A057198 -id:A057198 - cp's OEIS frontend

A246360 a(1) = 1, then A007051 ((3^n)+1)/2 interleaved with A057198 (5*3^(n-1)+1)/2.

1, 2, 3, 5, 8, 14, 23, 41, 68, 122, 203, 365, 608, 1094, 1823, 3281, 5468, 9842, 16403, 29525, 49208, 88574, 147623, 265721, 442868, 797162, 1328603, 2391485, 3985808, 7174454, 11957423, 21523361, 35872268, 64570082, 107616803, 193710245, 322850408, 581130734

Offset: 1

Antti Karttunen, Aug 24 2014

Also record values in A048673.

Antti Karttunen, Table of n, a(n) for n = 1..64
Index entries for linear recurrences with constant coefficients, signature (1,3,-3).

Even bisection: A007051 from A007051(1) onward: [2, 5, 14, 41, ...]
Odd bisection: 1 followed by A057198.
A029744 gives the corresponding record positions in A048673.
A247284 gives the maximum values of A048673 between these records and A247283 gives the positions where they occur.
Subsequence of A246361.
Cf. A000244, A193652, A246347.

LinearRecurrence[{1, 3, -3}, {1, 2, 3, 5}, 40] (* Hugo Pfoertner, Sep 27 2022 *)

def A246360(n): return 1 if n==1 else (3+((n&1)<<1))*3**((n>>1)-1)+1>>1 # Chai Wah Wu, Sep 02 2025

(define (A246360 n) (cond ((<= n 1) n) ((even? n) (/ (+ 1 (A000244 (/ n 2))) 2)) (else (/ (+ 1 (* 5 (A000244 (/ (- n 3) 2)))) 2))))

a(1) = 1, a(2n) = (3^n+1)/2, a(2n+1) = (5 * 3^(n-1)+1)/2.
a(n) = A048673(A029744(n)).
a(n) = A087503(n-3) + 2 for n >= 3. - Peter Kagey, Nov 30 2019
G.f.: x -x^2*(-2-x+4*x^2) / ( (x-1)*(3*x^2-1) ). - R. J. Mathar, Sep 23 2014

A048673 Permutation of natural numbers: a(n) = (A003961(n)+1) / 2 [where A003961(n) shifts the prime factorization of n one step towards larger primes].

Original entry on oeis.org

1, 2, 3, 5, 4, 8, 6, 14, 13, 11, 7, 23, 9, 17, 18, 41, 10, 38, 12, 32, 28, 20, 15, 68, 25, 26, 63, 50, 16, 53, 19, 122, 33, 29, 39, 113, 21, 35, 43, 95, 22, 83, 24, 59, 88, 44, 27, 203, 61, 74, 48, 77, 30, 188, 46, 149, 58, 47, 31, 158, 34, 56, 138, 365, 60, 98, 36, 86, 73

Offset: 1

Antti Karttunen, Jul 14 1999

nonn

Inverse of sequence A064216 considered as a permutation of the positive integers. - Howard A. Landman, Sep 25 2001
From Antti Karttunen, Dec 20 2014: (Start)
Permutation of natural numbers obtained by replacing each prime divisor of n with the next prime and mapping the generated odd numbers back to all natural numbers by adding one and then halving.
Note: there is a 7-cycle almost right in the beginning: (6 8 14 17 10 11 7). (See also comments at A249821. This 7-cycle is endlessly copied in permutations like A250249/A250250.)
The only 3-cycle in range 1 .. 402653184 is (2821 3460 5639).
For 1- and 2-cycles, see A245449.
(End)
The first 5-cycle is (1410, 2783, 2451, 2703, 2803). - Robert Israel, Jan 15 2015
From Michel Marcus, Aug 09 2020: (Start)
(5194, 5356, 6149, 8186, 10709), (46048, 51339, 87915, 102673, 137205) and (175811, 200924, 226175, 246397, 267838) are other 5-cycles.
(10242, 20479, 21413, 29245, 30275, 40354, 48241) is another 7-cycle. (End)
From Antti Karttunen, Feb 10 2021: (Start)
Somewhat artificially, also this permutation can be represented as a binary tree. Each child to the left is obtained by multiplying the parent by 3 and subtracting one, while each child to the right is obtained by applying A253888 to the parent:
                                       1
                                       |
                    ................../ \..................
                   2                                       3
         5......../ \........4                   8......../ \........6
        / \                 / \                 / \                 / \
       /   \               /   \               /   \               /   \
      /     \             /     \             /     \             /     \
    14       13         11       7          23       9          17       18
  41 10    38  12     32  28   20 15      68  25   26 63      50  16   53  19
etc.
Each node's (> 1) parent can be obtained with A253889. Sequences A292243, A292244, A292245 and A292246 are constructed from the residues (mod 3) of the vertices encountered on the path from n to the root (1).
(End)

			For n = 6, as 6 = 2 * 3 = prime(1) * prime(2), we have a(6) = ((prime(1+1) * prime(2+1))+1) / 2 = ((3 * 5)+1)/2 = 8.
For n = 12, as 12 = 2^2 * 3, we have a(12) = ((3^2 * 5) + 1)/2 = 23.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers.

