving a(1)=1, a(2)=3.
Because prime(2)=3, take 3 even numbers starting with 4 giving a(3)=4, a(4)=6, a(5)=8.
Because prime(3)=5, take 5 odd numbers starting with 9 giving a(6)=9, a(7)=11, a(8)=13, a(9)=15, a(10)=17.
Because prime(4)=7, take 7 even numbers starting with 18 gives a(11)=18, a(12)=20, ..., a(17)=30 etc.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A000040, A001614.

cp:=2:ct:=1:a := proc(n) option remember: global cp,ct: if(n=1)then return 1: elif(ct=cp)then ct:=1:cp:=nextprime(cp): return a(n-1)+1: else ct:=ct+1: return a(n-1)+2: fi: end: seq(a(n),n=1..100); # Nathaniel Johnston, Aug 11 2011

nxt[{p_,a_}]:={NextPrime[p],Range[Last[a]+1,Last[a]+2*NextPrime[p],2]}; Transpose[NestList[nxt,{2,{1,3}},10]][[2]]//Flatten (* Harvey P. Dale, Mar 23 2016 *)

A193666 F(2) odd Fib. numbers, F(3) even Fib. numbers, F(4) odd Fib. numbers..., F(n) = A000045(n), a(n) < a(n+1).

Original entry on oeis.org

1, 2, 8, 13, 21, 55, 144, 610, 2584, 10946, 46368, 75025, 121393, 317811, 514229, 1346269, 2178309, 5702887, 9227465, 14930352, 63245986, 267914296, 1134903170, 4807526976, 20365011074, 86267571272, 365435296162, 1548008755920, 6557470319842, 27777890035288

Offset: 1

Ctibor O. Zizka, Aug 08 2011

nonn

			Taking (F(2)=) 1 odd Fib. number gives a(1)=1.
Then taking F(3)=2 even Fib. numbers starting with 2 gives a(2)=2, a(3)=8.
Then taking F(4)=3 odd Fib. numbers starting with 13 gives a(4)=13, a(5)=21, a(6)=55.
Then taking F(5)=5 even Fib. numbers starting with 144 gives a(7)=144, a(8)=610, a(9)=2584, a(10)=10946, a(11)=46368, etc...

Cf. A000045, A001614, A014437, A014445.

cp's OEIS Frontend

A305848 Solution b() of the complementary equation a(n) + b(n) = 5n, where a(1) = 1. See Comments.

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A036571 Binary packing of Connell sequence (shifted once right).

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A118012 a(n) = 4*A117384(n) - n; a self-inverse permutation of the natural numbers.

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A190716 a(2*n) = 2*n and a(2*n-1) = A054569(n).

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A138585 The sequence is formed by concatenating subsequences S1, S2, ... each of finite length. S1 consists of the element 1. The n-th subsequence consist of numbers {(n/2)*(n/2 - 1)+ 1, ..., (n/2)*(n/2 + 1)} for n even, {((n-1)/2)^2, ..., (n-1)/2 * ((n-1)/2 + 2)} for n odd.

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A193732 Connell-like sequence.

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A193666 F(2) odd Fib. numbers, F(3) even Fib. numbers, F(4) odd Fib. numbers..., F(n) = A000045(n), a(n) < a(n+1).

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A190716 a(2n) = 2n and a(2*n-1) = A054569(n).

A138585 The sequence is formed by concatenating subsequences S1, S2, ... each of finite length. S1 consists of the element 1. The n-th subsequence consist of numbers {(n/2)(n/2 - 1)+ 1, ..., (n/2)(n/2 + 1)} for n even, {((n-1)/2)^2, ..., (n-1)/2 * ((n-1)/2 + 2)} for n odd.