A212700 -id:A212700 - cp's OEIS frontend

A212704 a(n) = 9n10^(n-1).

9, 180, 2700, 36000, 450000, 5400000, 63000000, 720000000, 8100000000, 90000000000, 990000000000, 10800000000000, 117000000000000, 1260000000000000, 13500000000000000, 144000000000000000, 1530000000000000000, 16200000000000000000, 171000000000000000000, 1800000000000000000000

Offset: 1

Stanislav Sykora, May 25 2012

Main transitions in systems of n particles with spin 9/2.
Please, refer to the general explanation in A212697.
This particular sequence is obtained for base b=10, corresponding to spin S = (b-1)/2 = 9/2.
Number of 0 needed to write all numbers of n+1 digits. - Bruno Berselli, Jun 30 2014
Essentially the same as A113119. - Bernard Schott, Nov 15 2022
From Bernard Schott, Nov 22 2022: (Start)
Number of nonzero digits needed to write all integers from 1 up to 10^n - 1.
a(n) is a square iff n in { A016754 union A033583\{0} } (see formulas). (End)

Stanislav Sykora, Table of n, a(n) for n = 1..100
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (20,-100).

Cf. A008591, A011557, A016754, A033583, A033713, A033714, A053541, A081045, A113119.

Cf. A001787, A212697, A212698, A212699, A212700, A212701, A212702, A212703 (for b=2..9).

Rest@ CoefficientList[Series[9 x/(10 x - 1)^2, {x, 0, 18}], x] (* or *)
Array[9 # 10^(# - 1) &, 18] (* Michael De Vlieger, Nov 18 2019 *)

mtrans(n, b) = n*(b-1)*b^(n-1);
a(n) = mtrans(n, 10);

def a(n): return 9*n*10**(n-1)
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Nov 14 2022

a(n) = n*(b-1)*b^(n-1) with b=10.
From R. J. Mathar, Oct 15 2013: (Start)
G.f.: 9*x/(10*x-1)^2.
a(n) = 9*A053541(n). (End)
From Bernard Schott, Nov 14 2022: (Start)
a(n+1) - a(n) = 9*A081045(n).
a(n) = A113119(n) for n > 1.
a(n) = A033713(n+1) - A033713(n) = A033714(n+1) - A033714(n).
a(A016754(n)) = (3 * (2n+1) * 10^(2*n*(n+1)))^2.
a(A033583(n)) = (3 * n * 10^(5*n^2))^2. (End)
From Elmo R. Oliveira, May 13 2025: (Start)
E.g.f.: 9*x*exp(10*x).
a(n) = A008591(n)*A011557(n-1).
a(n) = 20*a(n-1) - 100*a(n-2) for n > 2. (End)

A212697 a(n) = 2n3^(n-1).

Original entry on oeis.org

2, 12, 54, 216, 810, 2916, 10206, 34992, 118098, 393660, 1299078, 4251528, 13817466, 44641044, 143489070, 459165024, 1463588514, 4649045868, 14721978582, 46490458680, 146444944842, 460255540932, 1443528742014, 4518872583696, 14121476824050, 44059007691036

Offset: 1

Stanislav Sykora, May 24 2012

Main transitions in systems of n particles with spin 1.
Consider the set S of all b^n numbers which have n digits in base b. Define as "main transition" a pair (x,y) of elements of S such that x and y differ in base b in only one digit which in y exceeds that in x by 1. This particular sequence a(n) gives the number of such transitions for the case b=3.
The terminology originates from quantum theory of coupled spin systems (such as in magnetic resonance) with n particles, each with spin S = (b-1)/2. Then the i-th digit's value in base b can be intended as a label for the b = 2S+1 quantum states of the i-th particle. The most intense main quantum transitions then correspond to the above definition. Due to continuity, the correspondence holds regardless of how strongly coupled are the particles among themselves.
a(n) is the number of functions from {1,2,...,n} into {1,2,3} with a specially designated element of the domain that is restricted to be mapped into {1,2}. Hence the e.g.f. is 2*x*exp(x)^3. - Geoffrey Critzer, Mar 01 2015
a(n) is the distance spectral radius of the dimension-regular generalized recursive circulant graph (commonly known as multiplicative circulant graph) of order 3^n. - John Rafael M. Antalan, Sep 25 2020

			n=2, b=3, S={00, 01, 02, 10, 11, 12, 20, 21, 22}, main transitions = {(00,01), (00,10), (01,02), (01,12), (02,12), (10,11), (10,20), (11,12), (11,21), (12,22), (20,21), (21,22)}, main transitions count = 12.

