A027471 -id:A027471 - cp's OEIS frontend

A014915 a(1)=1, a(n) = n*3^(n-1) + a(n-1).

1, 7, 34, 142, 547, 2005, 7108, 24604, 83653, 280483, 930022, 3055786, 9964519, 32285041, 104029576, 333612088, 1065406345, 3389929279, 10750918570, 33996147910, 107218620331, 337346390797, 1059110761804, 3318547053652, 10379285465677, 32408789311195, 101039166676078

Offset: 1

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 1..600
Dillan Agrawal, Selena Ge, Jate Greene, Tanya Khovanova, Dohun Kim, Rajarshi Mandal, Tanish Parida, Anirudh Pulugurtha, Gordon Redwine, Soham Samanta, and Albert Xu, Chip-Firing on Infinite k-ary Trees, arXiv:2501.06675 [math.CO], 2025. See p. 14.
Alexander V. Kitaev, Meromorphic Solution of the Degenerate Third Painlevé Equation Vanishing at the Origin, arXiv:1809.00122 [math.CA], 2018.
Index entries for linear recurrences with constant coefficients, signature (7,-15,9).

Cf. A027261, A027471, A059045, A064017, A079272.

[((2*n - 1)*3^n + 1)/4: n in [1..30]]; // Vincenzo Librandi, Jun 09 2011

LinearRecurrence[{7, -15, 9}, {1, 7, 34}, 25] (* L. Edson Jeffery, May 08 2015 *)

From Henry Bottomley, Dec 18 2000: (Start)
a(n) = ((2*n-1)*3^n + 1)/4.
a(n) = 7*a(n-1) - 15*a(n-2) + 9*a(n-3) for n > 3.
a(n) = 1 + 2*3 + 3*3^2 + .. + n*3^(n-1).
a(n) = a(n-1) + A027471(n+1). (End)
G.f.: x/((1-x)*(1-3*x)^2). - Colin Barker, Jul 28 2012
a(n) = f^n(n)/2 with f(x) = 3*x-1. - Glen Gilchrist, Apr 10 2019
E.g.f.: exp(x)*(1 + exp(2*x)*(6*x - 1))/4. - Stefano Spezia, May 14 2024
a(n) = 6*a(n-1) - 9*a(n-2) + 1 for n > 2. - Elmo R. Oliveira, May 24 2025

A036219 Expansion of 1/(1-3*x)^6; 6-fold convolution of A000244 (powers of 3).

Original entry on oeis.org

1, 18, 189, 1512, 10206, 61236, 336798, 1732104, 8444007, 39405366, 177324147, 773778096, 3288556908, 13660159464, 55616363532, 222465454128, 875957725629, 3400777052442, 13036312034361, 49400761393368, 185252855225130, 688082033693340, 2533392942234570

Offset: 0

Wolfdieter Lang

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (18,-135,540,-1215,1458,-729).

Cf. A027465.

Sequences of the form 3^n*binomial(n+m, m): A000244 (m=0), A027471 (m=1), A027472 (m=2), A036216 (m=3), A036217 (m=4), this sequence (m=5), A036220 (m=6), A036221 (m=7), A036222 (m=8), A036223 (m=9), A172362 (m=10).

[3^n*Binomial(n+5, 5): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011

seq(3^n*binomial(n+5,5), n=0..30); # Zerinvary Lajos, Jun 13 2008

Table[3^n*Binomial[n+5, 5], {n, 0, 30}] (* G. C. Greubel, May 19 2021 *)
CoefficientList[Series[1/(1-3x)^6,{x,0,30}],x] (* or *) LinearRecurrence[ {18,-135,540,-1215,1458,-729},{1,18,189,1512,10206,61236},30] (* Harvey P. Dale, Jan 02 2022 *)

[3^n*binomial(n+5,5) for n in range(30)] # Zerinvary Lajos, Mar 10 2009

a(n) = 3^n*binomial(n+5, 5).
a(n) = A027465(n+6, 6).
G.f.: 1/(1-3*x)^6.
E.g.f.: (1/40)*(40 + 600*x + 1800*x^2 + 1800*x^3 + 675*x^4 + 81*x^5)*exp(3*x). - G. C. Greubel, May 19 2021
From Amiram Eldar, Sep 22 2022: (Start)
Sum_{n>=0} 1/a(n) = 240*log(3/2) - 385/4.
Sum_{n>=0} (-1)^n/a(n) = 3840*log(4/3) - 4415/4. (End)

A036220 Expansion of 1/(1-3*x)^7; 7-fold convolution of A000244 (powers of 3).

