Creighton Dement - cp's OEIS frontend

A308496 Numbers with digits 1,2,4,7 when written in base 8.

1, 2, 4, 7, 9, 10, 12, 15, 17, 18, 20, 23, 33, 34, 36, 39, 57, 58, 60, 63, 73, 74, 76, 79, 81, 82, 84, 87, 97, 98, 100, 103, 121, 122, 124, 127, 137, 138, 140, 143, 145, 146, 148, 151, 161, 162, 164, 167, 185, 186, 188, 191, 265, 266, 268, 271, 273

Offset: 1

Creighton Dement, Jun 01 2019

Floretions of all orders. - Creighton Dement, Oct 28 2022
For any natural number n, the set of terms of this sequence between indices (4^n-1)/3 and (4^(n+1)-4)/3 is "isomorphic" to the group of n-th order floretions. In this case, group multiplication is given by bitwise logical operations (see EXAMPLE). Note that the case of n = 1 is simply the quaternions.
In the table below, the left column is the binary representation, the middle column the terms of ((a(n)) and the right column the conventional notation. Multiply x*y (disregarding signs) using the bitwise XNOR operation, where x and y are any floretions of the same order. The XNOR operation returns a 1 if the number of 1's in its inputs is even, and a 0 if the number of 1's is odd. This operation is used to calculate the base vector of the result.
**** 1st-order floretions (= quaternions) ****
         | binary  | decimal | floretion
  1      | 001     |    1    |   i
  2      | 010     |    2    |   j
  4      | 100     |    4    |   k
  7      | 111     |    7    |   e (unit)
**** 2nd-order floretions ****
  1_1    | 001_001 |    9    |   ii
  1_2    | 001_010 |   10    |   ij
  1_4    | 001_100 |   12    |   ik
  1_7    | 001_111 |   15    |   ie
  2_1    | 010_001 |   17    |   ji
  2_2    | 010_010 |   18    |   jj
  2_4    | 010_100 |   20    |   jk
  2_7    | 010_111 |   23    |   je
  4_1    | 100_001 |   33    |   ki
  4_2    | 100_010 |   34    |   kj
  4_4    | 100_100 |   36    |   kk
  4_7    | 100_111 |   39    |   ke
  7_1    | 111_001 |   57    |   ei
  7_2    | 111_010 |   58    |   ej
  7_4    | 111_100 |   60    |   ek
  7_7    | 111_111 |   63    |   ee
**** 3rd-order floretions ****
  1_1_1
  1_1_2
  ...
Note that for a floretion of order n, two digits from any one of its "binary triplets" abc determine the other since XOR(a,b,c) = 1.
When working with a floretion algebra over the reals, i.e., elements of the form x = q_1*f_1 + ... q_m*f_m where q_1,...,q_m are real numbers and f_1,...,f_m are any floretions of the same order, then x may also be referred to as a "floretion". In this case f_1,...,f_m (i.e., terms of this sequence) may be referred to as "floretion base vectors" to avoid confusion.
Taking signs into account:
Given two binary representations (ab) and (cd) for quaternion elements, define multiplication as:
    Compute (XNOR(a,c))(XNOR(b,d)) to get the base vector of the result.
    Compute AND(b,c), AND(XNOR(a,b),d), and AND(a,XNOR(c,d)). These are all bitwise AND operations.
    The sign is negative if and only if the total number of 1's in the results is even.
For example, with k*j = (10)*(01) = -i, compute:
    The base vector as (XNOR(1,0) XNOR(0,1)) = (0)(0) = i.
    The signs as AND(0,1), AND(XNOR(1,0),1), AND(1, XNOR(0,1)) = 0, 0, 0. There are zero 1's in total, which is an even number, so the result is negative.
An example of image processing: take for example a quaternion x = .2i + .5j + .3k + e. Assume we have a square monitor (aspect ratio). Furthermore, assume the screen is divided into 4 squares- one for i (bottom left), one for j (top left), one for k (top right) one for e (bottom right) and that the coefficient is the amount the pixels are lit up on the screen (1 being full brightness, 0 being off- this could be modified later to accomodate negative numbers).  Now imagine we have square monitor of resolution 2^n x 2^n. Then we can represent any black and white image with that resolution with an n-th order floretion. This means we can multiply images together, with some parallels to Fourier analysis.
Multiplying an image by an idempotent floretion would allow one to repeatedly apply a specific transformation (e.g., a rotation, scaling, or some other operation) to an image, and then undo all of those transformations by continuing to apply the same operation a certain number of times. It could be used in applications such as data encryption, where an image could be "scrambled" using a specific floretion and then "unscrambled" by continuing to apply the same floretion.
A compact definition of multiplication is x*y = (ab)(cd) = (-1)^{m+1} (aqc)(bqd) where m = b&c + (aqb)&d + a&(cqd) and "q", "&" are the bitwise XNOR and AND operators respectively. - Creighton Dement, Jul 09 2023

