A152832 -id:A152832 - cp's OEIS frontend

A084964 Follow n+2 by n. Also solution of a(n+2)=a(n)+1, a(0)=2, a(1)=0.

2, 0, 3, 1, 4, 2, 5, 3, 6, 4, 7, 5, 8, 6, 9, 7, 10, 8, 11, 9, 12, 10, 13, 11, 14, 12, 15, 13, 16, 14, 17, 15, 18, 16, 19, 17, 20, 18, 21, 19, 22, 20, 23, 21, 24, 22, 25, 23, 26, 24, 27, 25, 28, 26, 29, 27, 30, 28, 31, 29, 32, 30, 33, 31, 34, 32, 35, 33, 36, 34, 37, 35, 38, 36, 39

Offset: 0

Michael Somos, Jun 15 2003

Stefano Spezia, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).
Index entries for two-way infinite sequences

Cf. A030451, A097065, A152832.

Cf. A217764(1,n) = a(n+2).

import Data.List (transpose)
a084964 n = a084964_list !! n
a084964_list = concat $ transpose [[2..], [0..]]
-- Reinhard Zumkeller, Apr 06 2015

&cat[ [n+2, n]: n in [0..37] ]; // Klaus Brockhaus, Nov 23 2009

A084964:=n->floor(n/2)+1+(-1)^n; seq(A084964(k), k=0..100); # Wesley Ivan Hurt, Nov 08 2013

lst={}; a=1; Do[a=n-a; AppendTo[lst, a], {n, 0, 100}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 14 2008 *)
Table[{n,n-2},{n,2,40}]//Flatten (* or *) LinearRecurrence[{1,1,-1},{2,0,3},80] (* Harvey P. Dale, Sep 12 2021 *)

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

G.f.: (2-2x+x^2)/((1-x)(1-x^2)).
a(2n+1)=n. a(2n)=n+2. a(n+2)=a(n)+1. a(n)=-a(-3-n).
a(n) = floor(n/2) + 1 + (-1)^n. - Reinhard Zumkeller, Aug 27 2005
A112032(n)=2^a(n); A112033(n)=3*2^a(n); a(n)=A109613(n+2)-A052938(n). - Reinhard Zumkeller, Aug 27 2005
a(n) = n + 1 - a(n-1) (with a(0)=2). - Vincenzo Librandi, Aug 08 2010
a(n) = floor(n/2)*3 - floor((n-1)/2)*2. - Ross La Haye, Mar 27 2013
a(n) = 3*n - 3 - 5*floor((n-1)/2). - Wesley Ivan Hurt, Nov 08 2013
a(n) = (3 + 5*(-1)^n + 2*n)/4. - Wolfgang Hintze, Dec 13 2014
E.g.f.: ((4 + x)*cosh(x) - (1 - x)*sinh(x))/2. - Stefano Spezia, Jul 01 2023

First part of definition adjusted to match offset by Klaus Brockhaus, Nov 23 2009

A152833 a(0) = -3; a(n) = n-a(n-1).

Original entry on oeis.org

-3, 4, -2, 5, -1, 6, 0, 7, 1, 8, 2, 9, 3, 10, 4, 11, 5, 12, 6, 13, 7, 14, 8, 15, 9, 16, 10, 17, 11, 18, 12, 19, 13, 20, 14, 21, 15, 22, 16, 23, 17, 24, 18, 25, 19, 26, 20, 27, 21, 28, 22, 29, 23, 30, 24, 31, 25, 32, 26, 33, 27, 34, 28, 35, 29, 36, 30, 37, 31, 38, 32, 39, 33, 40, 34

Offset: 0

Vladimir Joseph Stephan Orlovsky, Dec 14 2008

sign

Cf. A084964, A152832.

lst={};a=3;Do[a=n-a;AppendTo[lst,a],{n,0,6!}];lst
RecurrenceTable[{a[0]==-3,a[n]==n-a[n-1]},a,{n,80}] (* Harvey P. Dale, May 16 2016 *)

a(n) = (2n+1-13*(-1)^n)/4. G.f.: -(3-7x+3x^2)/((1+x)(1-x)^2). - R. J. Mathar, Jan 08 2009

Indices added to definition, offset corrected by R. J. Mathar, Jan 08 2009

A152835 a(0) = -22; a(n) = n-a(n-1).