M. H. Levitt, Spin Dynamics, Basics of Nuclear Magnetic Resonance, 2nd Edition, John Wiley & Sons, 2007, Section 6 (Mathematical techniques).
J. A. Pople, W. G. Schneider, H. J. Bernstein, High-Resolution Nuclear Magnetic Resonance, McGraw-Hill, 1959, Chapter 6.

Stanislav Sykora, Table of n, a(n) for n = 1..100
John Rafael M. Antalan and Francis Joseph H. Campeña, Distance eigenvalues and forwarding indices of dimension-regular generalized recursive circulant graph of order power of two and three, arXiv:2009.11608[math.CO], 2020.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (6,-9).

Cf. A001787 (b = 2).
Cf. A212698, A212699, A212700, A212701, A212702, A212703, A212704 (b = 4, 5, 6, 7, 8, 9, 10).
Row n=3 of A258997.

List([1..30], n-> 2*3^(n-1)*n) # G. C. Greubel, Jun 08 2019

[2*3^(n-1)*n: n in [1..30]]; // G. C. Greubel, Jun 08 2019

A212697:=n->2*n*3^(n-1): seq(A212697(n), n=1..30); # Wesley Ivan Hurt, Mar 01 2015

Table[Sum[Binomial[n, j] j 2^j, {j, n}], {n, 30}] (* Geoffrey Critzer, Mar 01 2015 *)
Table[2*3^(n-1)*n, {n,30}] (* G. C. Greubel, Jun 08 2019 *)

mtrans(n,b) = n*(b-1)*b^(n-1);
for (n=1,100,write("b212697.txt",n," ",mtrans(n,3)))

[2*3^(n-1)*n for n in (1..30)] # G. C. Greubel, Jun 08 2019

a(n) = n*(b-1)*b^(n-1). For this sequence, set b=3.
From R. J. Mathar, Oct 15 2013: (Start)
G.f.: 2*x/(1-3*x)^2.
a(n) = 2*A027471(n+1). (End)
a(n) = A005843(n)*A000244(n-1). - Omar E. Pol, Jan 21 2014
a(n) = Sum_{j=1..n} binomial(n,j)*j*2^j. - Geoffrey Critzer, Mar 01 2015
E.g.f.: 2*x*exp(3*x). - G. C. Greubel, Jun 08 2019

A212698 Main transitions in systems of n particles with spin 3/2.

Original entry on oeis.org

3, 24, 144, 768, 3840, 18432, 86016, 393216, 1769472, 7864320, 34603008, 150994944, 654311424, 2818572288, 12079595520, 51539607552, 219043332096, 927712935936, 3917010173952, 16492674416640, 69269232549888, 290271069732864, 1213860837064704, 5066549580791808

Offset: 1

Stanislav Sykora, May 25 2012

Please refer to the general explanation in A212697. This particular sequence is obtained for base b=4, corresponding to spin S = (b-1)/2 = 3/2.
Let P(A) be the power set of an n-element set A and let B be the Cartesian product of P(A) with itself. Then a(n) = the sum of the size of the union of x and y for every (x,y) in B. [See Relation (28): U(n) in document of Ross La Haye in reference.] - Bernard Schott, Jan 04 2013
A002697 is the analogous sequence if "union" is replaced by "intersection" and A002699 is the analogous sequence if "union" is replaced by "symmetric difference". Here, X union Y and Y union X are considered as two distinct Cartesian products, if we want to consider that X Union Y = Y Union X are the same Cartesian product, see A133224. - Bernard Schott Jan 11 2013

Stanislav Sykora, Table of n, a(n) for n = 1..100
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (8,-16).

Cf. A001787, A212697, A212699, A212700, A212701, A212702, A212703, A212704 (for b = 2, 3, 5, 6, 7, 8, 9, 10).

Cf. A000302, A002697, A002699, A008585, A133224.