Inverse: A064216.
Row 1 of A251722, Row 2 of A249822.
One more than A108228, half the terms of A243501.
Fixed points: A048674.
Positions of records: A029744, their values: A246360 (= A007051 interleaved with A057198).
Positions of subrecords: A247283, their values: A247284.
Cf. A246351 (Numbers n such that a(n) < n.)
Cf. A246352 (Numbers n such that a(n) >= n.)
Cf. A246281 (Numbers n such that a(n) <= n.)
Cf. A246282 (Numbers n such that a(n) > n.), A252742 (their char. function)
Cf. A246261 (Numbers n for which a(n) is odd.)
Cf. A246263 (Numbers n for which a(n) is even.)
Cf. A246260 (a(n) reduced modulo 2), A341345 (modulo 3), A341346, A292251 (3-adic valuation), A292252.
Cf. A246342 (Iterates starting from n=12.)
Cf. A246344 (Iterates starting from n=16.)
Cf. also A003602, A003961 (A045965), A108951, A151800, A245449, A249735, A249821, A250471, A250249, A250250.
Cf. A245447 (This permutation "squared", a(a(n)).)
Other permutations whose formulas refer to this sequence: A122111, A243062, A243066, A243500, A243506, A244154, A244319, A245605, A245608, A245610, A245612, A245708, A246265, A246267, A246268, A246363, A249745, A249824, A249826, and also A183209, A254103 that are somewhat similar.
Cf. also prime-shift based binary trees A005940, A163511, A245612 and A244154.
Cf. A253888, A253889, A292243, A292244, A292245 and A292246 for other derived sequences.
Cf. A323893 (Dirichlet inverse), A323894 (sum with it), A336840 (inverse Möbius transform).

a048673 = (`div` 2) . (+ 1) . a045965
-- Reinhard Zumkeller, Jul 12 2012

f:= proc(n)
local F,q,t;
  F:= ifactors(n)[2];
  (1 + mul(nextprime(t[1])^t[2], t = F))/2
end proc:
seq(f(n),n=1..1000); # Robert Israel, Jan 15 2015

Table[(Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ n, {n, 69}] (* Michael De Vlieger, Dec 18 2014, revised Mar 17 2016 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A048673(n) = (A003961(n)+1)/2; \\ Antti Karttunen, Dec 20 2014

A048673(n) = if(1==n,n,if(n%2,A253888(A048673((n-1)/2)),(3*A048673(n/2))-1)); \\ (Not practical, but demonstrates the construction as a binary tree). - Antti Karttunen, Feb 10 2021

from sympy import factorint, nextprime, prod
def a(n):
    f = factorint(n)
    return 1 if n==1 else (1 + prod(nextprime(i)**f[i] for i in f))//2 # Indranil Ghosh, May 09 2017

(define (A048673 n) (/ (+ 1 (A003961 n)) 2)) ;; Antti Karttunen, Dec 20 2014

From Antti Karttunen, Dec 20 2014: (Start)
a(1) = 1; for n>1: If n = product_{k>=1} (p_k)^(c_k), then a(n) = (1/2) * (1 + product_{k>=1} (p_{k+1})^(c_k)).
a(n) = (A003961(n)+1) / 2.
a(n) = floor((A045965(n)+1)/2).
Other identities. For all n >= 1:
a(n) = A108228(n)+1.
a(n) = A243501(n)/2.
A108951(n) = A181812(a(n)).
a(A246263(A246268(n))) = 2*n.
As a composition of other permutations involving prime-shift operations:
a(n) = A243506(A122111(n)).
a(n) = A243066(A241909(n)).
a(n) = A241909(A243062(n)).
a(n) = A244154(A156552(n)).
a(n) = A245610(A244319(n)).
a(n) = A227413(A246363(n)).
a(n) = A245612(A243071(n)).
a(n) = A245608(A245605(n)).
a(n) = A245610(A244319(n)).
a(n) = A249745(A249824(n)).
For n >= 2, a(n) = A245708(1+A245605(n-1)).
(End)
From Antti Karttunen, Jan 17 2015: (Start)
We also have the following identities:
a(2n) = 3*a(n) - 1. [Thus a(2n+1) = 0 or 1 when reduced modulo 3. See A341346]
a(3n) = 5*a(n) - 2.
a(4n) = 9*a(n) - 4.
a(5n) = 7*a(n) - 3.
a(6n) = 15*a(n) - 7.
a(7n) = 11*a(n) - 5.
a(8n) = 27*a(n) - 13.
a(9n) = 25*a(n) - 12.
and in general:
a(x*y) = (A003961(x) * a(y)) - a(x) + 1, for all x, y >= 1.
(End)
From Antti Karttunen, Feb 10 2021: (Start)
For n > 1, a(2n) = A016789(a(n)-1), a(2n+1) = A253888(a(n)).
a(2^n) = A007051(n) for all n >= 0. [A property shared with A183209 and A254103].
(End)
a(n) = A003602(A003961(n)). - Antti Karttunen, Apr 20 2022
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/4) * Product_{p prime} ((p^2-p)/(p^2-nextprime(p))) = 1.0319981... , where nextprime is A151800. - Amiram Eldar, Jan 18 2023