Original entry on oeis.org

1, 21, 252, 2268, 17010, 112266, 673596, 3752892, 19702683, 98513415, 472864392, 2192371272, 9865670724, 43257171636, 185387878440, 778629089448, 3211844993973, 13036312034361, 52145248137444, 205836505805700, 802762372642230, 3096369151620030

Offset: 0

Wolfdieter Lang

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (21,-189,945,-2835,5103,-5103,2187).

Cf. A027465.

Sequences of the form 3^n*binomial(n+m, m): A000244 (m=0), A027471 (m=1), A027472 (m=2), A036216 (m=3), A036217 (m=4), A036219 (m=5), this sequence (m=6), A036221 (m=7), A036222 (m=8), A036223 (m=9), A172362 (m=10).

[3^n*Binomial(n+6, 6): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011

seq(3^n*binomial(n+6,6), n=0..20); # Zerinvary Lajos, Jun 16 2008

Table[3^n*Binomial[n+6, 6], {n,0,30}] (* G. C. Greubel, May 19 2021 *)

[3^n*binomial(n+6,6) for n in range(30)] # Zerinvary Lajos, Mar 10 2009

a(n) = 3^n*binomial(n+6, 6).
a(n) = A027465(n+7,7).
G.f.: 1/(1-3*x)^7.
E.g.f.: (1/80)*(80 + 1440*x + 5400*x^2 + 7200*x^3 + 4050*x^4 + 972*x^5 + 81*x^6)*exp(3*x). - G. C. Greubel, May 19 2021
From Amiram Eldar, Sep 22 2022: (Start)
Sum_{n>=0} 1/a(n) = 1173/5 - 576*log(3/2).
Sum_{n>=0} (-1)^n/a(n) = 18432*log(4/3) - 26508/5. (End)

A061593 Number of ways to place 2n nonattacking kings on a 4 X 2n chessboard.

Original entry on oeis.org

12, 79, 408, 1847, 7698, 30319, 114606, 419933, 1501674, 5266069, 18174084, 61892669, 208424880, 695179339, 2299608732, 7552444115, 24648046806, 79994460139, 258339007890, 830619734681, 2660070154542, 8488515938929, 27000079296648, 85629004867577

Offset: 1

Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 22 2001

Bruno Berselli, Table of n, a(n) for n = 1..200
D. E. Knuth, Nonattacking kings on a chessboard, 1994.
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 90.
H. S. Wilf, The problem of the kings, Elec. J. Combin. 2, 1995.
Index entries for linear recurrences with constant coefficients, signature (9,-28,33,-9).

Column k=2 of A350819.

Cf. A061594, A001787.

[(17*n-109)*3^n+2*Fibonacci(2*n+10): n in [1..30]]; // Vincenzo Librandi, Jul 12 2011

with(combinat): A061593:=n->(17*n-109)*3^n+2*fibonacci(2*n+10): seq(A061593(n), n=1..30); # Wesley Ivan Hurt, Nov 08 2014

Table[(17 n - 109)*3^n + 2 Fibonacci[2 n + 10], {n, 30}] (* Wesley Ivan Hurt, Nov 08 2014 *)
CoefficientList[Series[x (12-29x+33x^2-9x^3)/((1-3x+x^2)(1-3x)^2),{x,0,30}],x] (* or *) LinearRecurrence[{9,-28,33,-9},{0,12,79,408,1847},30] (* Harvey P. Dale, Dec 20 2021 *)

G.f.: x*(12-29*x+33*x^2-9*x^3)/((1-3*x+x^2)*(1-3*x)^2).
a(n) = 9*a(n-1) - 28*a(n-2) + 33*a(n-3) - 9*a(n-4); a(1)=12, a(2)=79, a(3)=408, a(4)=1847.
a(n) = (17*n-109)*3^n + 2*Fibonacci(2*n+10).
a(n) = 17*A027471(n+2) - 126*A000244(n) + A025169(n+4).

