Further Transformations of Integer Sequences

This page was created by Christian G. Bower and is a sub-page of the On-Line Encyclopedia of Integer Sequences, maintained by N. J. A. Sloane.

File:BluePinLeft.gif Keywords: AFJ, AFK, AGJ, AGK, AIJ, BFJ, BFK, BGJ, BGK, BHJ, BHK, BIJ, BIK, CFJ, CFK, CGJ, CGK, CHJ, CHK, CIJ, CIK, DFJ, DFK, DGJ, DGK, DHJ, DHK, DIJ, DIK, EFJ, EFK, EGJ.

Table of Contents

Definition.
Algorithms.
Catalogue of Sequences.
Back to Integer Sequences.

Part 1: Definition

This is a generalization of transforms that count the ways objects can be partitioned.

Say we have boxes of different colors and sizes.

The sequence {a_n;n>=1} represents the number of colors a box holding n balls can be. The transformed sequence {b_n;n>=1} represents the number of ways we can have a collection of boxes so that the total number of balls is n, subject to the following rules.

**The boxes are ordered in one of the following ways:**
A. Linear (ordered) The boxes are in a line from beginning to end. B. Linear with turning over (reversible) The boxes are in a line that can be read in either direction. C. Circular (necklace) The boxes are in a circle. D. Circular with turning over (bracelet) The boxes are in a circle that can be read in either direction. E. None (unordered) The order of the boxes is not important.

**One of the following distinctness rules applies:**
F. Size No two boxes are the same size. G. Element No two boxes are the same size and color. H. Identity Any two boxes can be distinguished by size, color and position. I. None (indistinct) No restriction.

Distinctness H (identity) has different implications depending on the chosen order.

If order A is chosen, distinctness H is the same as distinctness I.
If order B is chosen, the boxes cannot form a palindrome of length greater than one.
Red 1 Blue 2 Red 1 is not allowed.
If order C is chosen, the sequence of boxes is aperiodic. It cannot be a repitition of a shorter subsequence.
Red 1 Blue 2 Red 1 Blue 2 is not allowed.
If order D is chosen, the boxes are aperiodic and cannot be a palindrome of length greater than two.
If order E is chosen, distinctness H is the same as distinctness G.

**One of the following labelling rules applies:**
J. Labeled The balls in the boxes are labeled. K. Unlabeled The balls in the boxes are not labeled.

Each transform is identified by a 3 letter code, e.g. BGJ to represent linear order with turning over, each object distinct, labeled.
An X is a wild card as in CXK, unlabeled necklace transforms.

AIK is the transform INVERT.
EGK is the transform WEIGH.
EIJ is the transform EXP.
EIK is the transform EULER.

There are 5×4×2=40 of these transforms.

However, the AHX and EHX transforms are redundant, leaving 36. Four of them are named. As far as I know, the other 32 are not. The new and old sequence listed illustrate the 32 new transforms.

Terminology:

XXX_k means the transform XXX with exactly k boxes.
These are denoted by XXX[k] in the On-Line Encyclopedia of Integer Sequences.
AIK₂ is the transform CONV.
Bracelet means necklace that can be turned over. More information about necklaces.
Compound windmill is a rooted planar tree where the sub-rooted tree extending from a node can be rotated independently of the rest of the tree. Much like some children's toys or carnival rides. Compound windmills can be dyslexic.
Dyslexic planar tree is a planar tree where each sub-rooted tree extending from a node can be read from left to right or right to left. It can be thought of as viewed by an observer who does not know left from right or as sub-rooted trees that can be turned around independent of the rest of the tree.
Eigensequence means a sequence that is stable under a given transform or is modified in some simple way. Eigensequences are covered in detail in: M. Bernstein & N. J. A. Sloane, Some canonical sequences of integers, Linear Algebra and its Applications, 226-228 (1995), 57-72.A000081, rooted trees, 1,1,2,4,9,20,48,115... is an eigensequence of the transform EULER. because the transformed sequence, 1,2,4,9,20,48,115,286,..., is the original sequence shifted left one place.
Identity bracelet means bracelet where every bead is distinguished by position and color, i.e. a bracelet generated by the transform DHK.