[3*n*4^(n-1): n in [1..30]]; // Vincenzo Librandi, Nov 29 2015

Table[Sum[Binomial[n,i] i 3^i,{i,0,n}],{n,1,21}] (* Geoffrey Critzer, Aug 08 2013 *)

mtrans(n, b) = n*(b-1)*b^(n-1);
for (n=1, 100, write("b212698.txt", n, " ", mtrans(n, 4)))

a(n) = n*(b-1)*b^(n-1). For this sequence, set b=4.
a(n) = 3*n*4^(n-1).
a(n) = 3*A002697(n).
From Geoffrey Critzer, Aug 08 2013: (Start)
a(n) = Sum_{i>=0} binomial(n,i)*i*3^i.
E.g.f.: 3*x*exp(4*x). (End)
G.f.: 3*x/(4*x-1)^2. - Colin Barker, Nov 03 2014
From Elmo R. Oliveira, May 24 2025: (Start)
a(n) = 8*a(n-1) - 16*a(n-2) for n > 2.
a(n) = A008585(n)*A000302(n-1). (End)

A212703 Main transitions in systems of n particles with spin 4.

Original entry on oeis.org

8, 144, 1944, 23328, 262440, 2834352, 29760696, 306110016, 3099363912, 30993639120, 306837027288, 3012581722464, 29372671794024, 284688972772848, 2745215094595320, 26354064908115072, 252010745683850376, 2401514164751985936, 22814384565143866392, 216136274827678734240

Offset: 1

Stanislav Sykora, May 25 2012

Please, refer to the general explanation in A212697.

This sequence is for base b=9 (see formula), corresponding to spin S=(b-1)/2=4.

Stanislav Sykora, Table of n, a(n) for n = 1..100
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (18,-81).

Cf. A001787, A212697, A212698, A212699, A212700, A212701, A212702, A212704 (b = 2, 3, 4, 5, 6, 7, 8, 10).

Cf. A001019, A008590, A053540.

LinearRecurrence[{18,-81},{8,144},30] (* Harvey P. Dale, Jun 28 2017 *)

mtrans(n, b) = n*(b-1)*b^(n-1);
for (n=1, 100, write("b212703.txt", n, " ", mtrans(n, 9)))

Vec(8*x/(9*x-1)^2 + O(x^100)) \\ Colin Barker, Jun 16 2015

a(n)=8*n*9^(n-1) \\ Charles R Greathouse IV, Jun 16 2015

a(n) = n*(b-1)*b^(n-1). For this sequence, set b=9.
From Colin Barker, Jun 16 2015: (Start)
a(n) = 18*a(n-1) - 81*a(n-2) for n > 2.
G.f.: 8*x/(9*x-1)^2. (End)
From Elmo R. Oliveira, May 13 2025: (Start)
E.g.f.: 8*x*exp(9*x).
a(n) = 8*A053540(n) = A008590(n)*A001019(n-1). (End)

A212699 Main transitions in systems of n particles with spin 2.

Original entry on oeis.org

4, 40, 300, 2000, 12500, 75000, 437500, 2500000, 14062500, 78125000, 429687500, 2343750000, 12695312500, 68359375000, 366210937500, 1953125000000, 10375976562500, 54931640625000, 289916992187500, 1525878906250000, 8010864257812500, 41961669921875000, 219345092773437500

Offset: 1

Stanislav Sykora, May 25 2012

Please, refer to the general explanation in A212697.

This particular sequence is obtained for base b=5, corresponding to spin S=(b-1)/2=2.

Stanislav Sykora, Table of n, a(n) for n = 1..100
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (10,-25).

Cf. A001787, A212697, A212698, A212700, A212701, A212702, A212703, A212704 (b = 2, 3, 4, 6, 7, 8, 9, 10).

Cf. A000351, A008586, A053464.

Join[{4},Table[4n*5^(n-1),{n,20}]] (* or *) Join[{4},LinearRecurrence[{10,-25},{4,40},20]] (* Harvey P. Dale, Aug 19 2014 *)

mtrans(n, b) = n*(b-1)*b^(n-1);
for (n=1, 100, write("b212699.txt", n, " ", mtrans(n, 5)))

a(n) = n*(b-1)*b^(n-1) where b=5.
a(n) = 10*a(n-1) - 25*a(n-2), a(0)=a(1)=4, a(2)=40. - Harvey P. Dale, Aug 19 2014
From Elmo R. Oliveira, May 13 2025: (Start)
G.f.: 4*x/(5*x-1)^2.
E.g.f.: 4*x*exp(5*x).
a(n) = 4*A053464(n) = A008586(n)*A000351(n-1). (End)

A212701 Main transitions in systems of n particles with spin 3.