Creighton Dement, Table of n, a(n) for n = 1..10000
Creighton Dement, Floretions and Actions of D3 (DRAFT)
Stefan Rakonjac, The Sounds of Sequences: Floretions

Cf. A007094, A108618, A047541.

A308496Q[n_]:=ContainsOnly[IntegerDigits[n,8],{1,2,4,7}];
Select[Range[1000],A308496Q] (* Paolo Xausa, Dec 31 2023 *)

is(n)=!#setminus(Set(digits(n,8)),[1,2,4,7]);
a(n) =
{
  local(total_count, index);
  until(total_count == n+1, if(is(index)==1, total_count++); index++);
  index-1;
}

a(n,b=8,d=[1,2,4,7]) = { for (w=1, oo, if (n>#d^w, n-=#d^w, return (fromdigits(apply(x -> d[1+x], digits(#d^w+n-1, #d))[2..-1],b)))) } \\ Rémy Sigrist, Jun 01 2019

A165660 Denominators of A007504 divided by A033955, starting from the second term of A033955.

Original entry on oeis.org

1, 3, 2, 8, 13, 18, 27, 29, 23, 56, 7, 74, 44, 98, 67, 49, 171, 200, 217, 28, 137, 309, 17, 116, 209, 448, 471, 174, 571, 629, 137, 739, 111, 793, 853, 318, 997, 1002, 560, 164, 610, 446, 1419, 1466, 385, 1615, 1573, 1633, 1707, 1825, 946, 662, 2221, 781, 1198

Offset: 1

Creighton Dement, Sep 24 2009

Conjecture: with the exception of the second term, 2 <= A165659(n)/a(n) < 3.

Cf. A007504, A033995, A002110, A102647, A165657, A165658, A165659

a1(n)=sum(i=1, n, prime(i)); b1(n)=sum(i=1, n, prime(n+1)%prime(i)); a(n)=if(n<0, 0, denominator(a1(n)/b1(n))); for(n=1, 50, print1(a(n) ", "))

Terms corrected by Creighton Dement, Oct 03 2009
Removed a conjecture - R. J. Mathar, Oct 09 2009
Typo in definition corrected by Creighton Dement, Oct 09 2009

A165657 Numerators of A002110 divided by A102647, starting from the second term of both.

Original entry on oeis.org

2, 3, 15, 105, 385, 5005, 17017, 323323, 1062347, 30808063, 434113615, 35336848261, 1448810778701, 33545541876077, 266186053068611, 5426100312552455, 9156001667401012567, 42962777054727828199

Offset: 1

Creighton Dement, Sep 24 2009

Cf. A002110, A102647, A165658, A165659, A165660.

a1(n)=prod(i=1, n, prime(i)); b1(n)=prod(i=1, n, prime(n+1)%prime(i));
a(n)=if(n<0, 0, numerator(a1(n)/b1(n))); for(n=1, 20, print1(a(n) ", "))

A165658 Denominators of A002110 divided by A102647, starting from the second term of both.