New name and crossrefs to derived sequences added by Antti Karttunen, Dec 20 2014

A278055 Relative of Hofstadter Q-sequence: a(1) = 1, a(2) = 2, a(3) = 3, a(4) = 4, a(5) = 5; a(n) = a(n-a(n-1)) + a(n-a(n-2)) + a(n-a(n-3)) for n > 5.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 15, 15, 16, 17, 17, 18, 18, 19, 20, 21, 21, 22, 23, 24, 24, 25, 26, 26, 27, 27, 28, 29, 30, 30, 31, 32, 33, 33, 34, 35, 35, 36, 36, 37, 38, 39, 39, 40, 41, 42, 42, 43, 44, 44, 45, 45, 46, 47, 48, 48, 49, 50, 50, 51

Offset: 1

Nathan Fox, Nov 10 2016

nonn

This sequence is monotonic, with successive terms increasing by 0 or 1. So the sequence hits every positive integer.

A number k appears twice in this sequence if and only if for some i, k is congruent to A057198(i) mod 3^i and k > A057198(i).

Nathan Fox, Table of n, a(n) for n = 1..10000
Nathan Fox, A Slow Relative of Hofstadter's Q-Sequence, arXiv preprint arXiv:1611.08244 [math.NT], 2016.

Cf. A005185, A057198, A063882, A267501, A274058.

a[n_] := a[n] = a[n - a[n -1]] + a[n - a[n -2]] + a[n - a[n -3]]; a[1] = 1; a[2] = 2; a[3] = 3; a[4] = 4; a[5] = 5; Array[a, 71] (* Robert G. Wilson v, Dec 02 2016 *)

A=Vecsmall([]);
a(n)=if(n<7, return(n)); if(#ACharles R Greathouse IV, Nov 19 2016

a(n) ~ 2n/3.

A176059 Periodic sequence: Repeat 3, 2.

Original entry on oeis.org

3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3

Offset: 0

Klaus Brockhaus, Apr 07 2010

Interleaving of A010701 and A007395.
Also continued fraction expansion of (3+sqrt(15))/2.
Also decimal expansion of 32/99.
a(n) = A010693(n+1).
Essentially first differences of A047218.
Binomial transform of 3 followed by -A122803.
Inverse binomial transform of 3 followed by A020714.
Second inverse binomial transform of A057198 without initial term 1.

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

Cf. A010701 (all 3's sequence), A007395 (all 2's sequence), A176058 (decimal expansion of (3+sqrt(15))/2), A010693 (repeat 2, 3), A047218 (congruent to {0, 3} mod 5), A122803 (powers of -2), A020714 (5*2^n), A057198 ((5*3^(n-1)+1)/2, n > 0).

Cf. A026532 (partial products).

a176059 = (3 -) . (`mod` 2) -- Reinhard Zumkeller, Nov 27 2012

a176059_list = cycle [3,2]  -- Reinhard Zumkeller, Apr 04 2012

&cat[ [3, 2]: n in [0..52] ];
[ (5+(-1)^n)/2: n in [0..104] ];

A176059:=n->(5+(-1)^n)/2; seq(A176059(n), n=0..100); # Wesley Ivan Hurt, Feb 26 2014

a[n_] := {3, 2}[[Mod[n, 2] + 1]]; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Jul 19 2013 *)
PadRight[{},120,{3,2}] (* Harvey P. Dale, Oct 06 2019 *)

a(n)=3-n%2 \\ Charles R Greathouse IV, Jul 13 2016

a(n) = (5+(-1)^n)/2.
a(n) = a(n-2) for n > 1; a(0) = 3, a(1) = 2.
a(n) = -a(n-1)+5 for n > 0; a(0) = 3.
a(n) = 3*((n+1) mod 2)+2*(n mod 2).
G.f.: (3+2*x)/((1-x)*(1+x)).
E.g.f.: 3*cosh(x) + 2*sinh(x). - Stefano Spezia, Aug 04 2025

A060816 a(0) = 1; a(n) = (5*3^(n-1) - 1)/2 for n > 0.