Part 2: Algorithms

a_n is the input sequence.

b_n is the output sequence.

A(x) is the generating function of a_n.

B(x) is the generating function of b_n.

(XXX a)_n = sum{k=1 to n} (XXX_k a)_n

MÖBIUS · XXX refers to the Möbius transform of the sequence transformed by XXX. Similarly for MÖBIUS^-1 · XXX. However, (MÖBIUS · XXX)_k and (MÖBIUS^-1 · XXX)_k are defined as follows:

(MÖBIUS · XXX)_ka_n = sum{d|k and d|n} (µ(d) × XXX_k/da_n/d)

(MÖBIUS^-1 · XXX)_ka_n = sum{d|k and d|n} (XXX_k/da_n/d)

AIK = INVERT
B(x) = A(x) / (1-A(x))

AIK_k
B(x) = A(x)^k

LPAL_k (Linear palindrome)
If n, k even: b_n = (AIK_k/2a)_n/2
If n odd, k even: b_n = 0
If n even, k odd: b_n = sum{i>0 and i<n/2} (a_2i × (AIK_(k-1)/2a)_n/2-i)
if n,k odd: b_n = sum{i>0 and i<n/2} (a_2i-1 × (AIK_(k-1)/2a)_(n+1)/2-i)

BIK_k
b_n = ((AIK_ka)_n + (LPAL_ka)_n) / 2

BHK_k
k=1: b_n = a_n
k>1: b_n = ((AIK_ka)_n - (LPAL_ka)_n) / 2

CHK_k
b_n = (MÖBIUS · AIK)_ka_n / n

CIK
CIK = MÖBIUS^-1 · CHK

CPAL_k (Circular palindrome)
CPAL₁ = IDENTITY
CPAL₂ = CIK₂
k>2:
If n, k even: b_n = (I+J)/2+K+L+M where:
(No boxes joined)
I=(AIK_k/2a)_n/2
(2 boxes joined are identical)
J=sum{i=1 to n/2}(AIK_k/2-1a)_(n-2i)/2
(2 boxes joined are even and different sizes)
K=sum{i,j even, j>i, i+j<n} (a_i × a_j × (AIK_k/2-1a)_(n-i-j)/2)
(2 boxes joined are odd and different sizes)
L=sum{i,j odd, j>i, i+j<n} (a_i × a_j × (AIK_k/2-1a)_(n-i-j)/2)
(2 boxes joined are the same size and different colors)
M=sum{i>0 and i<n/2} ((a_i²-a_i)/2 × (AIK_k/2-1a)_(n-2i)/2)
If n odd, k even:
b_n = sum{i odd, j even, i+j<n} (a_i × a_j × ((AIK_k/2-1a)_(n-i-j)/2)
If n even, k odd:
b_n = sum{i>0 and i<n/2} (a_2i × (AIK_(k-1)/2a)_n/2-i)
if n,k odd:
b_n = sum{i>0 and i<n/2} (a_2i-1 × (AIK_(k-1)/2a)_(n+1)/2-i)

DIK_k
b_n = ((CIK_ka)_n + (CPAL_ka)_n) / 2

DHK_k
DHK₁ = IDENTITY
DHK₂ = CHK₂
For k>2:
DHK_k = (MÖBIUS · (CIK - CPAL)/2)_k

If EXX is one of: {EFJ, EFK, EGJ, EGK, EIJ} then:
AXX_k = k! × EXX_k
BXX_k = max(1,k!/2) × EXX_k
CXX_k = (k-1!) × EXX_k
DXX_k = max(1,(k-1)!/2) × EXX_k

To calculate (EFX_ka)_n, enumerate the distinct partitions of n into k parts as terms of the following form:
p₁+p₂+...+p_k
Sum the terms calculated as follows:
EFJ_k: prod{i=1 to k}a_{p_i} × n! / prod{i=1 to k}p_i!
EFK_k: prod{i=1 to k}a_{p_i}

EFK can also be calculated as:
B(x)=prod{k=1 to infinity}(1+a_kx^k).

