A029744 -id:A029744 - cp's OEIS frontend

A246096 Paradigm shift sequence for (4,2) production scheme with replacement.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 21, 24, 28, 32, 36, 40, 45, 50, 55, 60, 66, 72, 84, 96, 112, 128, 144, 160, 180, 200, 225, 250, 275, 300, 336, 384, 448, 512, 576, 640, 720, 800, 900, 1000, 1125, 1250, 1375, 1536, 1792, 2048, 2304, 2560, 2880, 3200, 3600, 4000, 4500, 5000, 5625, 6250, 7168, 8192, 9216, 10240, 11520, 12800, 14400, 16000, 18000, 20000, 22500, 25000

Offset: 1

Jonathan T. Rowell, Aug 13 2014

nonn

This sequence is the solution to the following problem: "Suppose you have the choice of using one of three production options: apply a simple incremental action, bundle existing output as an integrated product (which requires p=4 steps), or implement the current bundled action (which requires q=2 steps). The first use of a novel bundle erases (or makes obsolete) all prior actions.  How large an output can be generated in n time steps?"
A production scheme with replacement R(p,q) eliminates existing output following a bundling action, while an additive scheme A(p,q) retains the output. The schemes correspond according to A(p,q)=R(p-q,q), with the replacement scheme serving as the default presentation.
This problem is structurally similar to the Copy and Paste Keyboard problem: Existing sequences (A178715 and A193286) should be regarded as Paradigm-Shift Sequences with production schemes R(1,1) and R(2,1) with replacement, respectively.
The ideal number of implementations per bundle, as measured by the geometric growth rate (p+zq root of z), is z = 4.
All solutions will be of the form a(n) = (qm+r) * m^b * (m+1)^d.

Paradigm shift sequences with q=2: A029744, A029747, A246080, A246084, A246088, A246092, A246096, A246100.

Paradigm shift sequences with p=4: A193456, A246096, A246097, A246098, A246099.

a(n) = (qd+r) * d^(C-R) * (d+1)^R, where r = (n-Cp) mod q, Q = floor( (R-Cp)/q ), R = Q mod (C+1), and d = floor ( Q/(C+1) ).

Recursive: a(n) = 4*a(n-12) for all n >= 67.

A246100 Paradigm shift sequence for (5,2) production scheme with replacement.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 21, 24, 28, 32, 36, 40, 45, 50, 55, 60, 66, 72, 78, 84, 96, 112, 128, 144, 160, 180, 200, 225, 250, 275, 300, 330, 360, 396, 448, 512, 576, 640, 720, 800, 900, 1000, 1125, 1250, 1375, 1500, 1650, 1800, 2048, 2304, 2560, 2880, 3200, 3600, 4000, 4500, 5000, 5625, 6250, 6875, 7500, 8250, 9216, 10240, 11520, 12800, 14400, 16000, 18000, 20000, 22500, 25000, 28125, 31250, 34375, 37500, 41250, 46080, 51200, 57600, 64000, 72000, 80000, 90000, 100000, 112500, 125000, 140625, 156250

Offset: 1

Jonathan T. Rowell, Aug 13 2014

nonn

This sequence is the solution to the following problem: "Suppose you have the choice of using one of three production options: apply a simple incremental action, bundle existing output as an integrated product (which requires p=5 steps), or implement the current bundled action (which requires q=2 steps). The first use of a novel bundle erases (or makes obsolete) all prior actions.  How large an output can be generated in n time steps?"
A production scheme with replacement R(p,q) eliminates existing output following a bundling action, while an additive scheme A(p,q) retains the output. The schemes correspond according to A(p,q)=R(p-q,q), with the replacement scheme serving as the default presentation.
This problem is structurally similar to the Copy and Paste Keyboard problem: Existing sequences (A178715 and A193286) should be regarded as Paradigm-Shift Sequences with production schemes R(1,1) and R(2,1) with replacement, respectively.
The ideal number of implementations per bundle, as measured by the geometric growth rate (p+zq root of z), is z = 5.
All solutions will be of the form a(n) = (qm+r) * m^b * (m+1)^d.