Original entry on oeis.org

1, 2, 7, 22, 67, 202, 607, 1822, 5467, 16402, 49207, 147622, 442867, 1328602, 3985807, 11957422, 35872267, 107616802, 322850407, 968551222, 2905653667, 8716961002, 26150883007, 78452649022, 235357947067, 706073841202

Offset: 0

Jason Earls, Apr 29 2001

From Erich Friedman's math magic page 2nd paragraph under "Answers" section.
Let A be the Hessenberg matrix of order n, defined by: A[1,j] = 1, A[i,i] = 2,(i>1),  A[i,i-1] = -1, and A[i,j] = 0 otherwise. Then, for n >= 1, a(n) = (-1)^n*charpoly(A,-1). - Milan Janjic, Jan 26 2010
If n > 0 and H = hex number (A003215), then 9*H(a(n)) - 2 = H(a(n+1)), for example 9*H(2) - 2 = 9*19 - 2 = 169 = H(7). For n > 2, this is a subsequence of A017209, see formula. - Klaus Purath, Mar 31 2021
Consider the Tower of Hanoi puzzle of order n (i.e., with n rings to be moved). Label each ring with a distinct symbol from an alphabet of size n. Construct words by performing moves according to the standard rules of the puzzle, recording the corresponding symbol each time a ring is moved. To ensure finiteness, we forbid returning to any previously encountered state. Additionally, we impose the constraint that the same ring cannot be moved twice in succession. Then, a(n) is the number of distinct words that can be formed under these rules. - Thomas Baruchel, Jul 22 2025

Harry J. Smith, Table of n, a(n) for n = 0..200
Erich Friedman, Math. Magic
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A005030 (first differences), A244762 (partial sums).

Cf. A003215, A003462, A007051, A008343, A017209, A057198, A134931.

LinearRecurrence[{4,-3},{1,2,7},30] (* Harvey P. Dale, Nov 15 2022 *)

A060816(n)=if(n, 3^n*5\6, 1) \\ M. F. Hasler, Apr 06 2019

The following is a summary of formulas added over the past 18 years.
a(n) = A057198(n) - 1.
a(n) = 3*a(n-1) + 1; with a(0)=1, a(1)=2. - Jason Earls, Apr 29 2001
From Henry Bottomley, May 01 2001: (Start)
For n>0, a(n) = a(n-1)+5*3^(n-2) = a(n-1)+A005030(n-2).
For n>0, a(n) = (5*A003462(n)+1)/3. (End)
From Colin Barker, Apr 24 2012: (Start)
a(n) = 4*a(n-1) - 3*a(n-2) for n > 2.
G.f.: (1-2*x+2*x^2)/((1-x)*(1-3*x)). (End)
a(n+1) = A134931(n) + 1. - Philippe Deléham, Apr 14 2013
For n > 0, A008343(a(n)) = 0. - Dmitry Kamenetsky, Feb 14 2017
For n > 0, a(n) = floor(3^n*5/6). - M. F. Hasler, Apr 06 2019
From Klaus Purath, Mar 31 2021: (Start)
a(n) = A017209(a(n-2)), n > 2.
a(n) = 2 + Sum_{i = 0..n-2} A005030(i).
a(n+1)*a(n+2) = a(n)*a(n+3) + 20*3^n, n > 1.
a(n) = 3^n - A007051(n-1). (End)
E.g.f.: (5*exp(3*x) - 3*exp(x) + 4)/6. - Stefano Spezia, Aug 28 2023

Edited by M. F. Hasler, Apr 06 2019 and by N. J. A. Sloane, Apr 09 2019

A193649 Q-residue of the (n+1)st Fibonacci polynomial, where Q is the triangular array (t(i,j)) given by t(i,j)=1. (See Comments.)

Original entry on oeis.org

1, 1, 3, 5, 15, 33, 91, 221, 583, 1465, 3795, 9653, 24831, 63441, 162763, 416525, 1067575, 2733673, 7003971, 17938661, 45954543, 117709185, 301527355, 772364093, 1978473511

Offset: 0

Clark Kimberling, Aug 02 2011

nonn

Suppose that p=p(0)*x^n+p(1)*x^(n-1)+...+p(n-1)*x+p(n) is a polynomial of positive degree and that Q is a sequence of polynomials: q(k,x)=t(k,0)*x^k+t(k,1)*x^(k-1)+...+t(k,k-1)*x+t(k,k), for k=0,1,2,... The Q-downstep of p is the polynomial given by D(p)=p(0)*q(n-1,x)+p(1)*q(n-2,x)+...+p(n-1)*q(0,x)+p(n).

Since degree(D(p))

Example: let p(x)=2*x^3+3*x^2+4*x+5 and q(k,x)=(x+1)^k.

D(p)=2(x+1)^2+3(x+1)+4(1)+5=2x^2+7x+14

D(D(p))=2(x+1)+7(1)+14=2x+23

D(D(D(p)))=2(1)+23=25;

the Q-residue of p is 25.