Original entry on oeis.org

1, 1, 1, 4, 6, 48, 64, 96, 576, 1728, 13824, 165888, 1036800, 9953280, 119439360, 297271296, 134369280000, 222953472000, 75246796800, 32105299968000, 229323571200, 568865783808000000, 125150472437760000, 6129819058176000

Offset: 1

Creighton Dement, Sep 24 2009

Cf. A002110, A102647, A165657, A165659, A165660.

a1(n)=prod(i=1, n, prime(i));
b1(n)=prod(i=1, n, prime(n+1)%prime(i));
a(n)=if(n<0, 0, denominator(a1(n)/b1(n)));
for(n=1, 25, print1(a(n) ", "))

A165659 Numerators of A007504 divided by A033955, starting from the second term of A033955.

Original entry on oeis.org

2, 5, 5, 17, 28, 41, 58, 77, 50, 129, 16, 197, 119, 281, 164, 127, 440, 501, 568, 71, 356, 791, 46, 321, 530, 1161, 1264, 457, 1480, 1593, 344, 1851, 284, 2127, 2276, 809, 2584, 2747, 1457, 441, 1633, 1149, 3638, 3831, 1007, 4227, 4438

Offset: 1

Creighton Dement, Sep 24 2009

Conjecture: with the exception of the second term, 2 <= a(n)/A165660(n) < 3.

Cf. A007504, A033995, A002110, A102647, A165657, A165658, A165660.

a1(n)=sum(i=1, n, prime(i));
b1(n)=sum(i=1, n, prime(n+1)%prime(i));
a(n)=if(n<0, 0, numerator(a1(n)/b1(n)));
for(n=1, 50, print1(a(n) ", "))

Typo in definition corrected by Creighton Dement, Oct 09 2009

A159582 Expansion of (1+6x+x^2-2x^3)/((x^2+2x-1)(x^2-2*x-1)), bisection is NSW numbers.

Original entry on oeis.org

1, 6, 7, 34, 41, 198, 239, 1154, 1393, 6726, 8119, 39202, 47321, 228486, 275807, 1331714, 1607521, 7761798, 9369319, 45239074, 54608393, 263672646, 318281039, 1536796802, 1855077841, 8957108166, 10812186007, 52205852194, 63018038201, 304278004998

Offset: 0

Creighton Dement, Apr 16 2009

Define c = [0, 7, 0, 41, 0, 239, 0, 1393, 0, 8119, 0, 47321, ...] where (c(2n+1)) = A002315(n+1) (NSW numbers). Then (a(n)) has the property c(2n) - a(2n) = -a(2n) = -A002315(n) and c(2n+1) - a(2n+1) = A002315(n) (NSW numbers).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

Cf. A002315.

Vec((1+6*x+x^2-2*x^3) / ((x^2+2*x-1)*(x^2-2*x-1)) + O(x^50)) \\ Colin Barker, Jun 29 2017

a(n) = 3*A078057(n)/2 - (-1)^n*A078057(n)/2. - R. J. Mathar, Nov 10 2009
From Colin Barker, Jun 29 2017: (Start)
a(n) = 6*a(n-2) - a(n-4) for n>3.
a(n) = ((-(-2+sqrt(2))*(-1+sqrt(2))^n - (-1-sqrt(2))^n*(2+sqrt(2)) - 3*(-(1-sqrt(2))^n*(-2+sqrt(2)) - (1+sqrt(2))^n*(2+sqrt(2))))) / (4*sqrt(2)).
(End)

A159288 Expansion of (1 + x + x^2)/(1 - x^2 - 2*x^3).