To calculate (AIJ_ka)_n, (BHJ_ka)_n, (CHJ_ka)_n or (EGX_ka)_n, enumerate the partitions of of n into k parts as terms of the following form:
p₁q₁+p₂q₂+...+p_jq_j where all the p_i's are distinct.
Sum the terms calculated as follows:
AIJ_k: prod{i=1 to j}a_{p_i}^q_i × n! × k! / ((prod{i=1 to j}p_i!q_i) × (prod{i=1 to j}q_i!))
BHJ_k:
term₁ = prod{i=1 to j}a_{p_i}^q_i × k! / (prod{i=1 to j}q_i!)
term₂ = prod{i=1 to j}a_{p_i}^[q_i/2] × [k/2]! / (prod{i=1 to j}[q_i/2]!)
If more than 1 q_i is odd: term₃ = term₁
otherwise: term₃ = term₁ - term₂
term = term₃ × n! / prod{i=1 to j}p_i!q_i / 2
CHJ_k:
term₂ = sum{d|q_m for all m} (µ(d) × prod{i=1 to j}a_{p_i}^[q_i/d] × [k/d]! / (prod{i=1 to j}[q_i/d]!))
term = term₂ × n! / prod{i=1 to j}p_i!q_i / k
EGJ_k: prod{i=1 to j}C(a_{p_i},q_i) × n! / prod{i=1 to j}p_i!q_i
EGK_k: prod{i=1 to j}C(a_{p_i},q_i)

DHJ:
Work is in progress.

Part 3: Catalogue of sequences

This table identifies a formula for each sequence, usually based on one of the transforms. This should provide a convenient way to browse the sequences and see how the transforms apply to a broad class of mathematics.

The base sequences:

These transforms have been applied to one of the base sequences defined in the following table or to sequences in the On-line Encyclopedia of Integer Sequences, identified by number.

s¹, s², s³...	s^k₁ = k, s^k_n=0 for n>1
all¹, all², all³,...	all^k_n = k for all n
codd (characteristic of odd)	codd_n = 1 if n is odd, 0 otherwise
noone	noone₁=0, noone_n=1 for n>1
twoone	twoone₁=2, twoone_n=1 for n>1
iden	iden_n=n
odd	odd_n=2n-1
even	even_n=2n

If T is a transform:

Left(n;k₁, k₂,..., k_n)T is the eigensequence that shifts left n places under T and has a_i=k_i for 1<=i<=n.
M2(n)T is the eigensequence that doubles the terms whose indices are greater than 1 under T.

AFJ sequences

A032000	AFJ all²
A032001	AFJ twoone
A032002	AFJ iden
A032003	AFJ odd
A032004	Left(1;1)AFJ

AFK sequences

A032005	AFK all²
A032006	AFK twoone
A032007	AFK iden
A032008	AFK odd
A032009	Left(1;1)AFK
A032010	(CFK A032009 )_n-1

AGJ sequences

A032011	AGJ all¹
A032012	AGJ codd
A032013	AGJ noone
A032014	AGJ all²
A032015	AGJ twoone
A032016	AGJ iden
A032017	AGJ odd
A032018	Left(1;1)AGJ
A032019	M2(2)AGJ

AGK sequences

A032020	AGK all¹
A032021	AGK codd
A032022	AGK noone
A032023	AGK all²
A032024	AGK twoone
A032025	AGK iden
A032026	AGK odd
A032027	Left(1;1)AGK
A032028	(CGK A032027 )_n-1
A032029	Left(2;1,1)AGK
A032030	M2(2)AGK