We may regard the sequence Q of polynomials as the triangular array formed by coefficients:

t(0,0)

t(1,0)....t(1,1)

t(2,0)....t(2,1)....t(2,2)

t(3,0)....t(3,1)....t(3,2)....t(3,3)

and regard p as the vector (p(0),p(1),...,p(n)). If P is a sequence of polynomials [or triangular array having (row n)=(p(0),p(1),...,p(n))], then the Q-residues of the polynomials form a numerical sequence.

Following are examples in which Q is the triangle given by t(i,j)=1 for 0<=i<=j:

Q.....P...................Q-residue of P

1.....1...................A000079, 2^n

1....(x+1)^n..............A007051, (1+3^n)/2

1....(x+2)^n..............A034478, (1+5^n)/2

1....(x+3)^n..............A034494, (1+7^n)/2

1....(2x+1)^n.............A007582

1....(3x+1)^n.............A081186

1....(2x+3)^n.............A081342

1....(3x+2)^n.............A081336

1.....A040310.............A193649

1....(x+1)^n+(x-1)^n)/2...A122983

1....(x+2)(x+1)^(n-1).....A057198

1....(1,2,3,4,...,n)......A002064

1....(1,1,2,3,4,...,n)....A048495

1....(n,n+1,...,2n).......A087323

1....(n+1,n+2,...,2n+1)...A099035

1....p(n,k)=(2^(n-k))*3^k.A085350

1....p(n,k)=(3^(n-k))*2^k.A090040

1....A008288 (Delannoy)...A193653

1....A054142..............A101265

1....cyclotomic...........A193650

1....(x+1)(x+2)...(x+n)...A193651

1....A114525..............A193662

More examples:

Q...........P.............Q-residue of P

(x+1)^n...(x+1)^n.........A000110, Bell numbers

(x+1)^n...(x+2)^n.........A126390

(x+2)^n...(x+1)^n.........A028361

(x+2)^n...(x+2)^n.........A126443

(x+1)^n.....1.............A005001

(x+2)^n.....1.............A193660

A094727.....1.............A193657

(k+1).....(k+1)...........A001906 (even-ind. Fib. nos.)

(k+1).....(x+1)^n.........A112091

(x+1)^n...(k+1)...........A029761

(k+1)......A049310........A193663

(In these last four, (k+1) represents the triangle t(n,k)=k+1, 0<=k<=n.)

A051162...(x+1)^n.........A193658

A094727...(x+1)^n.........A193659

A049310...(x+1)^n.........A193664

A075362....A075362........A193665

Changing the notation slightly leads to the Mathematica program below and the following formulation for the Q-downstep of p: first, write t(n,k) as q(n,k). Define r(k)=Sum{q(k-1,i)*r(k-1-i) : i=0,1,...,k-1} Then row n of D(p) is given by v(n)=Sum{p(n,k)*r(n-k) : k=0,1,...,n}.

			First five rows of Q, coefficients of Fibonacci polynomials (A049310):
1
1...0
1...0...1
1...0...2...0
1...0...3...0...1
To obtain a(4)=15, downstep four times:
D(x^4+3*x^2+1)=(x^3+x^2+x+1)+3(x+1)+1: (1,1,4,5) [coefficients]
DD(x^4+3*x^2+1)=D(1,1,4,5)=(1,2,11)
DDD(x^4+3*x^2+1)=D(1,2,11)=(1,14)
DDDD(x^4+3*x^2+1)=D(1,14)=15.

Cf. A192872 (polynomial reduction), A193091 (polynomial augmentation), A193722 (the upstep operation and fusion of polynomial sequences or triangular arrays).

q[n_, k_] := 1;
r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}];
f[n_, x_] := Fibonacci[n + 1, x];
p[n_, k_] := Coefficient[f[n, x], x, k]; (* A049310 *)
v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 24}]    (* A193649 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}]  (* 2^k *)
TableForm[Table[p[n, k], {n, 0, 6}, {k, 0, n}]]

Conjecture: G.f.: -(1+x)*(2*x-1) / ( (x-1)*(4*x^2+x-1) ). - R. J. Mathar, Feb 19 2015

A191450 Dispersion of (3*n-1), read by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 8, 4, 14, 23, 11, 6, 41, 68, 32, 17, 7, 122, 203, 95, 50, 20, 9, 365, 608, 284, 149, 59, 26, 10, 1094, 1823, 851, 446, 176, 77, 29, 12, 3281, 5468, 2552, 1337, 527, 230, 86, 35, 13, 9842, 16403, 7655, 4010, 1580, 689, 257, 104, 38, 15, 29525

Offset: 1

Clark Kimberling, Jun 05 2011

Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1. The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n))), s(s(s(t(n)))), ...). Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers. The sequence u given by u(n) = {index of the row of D that contains n} is a fractal sequence. In this case s(n) = A016789(n-1), t(n) = A032766(n) [from term A032766(1) onward] and u(n) = A253887(n). [Author's original comment edited by Antti Karttunen, Jan 24 2015]

For other examples of such sequences, please see the Crossrefs section.