Paradigm shift sequences with q=2: A029744, A029747, A246080, A246084, A246088, A246092, A246096, A246100.

Paradigm shift sequences with p=5: A193457, A246100, A246101, A246102, A246103.

a(n) = (qd+r) * d^(C-R) * (d+1)^R, where r = (n-Cp) mod q, Q = floor( (R-Cp)/q ), R = Q mod (C+1), and d = floor ( Q/(C+1) ).

Recursive: a(n) = 5*a(n-15) for all n >= 75.

A364494 Numbers k such that k divides A163511(k).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 105, 128, 144, 192, 210, 256, 288, 384, 420, 429, 512, 576, 768, 840, 858, 1024, 1152, 1365, 1536, 1617, 1680, 1716, 2048, 2304, 2730, 3072, 3234, 3360, 3432, 3887, 4096, 4235, 4608, 5460, 6144, 6468, 6720, 6864, 7774, 8192, 8470, 9216, 10829, 10920, 12288

Offset: 1

Antti Karttunen, Jul 27 2023

nonn

If n is present, then 2*n is also present, and vice versa.

A007283 is included as a subsequence, because it gives the known fixed points of map n -> A163511(n).

Positions of 1's in A364491.
Cf. A163511.
Subsequences: A007283, A029744, A364495 (odd terms).
Cf. also A364295, A364496, A364497.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A054429(n) = ((3<<#binary(n\2))-n-1);
A163511(n) = if(!n,1,A005940(1+A054429(n)))
isA364494(n) = !(A163511(n)%n);

A056486 a(n) = (9*2^n + (-2)^n)/4 for n>0.

Original entry on oeis.org

4, 10, 16, 40, 64, 160, 256, 640, 1024, 2560, 4096, 10240, 16384, 40960, 65536, 163840, 262144, 655360, 1048576, 2621440, 4194304, 10485760, 16777216, 41943040, 67108864, 167772160, 268435456, 671088640, 1073741824, 2684354560, 4294967296, 10737418240

Offset: 1

Marks R. Nester

Old name was: "Number of periodic palindromes using a maximum of four different symbols".

Number of necklaces with n beads and 4 colors that are the same when turned over and hence have reflection symmetry. - Herbert Kociemba, Nov 24 2016

			G.f. = 4*x + 10*x^2 + 16*x^3 + 40*x^4 + 64*x^5 + 160*x^6 + 256*x^7 + 640*x^8 + ...
For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,4).

Cf. A029744, A038754, A056450.

Column 4 of A284855.

[(9*2^n + (-2)^n)/4 : n in [1..50]]; // Wesley Ivan Hurt, Nov 24 2016

A056486:=n->(9*2^n + (-2)^n)/4: seq(A056486(n), n=1..50); # Wesley Ivan Hurt, Nov 24 2016

CoefficientList[Series[-1+(1+4*x+6*x^2)/(1-4*x^2),{x,0,30}],x] (* Herbert Kociemba, Nov 24 2016 *)
k=4; Table[(k^Floor[(n+1)/2] + k^Ceiling[(n+1)/2]) / 2, {n, 1, 30}] (* Robert A. Russell, Sep 21 2018 *)

a(n) = (9*2^n+(-2)^n)/4; \\ Altug Alkan, Sep 21 2018

[2^(n-2)*(9+(-1)^n) for n in range(1,51)] # G. C. Greubel, Mar 23 2024

a(n) = 4^((n+1)/2) for n odd, a(n) = 4^(n/2)*5/2 for n even.
From Colin Barker, Jul 08 2012: (Start)
a(n) = 4*a(n-2).
G.f.: 2*x*(2+5*x)/((1-2*x)*(1+2*x)). (End)
G.f.: -1 + (1+4*x+6*x^2)/(1-4*x^2). - Herbert Kociemba, Nov 24 2016
E.g.f.: 5*sinh(x)^2 + 2*sinh(2*x). - Ilya Gutkovskiy, Nov 24 2016
a(n) = ( 4^floor((n+1)/2) + 4^ceiling((n+1)/2) )/2. - Robert A. Russell, Sep 21 2018

Better name from Ralf Stephan, Jul 18 2013

A056493 Number of primitive (period n) periodic palindromes using a maximum of two different symbols.

