A108619 -id:A108619 - cp's OEIS frontend

A272693 a(0)=0; thereafter, a(n) = (A108618(n-1)-3*A108619(n))/2.

0, -1, -2, -5, -4, -3, -3, 0, 5, 2, -3, -6, -7, -5, 2, 9, 4, -5, -10, -9, -3, 6, 11, 5, -10, -21, -12, 5, 14, 9, -5, -19, -17, -4, 15, 16, 1, -15, -21, -12, 11, 20, 9, -11, -25, -17, 2, 21, 16, -5, -22, -21, -3, 18, 23, 5, -22, -33, -15, 18, 35, 17, -22, -45, -24, 17, 38, 21, -17, -43, -29, 8, 39

Offset: 0

N. J. A. Sloane, Jun 07 2016

sign

N. J. A. Sloane, Table of n, a(n) for n = 0..10000

Cf. A108618, A108619.

M:=1000;
a:=Array(0..M,0); # A108618 (with different offset)
b:=Array(0..M,0); # A108619 (with different offset)
c:=Array(0..M,0); # A272693
f:=n->sign(n)*(n mod 2);
a[0]:=0; b[0]:=0; c[0]:=0;
for n from 1 to M do
b[n]:=1+(a[n-1]+b[n-1])/2;
a[n]:=1+c[n-1]+f(c[n-1])+3*f(b[n]-1);
c[n]:=(a[n]-3*b[n])/2;
od:
[seq(a[n],n=0..M)];
[seq(b[n],n=0..M)];
[seq(c[n],n=0..M)];

A108620 2a(n) = A108618(n) + 3A108619(n).

Original entry on oeis.org

2, 4, 4, 2, 0, -3, -6, -4, 2, 6, 6, 2, -5, -10, -6, 4, 10, 8, 0, -9, -12, -4, 11, 20, 12, -6, -16, -10, 6, 19, 17, 1, -16, -18, -2, 16, 21, 9, -12, -22, -10, 12, 25, 17, -5, -22, -18, 4, 22, 20, 0, -21, -24, -4, 23, 32, 12, -21, -36, -16, 23, 44, 24, -18, -40, -22, 18, 43, 29, -11, -40, -30, 10, 40, 32, -6, -39, -35, 3, 40, 40, 2, -36

Offset: 0

Creighton Dement, Jun 22 2005

Floretion Algebra Multiplication Program, FAMP Code: 1vessum(*)seq[ + .5'i + .5'j + .5'k + .5e]

Creighton Dement, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Colored scatterplot of a(n) for n = 0..10000 (where the color is function of n mod 6)

Cf. A108618, A108619.

a[0] = b[0] = 1;
f[n_] := Sign[n]*Mod[n, 2];
a[n_] := a[n] = (1/2)*(a[n-1] - 3*b[n-1]) + 3*f[(1/2)*(a[n-1] + b[n-1])] + f[(1/2)*(a[n-1] - 3*b[n-1])] + 1;
b[n_] := b[n] = (1/2)*(a[n-1] + b[n-1]) + 1;
A108620 = Table[(a[n] + 3*b[n])/2, {n, 0, 100}] (* Jean-François Alcover, Feb 25 2015, after Benoit Jubin *)

A108618 A quaternion-generated sequence calculated using the rules given in the comment box with initial seed x = .5'i + .5'j + .5'k + .5e; version: "tes".

Original entry on oeis.org

1, 2, -1, -2, -3, -6, -6, 1, 4, 3, 0, -5, -10, -8, 3, 8, 5, -2, -9, -12, -6, 7, 16, 10, -9, -18, -11, 4, 15, 14, -2, -16, -20, -3, 14, 17, 6, -12, -24, -11, 10, 21, 14, -8, -22, -20, 3, 20, 17, -2, -21, -24, -6, 19, 28, 10, -21, -36, -18, 19, 40, 22, -21, -42, -23, 16, 39, 26, -14, -40, -32, 9, 38, 29, -8, -39, -36, 2, 36, 38, -1, -38

