A032174 -id:A032174 - cp's OEIS frontend

A282032 Additive number system based on U.S. coins.

1, 2, 3, 4, 5, 10, 15, 20, 25, 50, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200, 1300, 1400, 1500, 1600, 1700, 1800, 1900, 2000, 2100, 2200, 2300, 2400, 2500, 2600, 2700, 2800, 2900, 3000, 3100, 3200, 3300, 3400, 3500, 3600, 3700, 3800

Offset: 1

N. J. A. Sloane, Feb 20 2017

Any positive integer can be written uniquely as a sum of at most 5 numbers, one from each row of the following array:
1,2,3,4;
5,10,15,20;
25;
50;
100, 200, 300, 400, 500, ...
(the last row being infinite).

Colin Barker, Table of n, a(n) for n = 1..1000
Michael Maltenfort, Characterizing Additive Systems, The American Mathematical Monthly 124.2 (2017): 132-148. See Fig. 2.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

A032174 and A282034 are two other examples of additive number systems.

A282033 gives a very similar family of sets which is not an additive system.

Vec(x*(1 + 4*x^5 + 20*x^9 + 25*x^10 + 50*x^11) / (1 - x)^2 + O(x^50)) \\ Colin Barker, Apr 16 2020

From Colin Barker, Apr 16 2020: (Start)
G.f.: x*(1 + 4*x^5 + 20*x^9 + 25*x^10 + 50*x^11) / (1 - x)^2.
a(n) = 2*a(n-1) - a(n-2) for n>11.
(End)

A282034 Three-set non-British additive number system read by rows.

Original entry on oeis.org

1, 2, 3, 4, 50, 51, 52, 53, 54, 5, 10, 15, 20, 25, 100, 125, 200, 225, 300, 325, 400, 425, 500, 525, 600, 625, 700, 725, 800, 825, 900, 925, 1000, 1025, 1100, 1125, 1200, 1225, 1300, 1325, 1400, 1425, 1500, 1525, 1600, 1625, 1700, 1725, 1800, 1825, 1900, 1925

Offset: 1

N. J. A. Sloane, Feb 20 2017

Any positive integer can be written uniquely as a sum of at most 3 numbers, one from each row of the following array:
1, 2, 3, 4, 50, 51, 52, 53, 54;
5, 10, 15, 20;
25, 100, 125, 200, 225, 300, 325, 400, 425, 500, 525, 600, 625, 700, 725, 800, 825, 900, 925, 1000, ...
(the last row being infinite).

Colin Barker, Table of n, a(n) for n = 1..1000
Michael Maltenfort, Characterizing Additive Systems, The American Mathematical Monthly 124.2 (2017): 132-148. See Fig. 5.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

A032174 and A282032 are two other examples of additive number systems.

Vec(x*(1 + x + 45*x^4 - 45*x^6 - 50*x^9 + 4*x^10 + 54*x^11 + 70*x^14 + 20*x^15) / ((1 - x)^2*(1 + x)) + O(x^50)) \\ Colin Barker, Apr 16 2020

From Colin Barker, Apr 16 2020: (Start)
G.f.: x*(1 + x + 45*x^4 - 45*x^6 - 50*x^9 + 4*x^10 + 54*x^11 + 70*x^14 + 20*x^15) / ((1 - x)^2*(1 + x)).
a(n) = a(n-1) + a(n-2) - a(n-3) for n>16.
(End)

A032173 Sequence (a(n): n >= 1) that shifts left 2 places under the "CHK" (necklace, identity, unlabeled) transform and has initial terms a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 7, 12, 28, 55, 122, 258, 574, 1254, 2813, 6283, 14220, 32237, 73631, 168660, 388331, 896790, 2078822, 4832343, 11266422, 26332119, 61694574, 144858260, 340829231, 803427128, 1897269215, 4487725726

Offset: 1

Christian G. Bower

nonn

From Petros Hadjicostas, Dec 29 2018: (Start)
a(n+2) = (1/n)*Sum_{d|n} mu(n/d)*c(d), where c(n) = n*a(n) + Sum_{s=1..n-1} c(s)*a(n-s) with a(1) = a(2) = 1, c(1) = 1, and c(2) = 3.
G.f.: If A(x) = Sum_{n>=1} a(n)*x^n, then Sum_{n>=1} a(n+2)*x^n = -Sum_{n>=1} (mu(n)/n)*log(1-A(x^n)).
The g.f. of the auxiliary sequence (c(n): n>=1) is C(x) = Sum_{n>=1} c(n)*x^n = x*(dA(x)/dx)/(1-A(x)) = x + 3*x^2 + 7*x^3 + 15*x^4 + 36*x^5 + 81*x^6 + 197*x^7 + 455*x^8 + 1105*x^9 + 2618*x^10 + ... (The auxiliary sequence is given by sequence A322913.)
(End)
The first two terms of the sequence must be specified. In general, if the sequence (b(n): n >= 1) is such that (b(n+2): n >= 1) = CHK((b(n): n >= 1)), then b(3) = b(1), b(4) = (1/2)*(b(1)^2 + 2*b(2) - b(1)), b(5) = (b(1)/3)*(b(1)^2 + 3*b(2) + 2), and so on. - Petros Hadjicostas, Dec 31 2018

Andrew Howroyd, Table of n, a(n) for n = 1..200
C. G. Bower, Transforms (2)

Cf. A032171, A032174, A322913.

a[1] = a[2] = 1; c[1] = 1; c[2] = 3;
a[n_] := a[n] = 1/(n-2) Sum[MoebiusMu[(n-2)/d] c[d], {d, Divisors[n-2]}];
c[n_] := c[n] = n a[n] + Sum[c[s] a[n-s], {s, 1, n-1}];
Array[a, 32] (* Jean-François Alcover, Jan 02 2019 *)

CHK(p,n)={sum(d=1, n, moebius(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d)))}
seq(n)={my(p=1+O(x));for(i=1, n\2, p=1+x+x*CHK(x*p, 2*i)); Vec(p+O(x^n))} \\ Andrew Howroyd, Jun 20 2018

Name modified by Petros Hadjicostas, Jan 01 2019

A282033 An example of a collection of five sets (based on U.S. coinage) which is not an additive number system.