Original entry on oeis.org

2, 1, 2, 3, 6, 7, 14, 18, 28, 39, 62, 81, 126, 175, 246, 360, 510, 728, 1022, 1485, 2030, 3007, 4094, 6030, 8184, 12159, 16352, 24381, 32766, 48849, 65534, 97920, 131006, 196095, 262122, 392364, 524286, 785407, 1048446, 1571310, 2097150, 3143497

Offset: 1

Marks R. Nester

nonn

For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome.

Also number of aperiodic necklaces (Lyndon words) with two colors that are the same when turned over.

			a(1) = 2 with aaa... and bbb..., a(2) = 1 with ababab..., a(3) = 2 with aabaab... and abbabb..., a(4) = 3 with aaabaaab... and aabbaabb... and abbbabbb.... - _Michael Somos_, Nov 29 2016

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for a pdf file of Chap. 2]

Index entries for sequences related to Lyndon words

Column 2 of A284856.

Cf. A029744, A056391, A056458.

mx=40;gf[x_,k_]:=Sum[ MoebiusMu[n]*Sum[Binomial[k,i]x^(n i),{i,0,2}]/( 1-k x^(2n)),{n,mx}]; CoefficientList[Series[gf[x,2],{x,0,mx}],x] (* Herbert Kociemba, Nov 29 2016 *)

Sum_{d|n} mu(d)*b(n/d), where b(n) = A029744(n+1). [Corrected by Petros Hadjicostas, Oct 15 2017. The original formula referred to a previous version of sequence A029744 that had a different offset.]
More generally, let gf(k) be the g.f. for the number of necklaces with reflectional symmetry but no rotational symmetry and beads of k colors. Then gf(k): Sum_{n >= 1} mu(n)*Sum_{i=0..2} binomial(k,i)*x^(n*i)/(1 - k*x^(2*n)). - Herbert Kociemba, Nov 29 2016
G.f.: Sum_{n >= 1} mu(n)*x^n*(2 + 3*x^n)/(1 - 2*x^(2*n)). The g.f. by Herbet Kociemba above, with k = 2, becomes Sum_{n>=1} mu(n)*(x^n + 1)^2/(1 - 2*x^(2*n)). The two formulae differ by the "undetermined" constant Sum_{n >= 1} mu(n). - Petros Hadjicostas, Oct 15 2017

More terms and additional comments from Christian G. Bower, Jun 22 2000

A246088 Paradigm shift sequence for (2,2) production scheme with replacement.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 18, 21, 24, 28, 32, 36, 40, 45, 54, 63, 72, 84, 96, 112, 128, 144, 162, 189, 216, 252, 288, 336, 384, 448, 512, 576, 648, 756, 864, 1008, 1152, 1344, 1536, 1792, 2048, 2304, 2592, 3024, 3456, 4032, 4608, 5376, 6144, 7168, 8192, 9216, 10368, 12096, 13824, 16128, 18432, 21504, 24576, 28672, 32768, 36864, 41472

Offset: 1

Jonathan T. Rowell, Aug 13 2014

This sequence is the solution to the following problem: "Suppose you have the choice of using one of three production options: apply a simple incremental action, bundle existing output as an integrated product (which requires p=2 steps), or implement the current bundled action (which requires q=2 steps). The first use of a novel bundle erases (or makes obsolete) all prior actions. How large an output can be generated in n time steps?"
A production scheme with replacement R(p,q) eliminates existing output following a bundling action, while an additive scheme A(p,q) retains the output. The schemes correspond according to A(p,q)=R(p-q,q), with the replacement scheme serving as the default presentation.
This problem is structurally similar to the Copy and Paste Keyboard problem: Existing sequences (A178715 and A193286) should be regarded as Paradigm-Shift Sequences with production schemes R(1,1) and R(2,1) with replacement, respectively.
The ideal number of implementations per bundle, as measured by the geometric growth rate (p+zq root of z), is z = 4.
All solutions will be of the form a(n) = (qm+r) * m^b * (m+1)^d.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,4).