			The northwest corner of the square array:
  1,  2,  5,  14,  41,  122,  365,  1094,  3281,   9842,  29525,   88574, ...
  3,  8, 23,  68, 203,  608, 1823,  5468, 16403,  49208, 147623,  442868, ...
  4, 11, 32,  95, 284,  851, 2552,  7655, 22964,  68891, 206672,  620015, ...
  6, 17, 50, 149, 446, 1337, 4010, 12029, 36086, 108257, 324770,  974309, ...
  7, 20, 59, 176, 527, 1580, 4739, 14216, 42647, 127940, 383819, 1151456, ...
  9, 26, 77, 230, 689, 2066, 6197, 18590, 55769, 167306, 501917, 1505750, ...
  etc.
The leftmost column is A032766, and each successive column to the right of it is obtained by multiplying the left neighbor on that row by three and subtracting one, thus the second column is (3*1)-1, (3*3)-1, (3*4)-1, (3*6)-1, (3*7)-1, (3*9)-1, ... = 2, 8, 11, 17, 20, 26, ...

Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of array
Clark Kimberling, Interspersions and Dispersions.
Clark Kimberling, Interspersions and dispersions, Proceedings of the American Mathematical Society, 117 (1993) 313-321.
Index entries for sequences that are permutations of the natural numbers

Inverse: A254047.
Transpose: A254051.
Column 1: A032766.
Cf. A007051, A057198, A199109, A199113 (rows 1-4).
Cf. A253887 (row index of n in this array) & A254046 (column index, see also A253786).
Cf. A000244, A016789, A038722, A048673, A254053, A254103, A254104.
Examples of other arrays of dispersions: A114537, A035513, A035506, A191449, A191426-A191455.

A191450 := proc(r, c)
    option remember;
    if c = 1 then
        A032766(r) ;
    else
        A016789(procname(r, c-1)-1) ;
    end if;
end proc: # R. J. Mathar, Jan 25 2015

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;
f[n_] :=3n-1 (* complement of column 1 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191450 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191450 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

a(n,k)=3^(n-1)*(k*3\2*2-1)\2+1 \\ =3^(n-1)*(k*3\2-1/2)+1/2, but 30% faster. - M. F. Hasler, Jan 20 2015

(define (A191450 n) (A191450bi (A002260 n) (A004736 n)))
(define (A191450bi row col) (if (= 1 col) (A032766 row) (A016789 (- (A191450bi row (- col 1)) 1))))
(define (A191450bi row col) (/ (+ 3 (* (A000244 col) (- (* 2 (A032766 row)) 1))) 6)) ;; Another implementation based on L. Edson Jeffery's direct formula.
;; Antti Karttunen, Jan 21 2015

Conjecture: A(n,k) = (3 + (2*A032766(n) - 1)*A000244(k))/6. - L. Edson Jeffery, with slight changes by Antti Karttunen, Jan 21 2015

a(n) = A254051(A038722(n)). [When both this and transposed array A254051 are interpreted as one-dimensional sequences.] - Antti Karttunen, Jan 22 2015

Example corrected and description clarified by Antti Karttunen, Jan 24 2015

A254051 Square array A by downward antidiagonals: A(n,k) = (3 + 3^n(2floor(3*k/2) - 1))/6, n,k >= 1; read as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

Original entry on oeis.org

1, 3, 2, 4, 8, 5, 6, 11, 23, 14, 7, 17, 32, 68, 41, 9, 20, 50, 95, 203, 122, 10, 26, 59, 149, 284, 608, 365, 12, 29, 77, 176, 446, 851, 1823, 1094, 13, 35, 86, 230, 527, 1337, 2552, 5468, 3281, 15, 38, 104, 257, 689, 1580, 4010, 7655, 16403, 9842, 16, 44, 113, 311, 770, 2066, 4739, 12029, 22964, 49208, 29525, 18, 47

Offset: 1

L. Edson Jeffery & Antti Karttunen, Jan 24 2015

This is transposed dispersion of (3n-1), starting from its complement A032766 as the first row of square array A(row,col). Please see the transposed array A191450 for references and background discussion about dispersions.

For any odd number x = A135765(row,col), the result after one combined Collatz step (3x+1)/2 -> x (A165355) is found in this array at A(row+1,col).

			The top left corner of the array:
   1,   3,   4,   6,   7,   9,  10,  12,   13,   15,   16,   18,   19,   21
   2,   8,  11,  17,  20,  26,  29,  35,   38,   44,   47,   53,   56,   62
   5,  23,  32,  50,  59,  77,  86, 104,  113,  131,  140,  158,  167,  185
  14,  68,  95, 149, 176, 230, 257, 311,  338,  392,  419,  473,  500,  554
  41, 203, 284, 446, 527, 689, 770, 932, 1013, 1175, 1256, 1418, 1499, 1661
...