AIJ sequences

A000142	AIJ s¹
A000165	AIJ s²
A032031	AIJ s³
A000670	AIJ all¹
A000918	AIJ₂ all¹
A001117	AIJ₃ all¹
A000919	AIJ₄ all¹
A001118	AIJ₅ all¹
A000920	AIJ₆ all¹
A006154	AIJ codd
A032032	AIJ noone
A004123	AIJ all²
A006155	AIJ twoone
A032033	AIJ all³
A006153	AIJ iden
A000354	AIJ odd
A001147	Left(1;1)AIJ
A032034	Left(1;2)AIJ
A032035	Left(2;1,1)AIJ
A032036	Left(3;1,1,1)AIJ
A032037	M2(1)AIJ

BFJ sequences

A032038	BFJ all²
A032039	BFJ twoone
A032040	BFJ iden
A032041	BFJ odd
A032042	Left(1;1)BFJ

BFK sequences

A032043	BFK all²
A032044	BFK twoone
A032045	BFK iden
A032046	BFK odd
A032047	Left(1;1)BFK
A032048	(CFK A032047 )_n-1

BGJ sequences

A032049	BGJ all¹
A032050	BGJ codd
A032051	BGJ noone
A032052	BGJ all²
A032053	BGJ twoone
A032054	BGJ iden
A032055	BGJ odd
A032056	Left(1;1)BGJ
A032057	M2(2)BGJ

BGK sequences

A032058	BGK all¹
A032059	BGK codd
A032060	BGK noone
A032061	BGK all²
A032062	BGK twoone
A032063	BGK iden
A032064	BGK odd
A032065	Left(1;1)BGK
A032066	(CGK A032065 )_n-1
A032067	Left(2;1,1)BGK
A032068	M2(2)BGK

BHJ sequences

A032069	BHJ s²
A032070	BHJ s³
A032071	BHJ s⁴
A032072	BHJ s⁵
A032073	BHJ all¹
A032074	BHJ codd
A032075	BHJ noone
A032076	BHJ all²
A032077	BHJ twoone
A032078	BHJ all³
A032079	BHJ iden
A032080	BHJ odd
A032081	Left(1;1)BHJ
A032082	Left(1;2)BHJ
A032083	Left(2;1,1)BHJ
A032084	M2(2)BHJ

BHK sequences

A032085	BHK s²
A032086	BHK s³
A032087	BHK s⁴
A032088	BHK s⁵
A032089	BHK codd
A032090	BHK noone
A002620	(BHK₃ all¹)_n+2
A006584	(BHK₄ all¹)_n+2
A032091	BHK₅ all¹
A032092	BHK₆ all¹
A032093	BHK₇ all¹
A032094	BHK₈ all¹
A032095	(BHK_n all¹)_2n-1
A032096	BHK all²
A032097	BHK twoone
A032098	BHK all³
A032099	BHK iden
A032100	BHK odd
A032101	Left(1;1)BHK
A032102	(DHK A032101 )_n-1
A032103	Left(1;2)BHK
A032104	Left(1;1,1)BHK
A032105	M2(2)BHK
A032106	(BHK_n all¹)_2n

BIJ sequences

A001710	BIJ s¹
A032107	BIJ s²
A032108	BIJ s³
A032109	BIJ all¹
A009568	(-1)ⁿ⁺¹ × BIJ codd
A032110	BIJ noone
A032111	BIJ all²
A032112	BIJ twoone
A032113	BIJ all³
A032114	BIJ iden
A032115	BIJ odd
A032116	Left(1;1)BIJ
A032117	Left(1;2)BIJ
A032118	Left(2;1,1)BIJ
A032119	M2(1)BIJ