Original entry on oeis.org

6, 84, 882, 8232, 72030, 605052, 4941258, 39530064, 311299254, 2421216420, 18643366434, 142367525496, 1079620401678, 8138676874188, 61040076556410, 455765904954528, 3389758918099302, 25124095510618356, 185639150161791186, 1367867422244777160, 10053825553499112126

Offset: 1

Stanislav Sykora, May 25 2012

Please, refer to the general explanation in A212697.

This sequence is for base b=7 (see formula), corresponding to spin S=(b-1)/2=3.

Stanislav Sykora, Table of n, a(n) for n = 1..100
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (14,-49).

Cf. A001787, A212697, A212698, A212699, A212700, A212702, A212703, A212704 (b = 2, 3, 4, 5, 6, 8, 9, 10).

Cf. A000420, A008588, A027473.

LinearRecurrence[{14,-49},{6,84},20] (* Harvey P. Dale, Aug 02 2016 *)

mtrans(n, b) = n*(b-1)*b^(n-1);
for (n=1, 100, write("b212701.txt", n, " ", mtrans(n, 7)))

Vec(6*x/(7*x-1)^2 + O(x^100)) \\ Colin Barker, Jun 16 2015

a(n) = n*(b-1)*b^(n-1). For this sequence, set b=7.
From Colin Barker, Jun 16 2015: (Start)
a(n) = 14*a(n-1) - 49*a(n-2) for n > 2.
G.f.: 6*x/(7*x-1)^2. (End)
From Elmo R. Oliveira, May 14 2025: (Start)
E.g.f.: 6*x*exp(7*x).
a(n) = 6*A027473(n) = A008588(n)*A000420(n-1). (End)

A212702 Main transitions in systems of n particles with spin 7/2.

Original entry on oeis.org

7, 112, 1344, 14336, 143360, 1376256, 12845056, 117440512, 1056964608, 9395240960, 82678120448, 721554505728, 6253472382976, 53876069761024, 461794883665920, 3940649673949184, 33495522228568064, 283726776524341248, 2395915001761103872, 20176126330619822080

Offset: 1

Stanislav Sykora, May 25 2012

Please, refer to the general explanation in A212697.

This sequence is for base b=8 (see formula), corresponding to spin S=(b-1)/2=7/2.

Stanislav Sykora, Table of n, a(n) for n = 1..100
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (16,-64).

Cf. A001787, A212697, A212698, A212699, A212700, A212701, A212703, A212704 (b = 2, 3, 4, 5, 6, 7, 9, 10).

Cf. A001018, A008589, A053539.

LinearRecurrence[{16,-64},{7,112},30] (* Harvey P. Dale, Feb 11 2016 *)

mtrans(n, b) = n*(b-1)*b^(n-1);
for (n=1, 100, write("b212702.txt", n, " ", mtrans(n, 8)))

Vec(7*x/(8*x-1)^2 + O(x^100)) \\ Colin Barker, Jun 16 2015

a(n) = n*(b-1)*b^(n-1). For this sequence, set b=8.
From Colin Barker, Jun 16 2015: (Start)
a(n) = 16*a(n-1) - 64*a(n-2) for n > 2.
G.f.: 7*x/(8*x-1)^2. (End)
From Elmo R. Oliveira, May 14 2025: (Start)
E.g.f.: 7*x*exp(8*x).
a(n) = 7*A053539(n) = A008589(n)*A001018(n-1). (End)

A230539 a(n) = 3n2^(3*n-1).

Original entry on oeis.org

0, 12, 192, 2304, 24576, 245760, 2359296, 22020096, 201326592, 1811939328, 16106127360, 141733920768, 1236950581248, 10720238370816, 92358976733184, 791648371998720, 6755399441055744, 57420895248973824, 486388759756013568, 4107282860161892352

Offset: 0

Bruno Berselli, Oct 23 2013

Arithmetic derivative of 8^n: a(n) = A003415(8^n).

Sum of reciprocals of a(n), for n>0: (2/3)*log(8/7).

Bruno Berselli, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (16,-64).