Inverse: A254052.
Transpose: A191450.
Row 1: A032766.
Cf. A007051, A057198, A199109, A199113 (columns 1-4).
Cf. A254046 (row index of n in this array, see also A253786), A253887 (column index).
Array A135765(n,k) = 2*A(n,k) - 1.
Other related arrays: A254055, A254101, A254102.
Cf. also A000079, A000244, A003961, A007310, A032766, A002260, A004736, A038722, A165355, A254050.
Related permutations: A048673, A254053, A183209, A249745, A254103, A254104.

In A(n,k)-formulas below, n is the row, and k the column index, both starting from 1:
A(n,k) = (3 + ( A000244(n) * (2*A032766(k) - 1) )) / 6. - Antti Karttunen after L. Edson Jeffery's direct formula for A191450, Jan 24 2015
A(n,k) = A048673(A254053(n,k)). [Alternative formula.]
A(n,k) = (1/2) * (1 + A003961((2^(n-1)) * A254050(k))). [The above expands to this.]
A(n,k) = (1/2) * (1 + (A000244(n-1) * A007310(k))). [Which further reduces to this, equivalent to L. Edson Jeffery's original formula above.]
A(1,k) = A032766(k) and for n > 1: A(n,k) = (3 * A254051(n-1,k)) - 1. [The definition of transposed dispersion of (3n-1).]
A(n,k) = (1+A135765(n,k))/2, or when expressed one-dimensionally, a(n) = (1+A135765(n))/2.
A(n+1,k) = A165355(A135765(n,k)).
As a composition of related permutations. All sequences interpreted as one-dimensional:
a(n) = A048673(A254053(n)). [Proved above.]
a(n) = A191450(A038722(n)). [Transpose of array A191450.]

A046901 a(n) = a(n-1) - n if a(n-1) > n, else a(n) = a(n-1) + n.

Original entry on oeis.org

1, 3, 6, 2, 7, 1, 8, 16, 7, 17, 6, 18, 5, 19, 4, 20, 3, 21, 2, 22, 1, 23, 46, 22, 47, 21, 48, 20, 49, 19, 50, 18, 51, 17, 52, 16, 53, 15, 54, 14, 55, 13, 56, 12, 57, 11, 58, 10, 59, 9, 60, 8, 61, 7, 62, 6, 63, 5, 64, 4, 65, 3, 66, 2, 67, 1, 68, 136, 67, 137

Offset: 1

N. J. A. Sloane

Variation (1) on Recamán's sequence A005132.

a(A134931(n-1)) = 1. - Reinhard Zumkeller, Jan 31 2013

N. J. A. Sloane, First 10000 terms
Nick Hobson, Python program for this sequence
Kival Ngaokrajang, scatter plot in log-log scale looks, for both this sequence and A211346.
Index entries for sequences related to Recamán's sequence

Cf. A008343, A008344, A005132.
Cf. A076039, A003462, A076041, A076042, A057198.
Cf. A085059.
Cf. A238324.

a046901 n = a046901_list !! (n-1)
a046901_list = scanl1 (\u v -> if u > v then u - v else u + v) [1..]
-- Reinhard Zumkeller, Dec 07 2015, Jan 31 2013

A046901 := proc(n) option remember; if n = 1 then 1 else if A046901(n-1)>n then A046901(n-1)-n else A046901(n-1)+n; fi; fi; end;

a[1]=1;a[n_]:=a[n]=If[a[n-1]>n,a[n-1]-n,a[n-1]+n]; Table[a[i],{i,70}]  (* Harvey P. Dale, Apr 01 2011 *)
nxt[{n_,a_}]:={n+1,If[a>n+1,a-n-1,a+n+1]}; NestList[nxt,{1,1},70][[All,2]] (* Harvey P. Dale, Jun 01 2019 *)

a(n)=if(n<2,1,a(n-1)-if(sign(n-a(n-1))+1,-1,1)*n);

This is a concatenation S_0, S_1, S_2, ... where S_i = [b_0, b_1, ..., b_{k-1}], k=5*3^i, with b_0 = 1, b_{2j} = k+j, b_{2j+1} = (k+1)/2-j. E.g., S_0 = [1, 3, 6, 2, 7].
For any m>=1, for k such that 5*3^k+3>12m, a((5*3^k+3-12*m)/6)= m. For example, for k>=1, a((5*3^k-9)/6) = 1. - Benoit Cloitre, Oct 31 2002
a(n) = A008343(n+1) + 1. - Jon Maiga, Jul 09 2021

A237930 a(n) = 3^(n+1) + (3^n-1)/2.