Original entry on oeis.org

1, 1, 2, 3, 4, 7, 10, 15, 24, 35, 54, 83, 124, 191, 290, 439, 672, 1019, 1550, 2363, 3588, 5463, 8314, 12639, 19240, 29267, 44518, 67747, 103052, 156783, 238546, 362887, 552112, 839979, 1277886, 1944203, 2957844, 4499975, 6846250, 10415663

Offset: 0

Creighton Dement, Apr 08 2009

A floretion-generated sequence: 'i + 0.5('ij' + 'ik' + 'ji' + 'jk' + 'ki' + 'kj')

Starting with offset 1 the sequence appears to be the INVERT transform of (1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, ...). - Gary W. Adamson, Aug 27 2016

G. C. Greubel, Table of n, a(n) for n = 0..1000
Creighton Dement, Online Floretion Multiplier [broken link]
Index entries for linear recurrences with constant coefficients, signature (0,1,2).

Cf. A159284, A159285, A159286, A159287.

I:=[1, 1, 2]; [n le 3 select I[n] else Self(n-2) + 2*Self(n-2): n in [1..30]]; // G. C. Greubel, Jun 27 2018

CoefficientList[Series[(1+x+x^2)/(1-x^2-2x^3),{x,0,50}],x]  (* Harvey P. Dale, Mar 09 2011 *)
LinearRecurrence[{0, 1, 2}, {1, 1, 2}, 50] (* G. C. Greubel, Jun 27 2018 *)

a(n)=([0,1,0; 0,0,1; 2,1,0]^n*[1;1;2])[1,1] \\ Charles R Greathouse IV, Aug 27 2016

Vec((1 + x + x^2) / (1 - x^2 - 2*x^3) + O(x^40)) \\ Colin Barker, Apr 29 2019

a(n) = A159287(n) + A159287(n+1) + A159287(n+2). - R. J. Mathar, Apr 10 2009
a(n) = a(n-2) + 2*a(n-3) for n>2. - Colin Barker, Apr 29 2019
a(n)= A052947(n) + A052947(n-1) +A052947(n-2). - R. J. Mathar, Mar 23 2023

A159285 Expansion of (1+3x)/(1-x^2-2x^3).

Original entry on oeis.org

1, 3, 1, 5, 7, 7, 17, 21, 31, 55, 73, 117, 183, 263, 417, 629, 943, 1463, 2201, 3349, 5127, 7751, 11825, 18005, 27327, 41655, 63337, 96309, 146647, 222983, 339265, 516277, 785231, 1194807, 1817785, 2765269, 4207399, 6400839, 9737937, 14815637

Offset: 0

Creighton Dement, Apr 08 2009

A floretion-generated sequence: 'i + 0.5('ij' + 'ik' + 'ji' + 'jk' + 'ki' + 'kj')

G. C. Greubel, Table of n, a(n) for n = 0..1000
Creighton Dement, Online Floretion Multiplier [broken link]
Index entries for linear recurrences with constant coefficients, signature (0,1,2).

Cf. A159284, A159286, A159287, A159288.

I:=[1,3,1]; [n le 3 select I[n] else Self(n-2) + 2*Self(n-3): n in [1..30]]; // G. C. Greubel, Jun 27 2018

LinearRecurrence[{0, 1, 2}, {1, 3, 1}, 50] (* G. C. Greubel, Jun 27 2018 *)
CoefficientList[Series[(1+3x)/(1-x^2-2x^3),{x,0,40}],x] (* Harvey P. Dale, Jan 21 2019 *)

a(n)=([0,1,0; 0,0,1; 2,1,0]^n*[1;3;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

a(n) = A052947(n) + 3*A052947(n-1). - R. J. Mathar, Mar 23 2023

A159286 Expansion of (x-1)^2/(1-x^2-2*x^3).

Original entry on oeis.org

1, -2, 2, 0, -2, 4, -2, 0, 6, -4, 6, 8, -2, 20, 14, 16, 54, 44, 86, 152, 174, 324, 478, 672, 1126, 1628, 2470, 3880, 5726, 8820, 13486, 20272, 31126, 47244, 71670, 109496, 166158, 252836, 385150, 585152

Offset: 0

Creighton Dement, Apr 08 2009

A floretion-generated sequence: 'i + 0.5('ij' + 'ik' + 'ji' + 'jk' + 'ki' + 'kj').