Original entry on oeis.org

-22, 23, -21, 24, -20, 25, -19, 26, -18, 27, -17, 28, -16, 29, -15, 30, -14, 31, -13, 32, -12, 33, -11, 34, -10, 35, -9, 36, -8, 37, -7, 38, -6, 39, -5, 40, -4, 41, -3, 42, -2, 43, -1, 44, 0, 45, 1, 46, 2, 47, 3, 48, 4, 49, 5, 50, 6, 51, 7, 52, 8, 53, 9, 54, 10, 55, 11, 56, 12

Offset: 0

Vladimir Joseph Stephan Orlovsky, Dec 14 2008

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

Cf. A084964, A152832, A152833.

[(1-89*(-1)^n+2*n)/4 : n in [0..100]]; // Wesley Ivan Hurt, Oct 28 2014

A152835:=n->(1-89*(-1)^n+2*n)/4: seq(A152835(n), n=0..100); # Wesley Ivan Hurt, Oct 28 2014

lst={};a=-22;Do[a=n-a;AppendTo[lst,a],{n,0,6!}];lst

Vec(-(22*x^2-45*x+22)/((x-1)^2*(x+1)) + O(x^100)) \\ Colin Barker, Oct 28 2014

a(n) = a(n-1)+a(n-2)-a(n-3). G.f.: -(22*x^2-45*x+22) / ((x-1)^2*(x+1)). - Colin Barker, Oct 28 2014

Indices added to definition, offset corrected - R. J. Mathar, Jan 08 2009

Name and Mathematica code corrected by Colin Barker, Oct 28 2014

A152836 a(0)=-1; a(n)=n^a(n-1)-a(n-1)^n.

Original entry on oeis.org

-1, 2, 0, 1, 3, -118

Offset: 0

Vladimir Joseph Stephan Orlovsky, Dec 14 2008

Sequence is finite because followup terms are fractions. - R. J. Mathar, Jun 19 2021

Cf. A084964, A152832, A152833, A152835

lst={};a=1;Do[a=n^a-a^n;AppendTo[lst,a],{n,0,5}];lst
nxt[{n_,a_}]:={n+1,(n+1)^a-a^(n+1)}; NestList[nxt,{0,-1},5][[All,2]] (* Harvey P. Dale, Jun 06 2022 *)

Definition corrected by N. J. A. Sloane, Jan 11 2009

Offset corrected. R. J. Mathar, Jun 19 2021

A152837 a(0)=-1; a(n)=Floor[n^a(n-1)-a(n-1)^n].

Original entry on oeis.org

-1, 2, 0, 1, 3, -118, -2699554153024, 1044826807337428519663920677057429215016680080584103502827667086054551857192770337767423

Offset: 0

Vladimir Joseph Stephan Orlovsky, Dec 14 2008

sign

			a(1) = 1^(-1)-(-1)^1 = 2. - R. J. Mathar, Jan 08 2009

Cf. A084964, A152832, A152833, A152835, A152836

lst={};a=1;Do[a=n^a-a^n;AppendTo[lst,Floor[a]],{n,0,7}];lst

Indices added to definition, offset corrected - R. J. Mathar, Jan 08 2009

A152838 a(0)=1; a(n)=Floor[a(n-1)^n-n^a(n-1)].

Original entry on oeis.org

1, 0, -1, -2, 3, 118, -199068134034785153195409370979964879499768447341765846329146257207125965412281784631240438088

Offset: 0

Vladimir Joseph Stephan Orlovsky, Dec 14 2008

sign

			a(1) = 1^1-1^1 = 0. - R. J. Mathar, Jan 08 2009

Cf. A084964, A152832, A152833, A152835, A152836, A152837

lst={};a=1;Do[a=a^n-n^a;AppendTo[lst,Floor[a]],{n,0,6}];lst

Indices added to definition, offset corrected - R. J. Mathar, Jan 08 2009

A198392 a(n) = (6n(3n+7)+(2n+13)*(-1)^n+3)/16 + 1.

