A084964 -id:A084964 - cp's OEIS frontend

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

-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

A209350 Number of initially rising meander words, where each letter of the cyclic n-ary alphabet occurs twice.

Original entry on oeis.org

1, 0, 1, 5, 9, 11, 16, 19, 25, 29, 36, 41, 49, 55, 64, 71, 81, 89, 100, 109, 121, 131, 144, 155, 169, 181, 196, 209, 225, 239, 256, 271, 289, 305, 324, 341, 361, 379, 400, 419, 441, 461, 484, 505, 529, 551, 576, 599, 625, 649, 676, 701, 729, 755, 784, 811, 841

Offset: 0

Alois P. Heinz, Mar 06 2012

In a meander word letters of neighboring positions have to be neighbors in the alphabet, where in a cyclic alphabet the first and the last letters are considered neighbors too.  The words are not considered cyclic here.
A word is initially rising if it is empty or if it begins with the first letter of the alphabet that can only be followed by the second letter in this word position.
a(n) is also the number of (2*n-1)-step walks on n-dimensional cubic lattice from (1,0,...,0) to (2,2,...,2) with positive unit steps in all dimensions such that the indices of dimensions used in consecutive steps differ by 1 or are in the set {1,n}.

			a(0) = 1: the empty word.
a(1) = 0 = |{ }|.
a(2) = 1 = |{abab}|.
a(3) = 5 = |{abacbc, abcabc, abcacb, abcbac, abcbca}|.
a(4) = 9 = |{ababcdcd, abadcbcd, abadcdcb, abcbadcd, abcbcdad, abcdabcd, abcdadcb, abcdcbad, abcdcdab}|.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Row n=2 of A209349.
First differences for n>2 give: A084964(n+1), A097065(n+3).
Cf. A245578.

a:= n-> `if`(n<3, (n-1)^2, (n/2+1)^2 -(n mod 2)*5/4):
seq(a(n), n=0..60);

LinearRecurrence[{2,0,-2,1},{1,0,1,5,9,11,16},60] (* Harvey P. Dale, Jan 02 2020 *)

G.f.: -(3*x^6-5*x^5-2*x^4+5*x^3+x^2-2*x+1) / ((x+1)*(x-1)^3).

a(n) = (n-1)^2 if n<3, a(n) = (n/2+1)^2 - (n mod 2)*5/4 else.

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

A029177 Expansion of 1/((1-x^2)(1-x^4)(1-x^5)(1-x^12)).

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 2, 1, 3, 2, 4, 2, 6, 3, 7, 4, 9, 6, 10, 7, 13, 9, 15, 10, 19, 13, 21, 15, 25, 19, 28, 21, 33, 25, 37, 28, 43, 33, 47, 37, 54, 43, 59, 47, 67, 54, 73, 59, 82, 67, 89, 73, 99, 82, 107, 89, 118, 99, 127, 107, 140, 118, 150, 127, 164, 140, 175, 150, 190, 164, 203

Offset: 0

N. J. A. Sloane

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,1,-1,-1,0,-1,0,1,1,0,-1,0,-1,-1,1,1,0,1,0,-1).

Cf. A029011(n) = a(2n) = a(2n+5).

M := Matrix(23, (i,j)-> if (i=j-1) or (j=1 and member(i, [2, 4, 5, 11, 12, 18, 19, 21])) then 1 elif j=1 and member(i, [6, 7, 9, 14, 16, 17, 23]) then -1 else 0 fi); a := n -> (M^(n))[1,1]; seq (a(n), n=0..70); # Alois P. Heinz, Jul 25 2008

CoefficientList[Series[1/((1-x^2)(1-x^4)(1-x^5)(1-x^12)),{x,0,70}],x] (* Harvey P. Dale, May 31 2012 *)

a(n)=if(n<-22,-a(-23-n),polcoeff(1/((1-x^2)*(1-x^4)*(1-x^5)*(1-x^12))+x*O(x^n),n))

G.f.: 1/((1-x^2)(1-x^4)(1-x^5)(1-x^12)).
a(n) = -a(-23 - n).
a(n) = A029011(A084964(n) - 2).

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

A168198 a(n) = 3*n - a(n-1) + 1 with n > 1, a(1)=1.

Original entry on oeis.org

1, 6, 4, 9, 7, 12, 10, 15, 13, 18, 16, 21, 19, 24, 22, 27, 25, 30, 28, 33, 31, 36, 34, 39, 37, 42, 40, 45, 43, 48, 46, 51, 49, 54, 52, 57, 55, 60, 58, 63, 61, 66, 64, 69, 67, 72, 70, 75, 73, 78, 76, 81, 79, 84, 82, 87, 85, 90, 88, 93, 91, 96, 94, 99, 97, 102, 100, 105, 103, 108

Offset: 1

Vincenzo Librandi, Nov 20 2009

Alternately add 5 and subtract 2, starting with 1. Apparently this was a test question: Find the next two numbers after 1,6,4,9,7,12,10. - N. J. A. Sloane, Dec 18 2010

			From _Muniru A Asiru_, Mar 20 2018: (Start)
For n = 2, a(2) = 3*2 - a[2-1] + 1 = 6 - a[1] + 1 = 6 - 1 + 1 = 6.
For n = 3, a(3) = 3*3 - a[3-1] + 1 = 9 - a[2] + 1 = 9 - 6 + 1 = 4.
For n = 4, a(4) = 3*4 - a[4-1] + 1 = 12 - a[3] + 1 = 12 - 4 + 1 = 9.
... (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
David Klein, What do the NAEP math tests really measure?, Notices Amer. Math. Soc., 58 (No. 1, 2011), 53-55.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A084964, A103889, A168199, A168200.

a:=[1];; for n in [2..80] do a[n]:=3*n-a[n-1]+1; od; a; # Muniru A Asiru, Mar 20 2018

I:=[1,6,4]; [n le 3 select I[n] else Self(n-1)+Self(n-2)-Self(n-3): n in [1..60]]; // Vincenzo Librandi, Feb 28 2012

a:= proc(n) option remember: if n = 1 then 1 elif n >= 2 then 3*n - procname(n-1) + 1 fi; end:
seq(a(n), n = 1..70); # Muniru A Asiru, Mar 20 2018

LinearRecurrence[{1,1,-1},{1,6,4},100] (* Vincenzo Librandi, Feb 28 2012 *)

a(n)=(6*n+5+7*(-1)^n)/4 \\ Charles R Greathouse IV, Jan 11 2012

a(n) = (6*n + 5 + 7*(-1)^n)/4. - Jon E. Schoenfield, Jun 24 2010
G.f.: x*(1+5*x-3*x^2)/((1+x)(1-x)^2). - Bruno Berselli, Feb 28 2012
Sum_{n>=1} (-1)^(n+1)/a(n) = 1/3 + Pi/(6*sqrt(3)) + log(3)/2. - Amiram Eldar, Feb 23 2023

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.

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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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A209350 Number of initially rising meander words, where each letter of the cyclic n-ary alphabet occurs twice.

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Maple

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

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A029177 Expansion of 1/((1-x^2)(1-x^4)(1-x^5)(1-x^12)).

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Maple

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

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A168198 a(n) = 3*n - a(n-1) + 1 with n > 1, a(1)=1.