Paradigm shift sequences with q=2: A029744, A029747, A246080, A246084, A246088, A246092, A246096, A246100.

Paradigm shift sequences with p=2: A193286, A246088, A246089, A246090, A246091.

Vec(x*(1 +2*x +3*x^2 +4*x^3 +5*x^4 +6*x^5 +7*x^6 +8*x^7 +9*x^8 +10*x^9 +7*x^10 +4*x^11 +3*x^12 +2*x^13 +x^14 +x^20 +6*x^21 +3*x^22 +2*x^29 +9*x^30) / ((1 -2*x^5) * (1 +2*x^5)) + O(x^100)) \\ Colin Barker, Nov 19 2016

a(n) = (qd+r) * d^(C-R) * (d+1)^R, where r = (n-Cp) mod q, Q = floor( (R-Cp)/q ), R = Q mod (C+1), and d = floor ( Q/(C+1) ).
a(n) = 4*a(n-10) for all n >= 32.
G.f.: x*(1 +2*x +3*x^2 +4*x^3 +5*x^4 +6*x^5 +7*x^6 +8*x^7 +9*x^8 +10*x^9 +7*x^10 +4*x^11 +3*x^12 +2*x^13 +x^14 +x^20 +6*x^21 +3*x^22 +2*x^29 +9*x^30) / ((1 -2*x^5) * (1 +2*x^5)). - Colin Barker, Nov 19 2016

A193652 A020988 and A007583 interleaved.

Original entry on oeis.org

0, 1, 2, 3, 10, 11, 42, 43, 170, 171, 682, 683, 2730, 2731, 10922, 10923, 43690, 43691, 174762, 174763, 699050, 699051, 2796202, 2796203, 11184810, 11184811, 44739242, 44739243, 178956970, 178956971, 715827882, 715827883, 2863311530, 2863311531, 11453246122

Offset: 0

Reinhard Zumkeller, Aug 08 2011

a(2*n) = A020988(n), a(2*n+1) = a(2*n) + 1 = A007583(n);

apart from initial zero, record values in A048985: a(n)=A048985(A029744(n)) and a(n)<A048985(m) for m<A029744(n).

Index entries for linear recurrences with constant coefficients, signature (0,5,0,-4).

A193652 := proc(n)
        if type (n,'even') then
                A020988(n/2) ;
        else
                A007583(floor(n/2)) ;
        end if;
end proc: # R. J. Mathar, Jul 20 2016

LinearRecurrence[{0, 5, 0, -4}, {0, 1, 2, 3}, 35] (* Jean-François Alcover, Dec 16 2021 *)

a(n) = 2 * (4^floor(n/2) - 1) / 3 + n mod 2.

G.f.: ( -x*(-1-2*x+2*x^2) ) / ( (x-1)*(2*x+1)*(2*x-1)*(1+x) ). - R. J. Mathar, Feb 19 2015

Terms corrected by R. J. Mathar, Feb 19 2015

A209727 T(n,k) = 1/4 the number of (n+1) X (k+1) 0..2 arrays with every 2 X 2 subblock having distinct clockwise edge differences.

Original entry on oeis.org

2, 3, 3, 4, 4, 4, 6, 5, 5, 6, 8, 7, 6, 7, 8, 12, 9, 8, 8, 9, 12, 16, 13, 10, 10, 10, 13, 16, 24, 17, 14, 12, 12, 14, 17, 24, 32, 25, 18, 16, 14, 16, 18, 25, 32, 48, 33, 26, 20, 18, 18, 20, 26, 33, 48, 64, 49, 34, 28, 22, 22, 22, 28, 34, 49, 64, 96, 65, 50, 36, 30, 26, 26, 30, 36, 50, 65, 96