Original entry on oeis.org

2, 4, 12, 18, 31, 41, 59, 73, 96, 114, 142, 164, 197, 223, 261, 291, 334, 368, 416, 454, 507, 549, 607, 653, 716, 766, 834, 888, 961, 1019, 1097, 1159, 1242, 1308, 1396, 1466, 1559, 1633, 1731, 1809, 1912, 1994, 2102, 2188, 2301, 2391, 2509, 2603, 2726, 2824, 2952

Offset: 0

Bruno Berselli, Oct 25 2011

For an origin of this sequence, see the triangular spiral illustrated in the Links section.

First bisection gives A117625 (without the initial term).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A152832 (by Superseeker).

Cf. sequences related to the triangular spiral: A022266, A022267, A027468, A038764, A045946, A051682, A062708, A062725, A062728, A062741, A064225, A064226, A081266-A081268, A081270-A081272, A081275 [incomplete list].

[(6*n*(3*n+7)+(2*n+13)*(-1)^n+3)/16+1: n in [0..50]];

LinearRecurrence[{1,2,-2,-1,1},{2,4,12,18,31},60] (* Harvey P. Dale, Jun 15 2022 *)

for(n=0, 50, print1((6*n*(3*n+7)+(2*n+13)*(-1)^n+3)/16+1", "));

G.f.: (2+2*x+4*x^2+2*x^3-x^4)/((1+x)^2*(1-x)^3).
a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5).
a(n)-a(-n-1) = A168329(n+1).
a(n)+a(n-1) = A102214(n).
a(2n)-a(2n-1) = A016885(n).
a(2n+1)-a(2n) = A016825(n).

A152839 a(0) = 0; a(n) = n! - a(n-1)!.

Original entry on oeis.org

0, 0, 1, 5, -96

Offset: 0

Vladimir Joseph Stephan Orlovsky, Dec 14 2008

Cf. A084964, A152832, A152833, A152835, A152836, A152837, A152838

Indices added to definition, offset corrected - R. J. Mathar, Jan 08 2009

Definition corrected by Georg Fischer, Jun 03 2025

A162938 A 2-based alternate sum over the numbers from 0 to the n-th nonprime.

Original entry on oeis.org

2, 5, 8, 11, 14, 25, 17, 20, 23, 40, 26, 29, 32, 55, 35, 38, 65, 41, 70, 44, 47, 50, 85, 53, 90, 56, 59, 100, 62, 65, 68, 115, 71, 74, 125, 77, 130, 80, 83, 140, 86, 145, 89, 92, 95, 160, 98, 165, 101, 104, 175, 107, 110, 113, 190, 116, 195, 119, 122, 205, 125, 128, 215

Offset: 1

Juri-Stepan Gerasimov, Jul 18 2009

nonn

Define a 2-based sum S(n) = Sum_{i=0..n} (2 - (-1)^i*i) = 2*n - (-1)^n*A152832(n).

a(n) is this sum evaluated at A141468(n).

			a(1) = 2 - 0*(-1)^0 = 2.
a(2) = 2 - 0*(-1)^0 + 2 - 1*(-1)^1 = 2 + 3 = 5.
a(3) = 2 - 0*(-1)^0 + 2 - 1*(-1)^1 + 2 - 2*(-1)^2 + 2 - 3*(-1)^3 + 2 - 4*(-1)^4 = 2 + 3 + 0 + 5 - 2 = 8.

Cf. A141468.

A152832 := proc(n) option remember; if n = 0 then -2; else n-procname(n-1) ; fi; end:
A141468 := proc(n) option remember ; local a; if n <=2 then n-1; else for a from procname(n-1)+1 do if not isprime(a) then RETURN(a); fi; od: fi; end:
A162938 := proc(n) local npr; npr := A141468(n) ; 2*npr-(-1)^npr*A152832(npr) ; end:
seq(A162938(n),n=1..100) ; # R. J. Mathar, Jul 21 2009

a(n) = Sum_{x=0..n-th nonprime} (2 - x*(-1)^x). - Juri-Stepan Gerasimov, Jul 28 2009

Definition edited by R. J. Mathar, Jul 21 2009

A239304 Triangle of permutations corresponding to the compressed square roots of Gray code * bit-reversal permutation (A239303).