BIK sequences

A005418	(BIK s²)_n-1
A005418	BIK all¹
A032120	BIK s³
A032121	BIK s⁴
A032122	BIK s⁵
A001224	(BIK codd)_n+1
A001224	(BIK noone)_n+2
A002620	(BIK₃ all¹)_n+1
A005993	(BIK₄ all¹)_n+4
A005994	(BIK₅ all¹)_n+5
A005995	(BIK₆ all¹)_n+6
A018210	(BIK₇ all¹)_n+7
A018211	(BIK₈ all¹)_n+8
A018212	(BIK₉ all¹)_n+9
A018213	(BIK₁₀ all¹)_n+10
A018214	(BIK₁₁ all¹)_n+11
A032123	(BIK_n all¹)_2n-1
A005654	(BIK_n all¹)_2n
A005656	(BIK_n-3 all¹)_2n-3
A032124	BIK all²
A032125	BIK all³
A005207	BIK twoone
A032126	BIK iden
A032127	BIK odd
A032128	Left(1;1)BIK
A032129	(DIK A032128 )_n-1
A032130	Left(1;2)BIK
A032131	Left(2;1,1)BIK
A032132	M2(1)BIK
A032133	M2(2)BIK

CFJ sequences

A032134	CFJ all²
A032135	CFJ twoone
A032136	CFJ iden
A032137	CFJ odd
A032138	Left(1;1)CFJ

CFK sequences

A032139	CFK all²
A032140	CFK twoone
A032141	CFK iden
A032142	CFK odd
A032143	Left(1;1)CFK

CGJ sequences

A032144	CGJ all¹
A032145	CGJ codd
A032146	CGJ noone
A032147	CGJ all²
A032148	CGJ twoone
A032149	CGJ iden
A032150	CGJ odd
A032151	Left(1;1)CGJ
A032152	M2(2)CGJ

CGK sequences

A032153	CGK all¹
A032154	CGK codd
A032155	CGK noone
A032156	CGK all²
A032157	CGK twoone
A032158	CGK iden
A032159	CGK odd
A032160	Left(1;1)CGK
A032161	Left(1;2)CGK
A032162	Left(2;1,1)CGK
A032163	M2(2)CGK

CHJ sequences

A032321	CHJ s²
A032322	CHJ s³
A032323	CHJ s⁴
A032264	CHJ s⁵
A032325	CHJ all¹
A032326	CHJ codd
A032327	CHJ noone
A032328	CHJ all²
A032329	CHJ twoone
A032330	CHJ all³
A032331	CHJ iden
A032332	CHJ odd
A032333	Left(1;1)CHJ
A032334	Left(1;2)CHJ
A032335	Left(2;1,1)CHJ
A032336	M2(2)CHJ

CHK sequences

A001037	CHK s²
A001037	(CHK all¹) + s¹
A027376	CHK s³
A027376	(CHK all²) + s¹
A027376	(CHK odd) + s²
A027377	CHK s⁴
A027377	(CHK all³) + s¹
A001692	CHK s⁵
A027378	CHK s⁵
A032164	CHK s⁶
A001693	CHK s⁷
A027379	CHK s⁷
A027380	CHK s⁸
A027381	CHK s⁹
A032165	CHK s¹⁰
A032166	CHK s¹¹
A032167	CHK s¹²
A006206	(CHK codd) + CHAR({2})
A006206	(CHK noone) + s¹
A001840	(CHK₃ all¹)_n+4
A006918	(CHK₄ all¹)_n+4
A011795	(CHK₅ all¹)_n+1
A011796	(CHK₆ all¹)_n+6
A011797	(CHK₇ all¹)_n+1
A031164	(CHK₈ all¹)_n+9
A011845	CHK₉ all¹
A032168	CHK₁₀ all¹
A032169	CHK₁₁ all¹
A000108	(CHK_n+1 all¹)_2n+1
A022553	(CHK_n+1 all¹)_2n+2
A022553	(CHK A000108 )_n-1
A032170	CHK iden
A032170	CHK twoone + s¹
A032171	Left(1;1)CHK
A032172	Left(1;2)CHK
A032173	Left(2;1,1)CHK
A032174	M2(2)CHK
A032175	CHK A004111
A032176	WEIGH A032175
A032177	A032176 - A004111
A032178	WEIGH A032177