Cf. A001018, A003415.
Cf. arithmetic derivative of k^n: A001787 (k=2), A027471 (k=3), A018215 (k=4), A053464 (k=5), A212700 (k=6), A027473 (k=7), this sequence, A230540 (k=9), A085708 (k=10), A081127 (k=11).
Row n=8 of A258997.

Magma
```
[3*n*2^(3*n-1): n in [0..20]];
```

A230539:=n->3*n*2^(3*n-1): seq(A230539(n), n=0..30); # Wesley Ivan Hurt, May 03 2017

Table[3 n 2^(3 n - 1), {n,0,20}]
LinearRecurrence[{16,-64},{0,12},20] (* Harvey P. Dale, Dec 25 2022 *)

a(n) = 3*n*2^(3*n-1); \\ Michel Marcus, Oct 23 2013

G.f.: 12*x/(1-8*x)^2.

a(n) = 12*A053539(n).

A230540 a(n) = 2n3^(2*n-1).

Original entry on oeis.org

0, 6, 108, 1458, 17496, 196830, 2125764, 22320522, 229582512, 2324522934, 23245229340, 230127770466, 2259436291848, 22029503845518, 213516729579636, 2058911320946490, 19765548681086304, 189008059262887782, 1801135623563989452, 17110788423857899794

Offset: 0

Bruno Berselli, Oct 23 2013

Arithmetic derivative of 9^n: a(n) = A003415(9^n).

Sum of reciprocals of a(n), for n>0: (3/2)*log(9/8).

Bruno Berselli, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (18,-81).

Cf. A001019, A003415.

Cf. arithmetic derivative of k^n: A001787 (k=2), A027471 (k=3), A018215 (k=4), A053464 (k=5), A212700 (k=6), A027473 (k=7), A230539 (k=8), this sequence, A085708 (k=10), A081127 (k=11).

Magma
```
[2*n*3^(2*n-1): n in [0..20]];
```
Mathematica
```
Table[2 n 3^(2 n - 1), {n, 0, 20}]
```

a(n) = 2*n*3^(2*n-1); \\ Michel Marcus, Oct 23 2013

G.f.: 6*x/(1-9*x)^2.

a(n) = 6*A053540(n), with A053540(0)=0.

A203233 (n-1)-st elementary symmetric function of the first n terms of the periodic sequence (3,2,3,2,3,2,...).

Original entry on oeis.org

1, 5, 21, 60, 216, 540, 1836, 4320, 14256, 32400, 104976, 233280, 746496, 1632960, 5178816, 11197440, 35271936, 75582720, 236825856, 503884800, 1572120576, 3325639680, 10339716096, 21767823360, 67480252416, 141490851840

Offset: 1

Clark Kimberling, Dec 30 2011

nonn

Robert Israel, Table of n, a(n) for n = 1..2559

Cf. A203232, A212700 (bisection)

f:= proc(n) if n::even then (5/12)*n*6^(n/2) else (5*n-1)*6^((n+1)/2)/24 fi
end proc:
map(f, [$1..100]); # Robert Israel, May 04 2017

r = {3, 2, 3, 2, 3, 2};
s = Flatten[{r, r, r, r, r, r, r, r, r}];
t[n_] := Part[s, Range[n]]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 32}]     (* A203233 *)

Conjecture: a(n)=12*a(n-2)-36*a(n-4) with G.f. x*(1+5*x+9*x^2) / (-1+6*x^2)^2 . - R. J. Mathar, Oct 15 2013, verified by Robert Israel, May 04 2017

a(n) = (5/12)*n*6^(n/2) if n is even, (5*n-1)*6^((n+1)/2)/24 if n is odd. - Robert Israel, May 04 2017

cp's OEIS Frontend

A212704 a(n) = 9*n*10^(n-1).

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A212697 a(n) = 2*n*3^(n-1).

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A212698 Main transitions in systems of n particles with spin 3/2.

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A212703 Main transitions in systems of n particles with spin 4.

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A212699 Main transitions in systems of n particles with spin 2.

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A212701 Main transitions in systems of n particles with spin 3.

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A212702 Main transitions in systems of n particles with spin 7/2.

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A212704 a(n) = 9n10^(n-1).

A212697 a(n) = 2n3^(n-1).

A230539 a(n) = 3n2^(3*n-1).

A230540 a(n) = 2n3^(2*n-1).