Original entry on oeis.org

1, 2, 3, 4, 5, 10, 20, 25, 50, 75, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200, 1300, 1400, 1500, 1600, 1700, 1800, 1900, 2000, 2100, 2200, 2300, 2400, 2500, 2600, 2700, 2800, 2900, 3000, 3100, 3200, 3300, 3400, 3500, 3600, 3700

Offset: 1

N. J. A. Sloane, Feb 20 2017

The five sets are the following:
1, 2, 3, 4;
5;
10, 20;
25, 50, 75;
100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200, ...
(the last set being infinite).
In contrast to A282032 this is not an additive number system because 26 for example can be represented in two ways as a sum of numbers from distinct sets (26 = 1+5+20 = 1+25).

Colin Barker, Table of n, a(n) for n = 1..1000
Michael Maltenfort, Characterizing Additive Systems, The American Mathematical Monthly 124.2 (2017): 132-148. See Fig. 3.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A032174, A282032, A282034 are legitimate examples of additive number systems.

LinearRecurrence[{2,-1},{1,2,3,4,5,10,20,25,50,75,100,200,300,400},50] (* or *) CoefficientList[Series[x (1+4x^5+5x^6-5x^7+ 20x^8+ 75x^11)/ (1-x)^2, {x,0,50}],x] (* Harvey P. Dale, Aug 04 2021 *)

Vec(x*(1 + 4*x^5 + 5*x^6 - 5*x^7 + 20*x^8 + 75*x^11) / (1 - x)^2 + O(x^50)) \\ Colin Barker, Apr 16 2020

From Colin Barker, Apr 16 2020: (Start)
G.f.: x*(1 + 4*x^5 + 5*x^6 - 5*x^7 + 20*x^8 + 75*x^11) / (1 - x)^2.
a(n) = 2*a(n-1) - a(n-2) for n>12.
(End)

A322913 Inverse Moebius transform of the sequence (n*A032173(n+2): n >= 1).

Original entry on oeis.org

1, 3, 7, 15, 36, 81, 197, 455, 1105, 2618, 6315, 15141, 36570, 88161, 213342, 516247, 1251728, 3037059, 7378290, 17938430, 43655465, 106317863, 259127707, 631986437, 1542364386, 3766351332, 9202390342, 22496047757, 55020807236, 134631987776, 329579227722, 807142635031

Offset: 1

Petros Hadjicostas, Dec 30 2018

nonn

The sequence (A032173(n): n >= 1) shifts two places to the left under Bower's "CHK" (necklace, identity, unlabeled) transform. The current sequence satisfies A032173(n+2) = (1/n)*Sum_{d|n} mu(d)*a(n/d).

Andrew Howroyd, Table of n, a(n) for n = 1..200
C. G. Bower, Transforms (2)

Cf. A032171, A032173, A032174.

(* b = A032173 *) b[1] = b[2] = 1; c[1] = 1; c[2] = 3;
b[n_] := b[n] = 1/(n-2) Sum[MoebiusMu[(n-2)/d] c[d], {d, Divisors[n-2]}];
c[n_] := c[n] = n b[n] + Sum[c[s] b[n-s], {s, 1, n-1}];
a[n_] := Sum[d b[d+2], {d, Divisors[n]}];
Array[a, 26] (* Jean-François Alcover, Jan 02 2019 *)

CHK(p, n)={sum(d=1, n, moebius(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d)))}
seq(n)={my(p=1+O(x)); for(i=1, n\2, p=1+x+x*CHK(x*p, 2*i)); Vec(deriv(x*p)/(1-x*p)+O(x^n))} \\ Andrew Howroyd, Apr 27 2020

a(n) = Sum_{d|n} d*A032173(d+2).
a(n) = n*A032173(n) + Sum_{s=1..n-1} a(s)*A032173(n-s).
G.f.: If A(x) = Sum_{n>=1} a(n)*x^n and B(x) = Sum_{n>=1} A032173(n)*x^n, then A(x) = x*(dB(x)/dx)/(1-B(x)), while (B(x) - x - x^2)/x^2 = Sum_{n>=1} A032173(n+2)*x^n = -Sum_{n>=1} (mu(n)/n)*log(1-B(x^n)).

Terms a(27) and beyond from Andrew Howroyd, Apr 27 2020

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A282032 Additive number system based on U.S. coins.

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A282034 Three-set non-British additive number system read by rows.

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A032173 Sequence (a(n): n >= 1) that shifts left 2 places under the "CHK" (necklace, identity, unlabeled) transform and has initial terms a(1) = a(2) = 1.

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A282033 An example of a collection of five sets (based on U.S. coinage) which is not an additive number system.

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A322913 Inverse Moebius transform of the sequence (n*A032173(n+2): n >= 1).

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