CIJ sequences

A000142	(CIJ s¹)_n+1
A000165	(CIJ s²)_n+1 × 2
A032179	CIJ s³
A000629	CIJ all¹
A000225	(CIJ₂ all¹)_n+1
A028243	CIJ₃ all¹
A028244	CIJ₄ all¹
A028245	CIJ₅ all¹
A032180	CIJ₆ all¹
A003704	(-1)ⁿ⁺¹ × (CIJ codd)
A032181	CIJ noone
A027882	CIJ all²
A032182	CIJ twoone
A032183	CIJ all³
A009444	(-1)ⁿ⁺¹ × (CIJ iden)
A032184	CIJ odd
A029768	Left(1;1)CIJ
A032185	Left(1;2)CIJ
A032186	Left(2;1,1)CIJ
A032187	Left(3;1,1,1)CIJ
A032188	M2(1)CIJ

CIK sequences

A000031	CIK s²
A000031	(CIK all¹) + all¹
A008965	CIK all¹
A008965	(CIK s²) - all¹
A001867	CIK s³
A001867	(CIK all²) + all¹
A001868	CIK s⁴
A001868	(CIK all³) + all¹
A001869	CIK s⁵
A001869	(CIK all⁴) + all¹
A032189	CIK codd
A032190	CIK noone
A000358	(CIK noone) + all¹
A007997	(CIK₃ all¹)_n+3
A008610	(CIK₄ all¹)_n+4
A008646	(CIK₅ all¹)_n+5
A032191	CIK₆ all¹
A032192	CIK₇ all¹
A032193	CIK₈ all¹
A032194	CIK₉ all¹
A032195	CIK₁₀ all¹
A032196	CIK₁₁ all¹
A032197	CIK₁₂ all¹
A000108	(CHK_n+1 all¹)_2n+1
A003239	(CIK_n-1 all¹)_2n-2
A003239	(CIK A000108 _n-1)_n-1
A005594	CIK twoone
A032198	CIK iden
A032199	CIK odd
A032200	Left(1;1)CIK
A032201	Left(1;2)CIK
A032202	Left(2;1,1)CIK
A032203	M2(1)CIK
A032204	M2(2)CIK
A002861	CIK A000081
A027852	CIK₂ A000081
A029852	CIK₃ A000081
A029853	CIK₄ A000081
A029868	CIK₅ A000081
A029869	CIK₆ A000081
A029870	CIK₇ A000081
A029871	CIK₈ A000081
A032205	CIK₉ A000081
A032206	CIK₁₀ A000081
A032207	CIK₁₁ A000081
A032208	CIK₁₂ A000081

DFJ sequences

A032209	DFJ all²
A032210	DFJ twoone
A032211	DFJ iden
A032212	DFJ odd
A032213	Left(1;1)DFJ

DFK sequences

A032214	DFK all²
A032215	DFK twoone
A032216	DFK iden
A032217	DFK odd
A032218	Left(1;1)DFK

DGJ sequences

A032219	DGJ all¹
A032220	DGJ codd
A032221	DGJ noone
A032222	DGJ all²
A032223	DGJ twoone
A032224	DGJ iden
A032225	DGJ odd
A032226	Left(1;1)DGJ
A032227	M2(2)DGJ

DGK sequences

A032228	DGK all¹
A032229	DGK codd
A032230	DGK noone
A032231	DGK all²
A032232	DGK twoone
A032233	DGK iden
A032234	DGK odd
A032235	Left(1;1)DGK
A032236	Left(1;2)DGK
A032237	Left(2;1,1)DGK
A032238	M2(2)DGK