Original entry on oeis.org

1, 1, 2, 3, 1, 2, 4, 2, 1, 3, 2, 5, 4, 1, 3, 2, 5, 6, 3, 1, 4, 6, 2, 3, 7, 5, 1, 4, 7, 3, 2, 6, 8, 4, 1, 5, 3, 8, 7, 2, 4, 9, 6, 1, 5, 3, 8, 9, 4, 2, 7, 10, 5, 1, 6, 9, 3, 4, 10, 8, 2, 5, 11, 7, 1, 6, 10, 4, 3, 9

Offset: 1

Tilman Piesk, Mar 14 2014

The symmetrical binary matrices corresponding to the rows of A239303 can be interpreted as adjacency matrices of undirected graphs. These graphs are chains where one end is connected to itself, so they can be interpreted as permutations. The end connected to itself is always the first element of the permutation, i.e., on the left side of the triangle.
Columns of the square array:
T(m,1) = A008619(m) = 1,2,2,3,3...
T(m,2) = 1,1,1...
T(m,3) = A028242(m+3) = 3,2,4,3,5,4,6,5,7,6,8,7,9,8,10,9,11,10,12...
T(m,4) = m+3 = 4,5,6...
T(m,5) = A084964(m+4) = 2,5,3,6,4,7,5,8,6,9,7,10,8,11,9,12,10,13...
T(m,6) = 2,2,2...
T(m,7) = A168230(m+5) = 6,3,7,4,8,5,9,6,10,7,11,8,12,9,13,10,14...
T(m,8) = m+6 = 7,8,9...
T(m,9) = A152832(m+9) = 3,8,4,9,5,10,6,11,7,12,8,13,9,14,10,15...
T(m,10) = 3,3,3...
Diagonals of the square array:
T(n,n) = a(A001844(n)) = 1,1,4,7,4,2,9,14,7,3,14,21,10,4,19,28,13,5,24...
T(n,2n-1) = a(A064225(n)) = 1,2,3...
T(2n-1,n) = a(A081267(n)) = 1,1,5,10,6,2,12,21,11,3,19,32,16,4,26,43,21...

			Triangular array begins:
  1
  1 2
  3 1 2
  4 2 1 3
  2 5 4 1 3
  2 5 6 3 1 4
Square array begins:
  1 1 3 4 2 2
  2 1 2 5 5 2
  2 1 4 6 3 2
  3 1 3 7 6 2
  3 1 5 8 4 2
  4 1 4 9 7 2
Row 5 of A239303 is the vector (12,18,1,17,10), which corresponds to the following binary matrix:
  0 0 1 1 0
  0 1 0 0 1
  1 0 0 0 0
  1 0 0 0 1
  0 1 0 1 0
Interpreted as an adjacency matrix it describes the following graph, where each number is connected to its neighbors, and only the 2 is connected to itself:
  2 5 4 1 3
This is row 5 of the triangular array.

Tilman Piesk, First 140 rows of the triangle, flattened
Tilman Piesk, Sequency ordered Walsh matrix (Wikiversity)
Tilman Piesk, Calculation in MATLAB

Cf. A239303, A028242, A084964, A168230, A152832.

cp's OEIS Frontend

A084964 Follow n+2 by n. Also solution of a(n+2)=a(n)+1, a(0)=2, a(1)=0.

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A152833 a(0) = -3; a(n) = n-a(n-1).

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A152835 a(0) = -22; a(n) = n-a(n-1).

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A152836 a(0)=-1; a(n)=n^a(n-1)-a(n-1)^n.

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A152837 a(0)=-1; a(n)=Floor[n^a(n-1)-a(n-1)^n].

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A152838 a(0)=1; a(n)=Floor[a(n-1)^n-n^a(n-1)].

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A198392 a(n) = (6*n*(3*n+7)+(2*n+13)*(-1)^n+3)/16 + 1.

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Magma

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A152839 a(0) = 0; a(n) = n! - a(n-1)!.

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A198392 a(n) = (6n(3n+7)+(2n+13)*(-1)^n+3)/16 + 1.