G. C. Greubel, Table of n, a(n) for n = 0..1000
Creighton Dement, Online Floretion Multiplier [broken link].
Index entries for linear recurrences with constant coefficients, signature (0,1,2).

Cf. A159284, A159285, A159287, A159288.

I:=[1, -2, 2]; [n le 3 select I[n] else Self(n-2) + 2*Self(n-3): n in [1..30]]; // G. C. Greubel, Jun 27 2018

CoefficientList[Series[(x-1)^2/(1-x^2-2*x^3),{x,0,40}],x] (* or *) LinearRecurrence[{0,1,2},{1,-2,2},40]  (* Harvey P. Dale, Apr 24 2011 *)

a(n)=([0,1,0; 0,0,1; 2,1,0]^n*[1;-2;2])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

a(n) = A159288(n) - 3*A159287(n+1). - R. J. Mathar, Apr 10 2009
a(1)=1, a(2)=-2, a(3)=2, a(n) = 1*a(n-2) + 2*a(n-3) for n >= 3. - Harvey P. Dale, Apr 24 2011
a(n) = A078026(n)-A078026(n-1). - R. J. Mathar, Mar 23 2023

A159290 A generalized Jacobsthal sequence.

Original entry on oeis.org

3, 5, 13, 25, 53, 105, 213, 425, 853, 1705, 3413, 6825, 13653, 27305, 54613, 109225, 218453, 436905, 873813, 1747625, 3495253, 6990505, 13981013, 27962025, 55924053, 111848105, 223696213, 447392425, 894784853, 1789569705, 3579139413

Offset: 0

Creighton Dement, Apr 08 2009

Sequence generated by the floretion: X*Y with X = 0.5('i + 'j + 'k + 'ee') and Y = 0.5(i' + j' + k' + 'ij' + 'ik' + 'ji' + 'jk' + 'ki' + 'kj' + 'ee')

G. C. Greubel, Table of n, a(n) for n = 0..1000
Creighton Dement, Online Floretion Multiplier [broken link]
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

A083943, A068156

[-1 + (2*(-1)^n + 5*2^(n+1))/3: n in [0..50]]; // G. C. Greubel, Jun 27 2018

LinearRecurrence[{2, 1, -2}, {3, 5, 13}, 50] (* or *) Table[-1 + (2*(-1)^n + 5*2^(n+1))/3, {n,0,30}] (* G. C. Greubel, Jun 27 2018 *)

x='x+O('x^50); Vec((3-x)/(-x^2+1-2*x+2*x^3)) \\ G. C. Greubel, Jun 27 2018

a(n) = -1 + (2*(-1)^n + 5*2^(n+1))/3.
G.f.: (3-x)/((1-x)*(1+x)*(1-2*x)).
a(n) = 3*A000975(n+1) - A000975(n). - R. J. Mathar, Sep 11 2019
a(n)+a(n+1) = A051633(n+1). - R. J. Mathar, Mar 23 2023

cp's OEIS Frontend

User: Creighton Dement

A308496 Numbers with digits 1,2,4,7 when written in base 8.

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A165660 Denominators of A007504 divided by A033955, starting from the second term of A033955.

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A165657 Numerators of A002110 divided by A102647, starting from the second term of both.

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A165658 Denominators of A002110 divided by A102647, starting from the second term of both.

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A165659 Numerators of A007504 divided by A033955, starting from the second term of A033955.

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Extensions

A159582 Expansion of (1+6*x+x^2-2*x^3)/((x^2+2*x-1)*(x^2-2*x-1)), bisection is NSW numbers.

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A159288 Expansion of (1 + x + x^2)/(1 - x^2 - 2*x^3).

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Magma

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A159285 Expansion of (1+3*x)/(1-x^2-2*x^3).

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A159582 Expansion of (1+6x+x^2-2x^3)/((x^2+2x-1)(x^2-2*x-1)), bisection is NSW numbers.

A159285 Expansion of (1+3x)/(1-x^2-2x^3).