DHJ sequences

A032337	DHJ s²
A032338	DHJ s³
A032339	DHJ s⁴
A032340	DHJ s⁵

DHK sequences

A032239	DHK s²
A032240	DHK s³
A032241	DHK s⁴
A032242	DHK s⁵
A032243	DHK codd
A032244	DHK noone
A032245	DHK all¹
A001399	(DHK₃ all¹)_n+6
A018845	(DHK₃ all¹)_n+6
A026809	(DHK₃ all¹)_n+3
A008804	(DHK₄ all¹)_n+7
A032246	DHK₅ all¹
A032247	DHK₆ all¹
A032248	DHK₇ all¹
A032249	DHK₈ all¹
A032250	(DHK_n all¹)_2n
A032251	DHK all²
A032252	DHK twoone
A032253	DHK all³
A032254	DHK iden
A032255	DHK odd
A032256	Left(1;1)DHK
A032257	Left(1;2)DHK
A032258	Left(2;1,1)DHK
A032259	M2(2)DHK
A032260	(DHK_n all¹)_2n-1

DIJ sequences

A001710	(DIJ s¹)_n+1
A000165	(DIJ s²)_n+1 - s²
A032261	DIJ s³
A032262	DIJ all¹
A000225	(DIJ₂ all¹)_n+1
A000392	DIJ₃ all¹
A032263	DIJ₄ all¹
A032264	DIJ codd
A032265	DIJ noone
A032266	DIJ all²
A032267	DIJ twoone
A032268	DIJ all³
A032269	DIJ iden
A032270	DIJ odd
A032271	Left(1;1)DIJ
A032272	Left(1;2)DIJ
A032273	Left(2;1,1)DIJ
A032274	M2(1)DIJ

DIK sequences

A000029	DIK s²
A000029	(DIK all¹) + all¹
A027671	DIK s³
A032275	DIK s⁴
A032276	DIK s⁵
A032277	DIK codd
A032278	DIK noone
A001399	(DIK₃ all¹)_n+3
A018845	(DIK₃ all¹)_n+3
A026809	DIK₃ all¹
A005232	DIK₄ all¹
A032279	DIK₅ all¹
A005513	DIK₆ all¹
A032280	DIK₇ all¹
A005514	DIK₈ all¹
A032281	DIK₉ all¹
A005515	DIK₁₀ all¹
A032282	DIK₁₁ all¹
A005516	DIK₁₂ all¹
A005648	(DIK_n all¹)_2n
A007123	(DIK_n all¹)_2n-1
A032283	DIK all²
A032284	DIK all³
A032285	DIK all⁴
A032286	DIK all⁵
A005595	DIK twoone
A032287	DIK iden
A032288	DIK odd
A032289	Left(1;1)DIK
A032290	Left(1;2)DIK
A032291	Left(2;1,1)DIK
A032292	M2(1)DIK
A032293	M2(2)DIK
A001371	MÖBIUS A000029
A032294	MÖBIUS A027671
A032295	MÖBIUS A032275
A032296	MÖBIUS A032276

EFJ sequences

A032297	EFJ all²
A032298	EFJ twoone
A032299	EFJ iden
A032300	EFJ odd
A032301	Left(1;1)EFJ

EFK sequences

A032302	EFK all²
A032303	EFK twoone
A022629	EFK iden
A032304	EFK odd
A032305	Left(1;1)EFK
A032306	Left(1;2)EFK
A032307	Left(2;1,1)EFK
A032308	EFK all³
A032309	EFK even

EGJ sequences

A007837	EGJ all¹
A032310	EGJ codd
A032311	EGJ noone
A032312	EGJ all²
A032313	EGJ twone
A032314	EGJ all³
A032315	EGJ iden
A032316	EGJ odd
A032317	Left(1;1)EGJ
A032318	Left(1;2)EGJ
A032319	Left(2;1,1)EGJ
A032320	M2(2)EGJ

More Transformations of Integer Sequences

Further Transformations of Integer Sequences

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