Original entry on oeis.org

3, 10, 31, 94, 283, 850, 2551, 7654, 22963, 68890, 206671, 620014, 1860043, 5580130, 16740391, 50221174, 150663523, 451990570, 1355971711, 4067915134, 12203745403, 36611236210, 109833708631, 329501125894, 988503377683, 2965510133050, 8896530399151

Offset: 0

Philippe Deléham, Feb 16 2014

a(n-1) agrees with the graph radius of the n-Sierpinski carpet graph for n = 2 to at least n = 5. See A100774 for the graph diameter of the n-Sierpinski carpet graph.
The inverse binomial transform gives 3, 7, 14, 28, 56, ... i.e., A005009 with a leading 3. - R. J. Mathar, Jan 08 2020
First differences of A108765. The digital root of a(n) for n > 1 is always 4. a(n) is never divisible by 7 or by 12. a(n) == 10 (mod 84) for odd n. a(n) == 31 (mod 84) for even n > 0. Conjecture: This sequence contains no prime factors p == {11, 13, 23, 61 71, 73} (mod 84). - Klaus Purath, Apr 13 2020
This is a subsequence of A017209 for n > 1. See formula. - Klaus Purath, Jul 03 2020

			Ternary....................Decimal
10...............................3
101.............................10
1011............................31
10111...........................94
101111.........................283
1011111........................850
10111111......................2551
101111111.....................7654, etc.

Colin Barker, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Graph Radius.
Eric Weisstein's World of Mathematics, Sierpinski Carpet Graph.
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A000244, A003462, A005009, A005032 (first differences), A017209, A060816, A100774, A108765 (partial sums), A199109, A329774.

Cf. A024023, A029858, A057198, A116952, A171498.

[3^(n+1) + (3^n-1)/2: n in [0..40]]; // Vincenzo Librandi, Jan 09 2020

(* Start from Eric W. Weisstein, Mar 13 2018 *)
Table[(7 3^n - 1)/2, {n, 0, 20}]
(7 3^Range[0, 20] - 1)/2
LinearRecurrence[{4, -3}, {10, 31}, {0, 20}]
CoefficientList[Series[(3 - 2 x)/((x - 1) (3 x - 1)), {x, 0, 20}], x]
(* End *)

Vec((3 - 2*x) / ((1 - x)*(1 - 3*x)) + O(x^30)) \\ Colin Barker, Nov 27 2019

G.f.: (3-2*x)/((1-x)*(1-3*x)).
a(n) = A000244(n+1) + A003462(n).
a(n) = 3*a(n-1) + 1 for n > 0, a(0)=3. (Note that if a(0) were 1 in this recurrence we would get A003462, if it were 2 we would get A060816. - N. J. A. Sloane, Dec 06 2019)
a(n) = 4*a(n-1) - 3*a(n-2) for n > 1, a(0)=3, a(1)=10.
a(n) = 2*a(n-1) + 3*a(n-2) + 2 for n > 1.
a(n) = A199109(n) - 1.
a(n) = (7*3^n - 1)/2. - Eric W. Weisstein, Mar 13 2018
From Klaus Purath, Apr 13 2020: (Start)
a(n) = A057198(n+1) + A024023(n).
a(n) = A029858(n+2) - A024023(n).
a(n) = A052919(n+1) + A029858(n+1).
a(n) = (A000244(n+1) + A171498(n))/2.
a(n) = 7*A003462(n) + 3.
a(n) = A116952(n) + 2. (End)
a(n) = A017209(7*(3^(n-2)-1)/2 + 3), n > 1. - Klaus Purath, Jul 03 2020
E.g.f.: exp(x)*(7*exp(2*x) - 1)/2. - Stefano Spezia, Aug 28 2023

Showing 1-10 of 11 results. Next

cp's OEIS Frontend

A246360 a(1) = 1, then A007051 ((3^n)+1)/2 interleaved with A057198 (5*3^(n-1)+1)/2.

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A048673 Permutation of natural numbers: a(n) = (A003961(n)+1) / 2 [where A003961(n) shifts the prime factorization of n one step towards larger primes].

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A278055 Relative of Hofstadter Q-sequence: a(1) = 1, a(2) = 2, a(3) = 3, a(4) = 4, a(5) = 5; a(n) = a(n-a(n-1)) + a(n-a(n-2)) + a(n-a(n-3)) for n > 5.

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A176059 Periodic sequence: Repeat 3, 2.

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A060816 a(0) = 1; a(n) = (5*3^(n-1) - 1)/2 for n > 0.

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A193649 Q-residue of the (n+1)st Fibonacci polynomial, where Q is the triangular array (t(i,j)) given by t(i,j)=1. (See Comments.)

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A191450 Dispersion of (3*n-1), read by antidiagonals.

A254051 Square array A by downward antidiagonals: A(n,k) = (3 + 3^n(2floor(3*k/2) - 1))/6, n,k >= 1; read as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...