Offset: 1

R. H. Hardin, Mar 12 2012

Table starts
..2..3..4..6..8.12.16.24.32.48.64..96.128.192.256.384.512.768.1024.1536.2048
..3..4..5..7..9.13.17.25.33.49.65..97.129.193.257.385.513.769.1025.1537.2049
..4..5..6..8.10.14.18.26.34.50.66..98.130.194.258.386.514.770.1026.1538.2050
..6..7..8.10.12.16.20.28.36.52.68.100.132.196.260.388.516.772.1028.1540.2052
..8..9.10.12.14.18.22.30.38.54.70.102.134.198.262.390.518.774.1030.1542.2054
.12.13.14.16.18.22.26.34.42.58.74.106.138.202.266.394.522.778.1034.1546.2058
.16.17.18.20.22.26.30.38.46.62.78.110.142.206.270.398.526.782.1038.1550.2062
.24.25.26.28.30.34.38.46.54.70.86.118.150.214.278.406.534.790.1046.1558.2070

			Some solutions for n=4, k=3
..2..1..2..1....0..2..0..1....1..2..0..2....0..1..0..2....2..1..2..1
..0..2..0..2....2..1..2..0....2..0..1..0....2..0..2..1....0..2..0..2
..1..0..1..0....0..2..0..1....1..2..0..2....0..1..0..2....1..0..1..0
..0..2..0..2....2..1..2..0....2..0..1..0....2..0..2..1....0..2..0..2
..1..0..1..0....0..2..0..1....1..2..0..2....0..1..0..2....2..1..2..1

R. H. Hardin, Table of n, a(n) for n = 1..1512

Column 1 is A029744(n+1).

Diagonal is A027383.

Empirical for column k:
k=1: a(n) = 2*a(n-2).
k=2..7: a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3).

A275717 Numbers n for which A003961(n) > A003961(n-1).

Original entry on oeis.org

2, 3, 4, 6, 8, 12, 14, 15, 16, 18, 20, 24, 26, 27, 30, 32, 35, 36, 38, 39, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 78, 80, 81, 84, 86, 87, 88, 90, 92, 95, 96, 98, 100, 102, 104, 108, 110, 112, 114, 116, 117, 119, 120, 122, 123, 124, 125, 126, 128, 130, 132, 135, 138, 140, 143, 144

Offset: 1

Antti Karttunen, Aug 07 2016

nonn

One more than the positions of ascents in permutation A048673.

Antti Karttunen, Table of n, a(n) for n = 1..10000

One more than A275721.
Cf. A003961, A048673.
Complement: A275718 (apart from 1 which is in neither sequence).
Cf. A029744 (a subsequence, apart from its initial 1).
Cf. also A275719, A275720, A246281, A246282.

f[n_] := Times @@ Map[Prime[PrimePi@ First[#] + 1]^Last[#] &, FactorInteger@ n]; Select[Range@ 145, f[# - 1] < f@ # &] (* Michael De Vlieger, Aug 07 2016 *)

A116451 Numbers having fewer prime factors than at least one smaller number.

Original entry on oeis.org

5, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82

Offset: 0

Reinhard Zumkeller, Feb 16 2006

nonn

Complement of A029744.

Numbers whose odd part is greater than 3. - Peter Munn, Aug 12 2020

Cf. A029744, A116452, A001222.

A116452(n) = A001222(a(n)).

The offset should really be 1, since this is a list, but that change would also require a complete rewrite of A116454, plus changes to A116453. So for the moment let's leave this with offset 0. - N. J. A. Sloane, Aug 17 2020

cp's OEIS Frontend

A246096 Paradigm shift sequence for (4,2) production scheme with replacement.

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A246100 Paradigm shift sequence for (5,2) production scheme with replacement.

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A364494 Numbers k such that k divides A163511(k).

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A056486 a(n) = (9*2^n + (-2)^n)/4 for n>0.

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A056493 Number of primitive (period n) periodic palindromes using a maximum of two different symbols.

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Mathematica

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A246088 Paradigm shift sequence for (2,2) production scheme with replacement.

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A193652 A020988 and A007583 interleaved.

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Maple

Mathematica

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A209727 T(n,k) = 1/4 the number of (n+1) X (k+1) 0..2 arrays with every 2 X 2 subblock having distinct clockwise edge differences.

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