Offset: 0

Creighton Dement, Jun 12 2005

Set y = x = .5'i + .5'j + .5'k + .5e Define a(0) = 1 (this is twice the coefficient of the unit e in x), then "loop" steps 1-5, below. a(n) is given by twice the coefficient of e (the unit) in y from step 4 inside the n-th loop. Step 1 (Loop 1): Calculate x*y Result: x*y = .5'i + .5'j + .5'k - .5e Step 2 (Loop 1): Add the fractional parts of the real coefficient basis vectors of x*y (i.e. 'i, 'j, 'k, e) Result: .5 + .5 + .5 - .5 = 1 = s Step 3 (Loop 1): Calculate x + x*y + se Result .5'i + .5'j + .5'k + .5e + (.5'i + .5'j + .5'k - .5e) + se = 'i + 'j + 'k + e. Step 4 (Loop 1): Set y equal to the result from Step 3. Result: y = 'i + 'j + 'k + e; thus a(1) = 2*1 = 2 Step 5 (Loop 1): Return to Step 1 Step 1 (Loop 2): Result: x*y = 'i + 'j + 'k - e Step 2 (Loop 2): Result: s = 0 Step 3 (Loop 2): 1.5'i + 1.5'j + 1.5'k -.5e Step 4 (Loop 2): y = 1.5'i + 1.5'j + 1.5'k -.5e; thus a(2) = 2*(-.5) = -1 **Loop 1** + 'i + 'j + 'k + e **Loop 2** + 1.5'i + 1.5'j + 1.5'k - .5e **Loop 3** + 'i + 'j + 'k - e **Loop 4** + .5'i + .5'j + .5'k - 1.5e **Loop 5** - 3e **Loop 6** - 'i - 'j - 'k - 3e **Loop 7** - 1.5'i - 1.5'j - 1.5'k + .5e **Loop 8** + 2e **Loop 9** + 1.5'i + 1.5'j + 1.5'k + 1.5e **Loop 10** + 2'i + 2'j + 2'k **Loop 11** + 1.5'i + 1.5'j + 1.5'k - 2.5e **Loop 12** - 5e
Notice the horizontal line segments in the graph of (a(n)) against the natural numbers. These may be referred to as "Gerald's diamonds" (after Gerald McGarvey, who pointed them out shortly after this sequence was submitted). It could be an interesting task to find the approximate area of these diamonds and compare to the approximate area of the other diamonds.
From Benoit Jubin, Aug 12 2009: (Start)
Define the function f on the integers to be the odd function such that for n>=0, f(2n)=0 and f(2n+1)=1. Define the sequences a and b by
a(0)=b(0)=0,
a(n+1) = 1 + (a(n)-3b(n))/2 + f((a(n)-3b(n))/2) + 3 f((a(n)+b(n))/2),
b(n+1) = 1 + (a(n)+b(n))/2.
Then (with an offset shifted by 1), a=A108618 and b=A108619. (End)

Creighton Dement, Table of n, a(n) for n = 0..10000
Creighton Dement, Plot of A108618 against A108619 (patch on)
Creighton Dement, Plot of A108618 against A108619 (patch off)
Creighton Dement, Floretions 2009, DRAFT. See section 4.1 Algorithms.
Rémy Sigrist, Colored scatterplot of a(n) for n = 0..10000 (where the color is function of n mod 6)

Cf. A108619, A108620, A108621, A272693.

a[0] = b[0] = 0;
f[n_] := Sign[n]*Mod[n, 2];
a[n_] := a[n] = (1/2)*(a[n-1] - 3*b[n-1]) + 3*f[(1/2)*(a[n-1] + b[n-1])] + f[(1/2)*(a[n-1] - 3*b[n-1])] + 1;
b[n_] := b[n] = (1/2)*(a[n-1] + b[n-1]) + 1;
A108618 = Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Feb 25 2015, after Benoit Jubin *)

A272694 a(n) = (c(n)+f(c(n)))/2, where c() = A272693() and f(k) = sign(k)*(k mod 2).

Original entry on oeis.org

0, -1, -1, -3, -2, -2, -2, 0, 3, 1, -2, -3, -4, -3, 1, 5, 2, -3, -5, -5, -2, 3, 6, 3, -5, -11, -6, 3, 7, 5, -3, -10, -9, -2, 8, 8, 1, -8, -11, -6, 6, 10, 5, -6, -13, -9, 1, 11, 8, -3, -11, -11, -2, 9, 12, 3, -11, -17, -8, 9, 18, 9, -11, -23, -12, 9, 19, 11, -9, -22, -15, 4, 20, 14, -6, -20, -17, 2, 19, 17

Offset: 0

N. J. A. Sloane, Jun 08 2016

sign

Created in an attempt to simplify the definition of A108618.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000

Cf. A108618, A108619, A272693.

M:=1000;
a:=Array(0..M,0); # A108618 (with different offset)
b:=Array(0..M,0); # A108619 (with different offset)
c:=Array(0..M,0); # A272693
f:=n->sign(n)*(n mod 2);
a[0]:=0; b[0]:=0; c[0]:=0;
for n from 1 to M do
b[n]:=1+(a[n-1]+b[n-1])/2;
a[n]:=1+c[n-1]+f(c[n-1])+3*f(b[n]-1);
c[n]:=(a[n]-3*b[n])/2;
od:
[seq(a[n],n=0..M)];
[seq(b[n],n=0..M)];
[seq(c[n],n=0..M)];
[seq((c[n]+f(c[n]))/2,n=0..M)]; # A272694

cp's OEIS Frontend

A272693 a(0)=0; thereafter, a(n) = (A108618(n-1)-3*A108619(n))/2.

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A108620 2a(n) = A108618(n) + 3A108619(n).

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A108618 A quaternion-generated sequence calculated using the rules given in the comment box with initial seed x = .5'i + .5'j + .5'k + .5e; version: "tes".

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Mathematica

A272694 a(n) = (c(n)+f(c(n)))/2, where c() = A272693() and f(k) = sign(k)*(k mod 2).

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Author

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Programs

Maple

cp's OEIS Frontend

A272693 a(0)=0; thereafter, a(n) = (A108618(n-1)-3*A108619(n))/2.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

A108620 2*a(n) = A108618(n) + 3*A108619(n).

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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A108618 A quaternion-generated sequence calculated using the rules given in the comment box with initial seed x = .5'i + .5'j + .5'k + .5e; version: "tes".

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A272694 a(n) = (c(n)+f(c(n)))/2, where c() = A272693() and f(k) = sign(k)*(k mod 2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

A108620 2a(n) = A108618(n) + 3A108619(n).