A036221 Expansion of 1/(1-3*x)^8; 8-fold convolution of A000244 (powers of 3).

Original entry on oeis.org

1, 24, 324, 3240, 26730, 192456, 1250964, 7505784, 42220035, 225173520, 1148384952, 5637526128, 26778249108, 123591918960, 556163635320, 2447119995408, 10553204980197, 44695926974952, 186233029062300

Offset: 0

Wolfdieter Lang

With a different offset, number of n-permutations (n>=7) of 4 objects: u, v, z, x with repetition allowed, containing exactly seven (7) u's. Example: a(1)=24 because we have uuuuuuuv, uuuuuuuz, uuuuuuux, uuuuuuvu, uuuuuuzu, uuuuuuxu, uuuuuvuu, uuuuuzuu, uuuuuxuu, uuuuvuuu, uuuuzuuu, uuuuxuuu, uuuvuuuu, uuuzuuuu, uuuxuuuu, uuvuuuuu, uuzuuuuu, uuxuuuuu, uvuuuuuu, uzuuuuuu, uxuuuuuu, vuuuuuuu, zuuuuuuu, xuuuuuuu. - Zerinvary Lajos, Jun 23 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (24,-252,1512,-5670,13608,-20412,17496,-6561).

Cf. A027465.

Sequences of the form 3^n*binomial(n+m, m): A000244 (m=0), A027471 (m=1), A027472 (m=2), A036216 (m=3), A036217 (m=4), A036219 (m=5), A036220 (m=6), this sequence (m=7), A036222 (m=8), A036223 (m=9), A172362 (m=10).

[3^n*Binomial(n+7, 7): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011

seq(3^n*binomial(n+7,7), n=0..30); # Zerinvary Lajos, Jun 23 2008

Table[3^n*Binomial[n+7,7], {n,0,30}] (* G. C. Greubel, May 19 2021 *)

[3^n*binomial(n+7, 7) for n in range(30)] # Zerinvary Lajos, Mar 13 2009

a(n) = 3^n*binomial(n+7, 7).
a(n) = A027465(n+8, 8.)
G.f.: 1/(1-3*x)^8.
E.g.f.: (1/560)*(560 +11760*x +52920*x^2 +88200*x^3 +66150*x^4 +23814*x^5 +3969*x^6 +243*x^7)*exp(3*x). - G. C. Greubel, May 19 2021
From Amiram Eldar, Sep 22 2022: (Start)
Sum_{n>=0} 1/a(n) = 1344*log(3/2) - 5439/10.
Sum_{n>=0} (-1)^n/a(n) = 86016*log(4/3) - 247443/10. (End)

A036222 Expansion of 1/(1-3*x)^9; 9-fold convolution of A000244 (powers of 3).

Original entry on oeis.org

1, 27, 405, 4455, 40095, 312741, 2189187, 14073345, 84440070, 478493730, 2583866142, 13389124554, 66945622770, 324428787270, 1529449997130, 7035469986798, 31659614940591, 139674771796725, 605257344452475

Offset: 0

Wolfdieter Lang

With a different offset, number of n-permutations (n>=8) of 4 objects: u, v, z, x with repetition allowed, containing exactly eight (8) u's. Example: a(1)=27 because we have uuuuuuuuv, uuuuuuuuz, uuuuuuuux, uuuuuuuvu, uuuuuuuzu, uuuuuuuxu, uuuuuuvuu, uuuuuuzuu, uuuuuuxuu, uuuuuvuuu, uuuuuzuuu, uuuuuxuuu, uuuuvuuuu, uuuuzuuuu, uuuuxuuuu, uuuvuuuuu, uuuzuuuuu, uuuxuuuuu, uuvuuuuuu, uuzuuuuuu, uuxuuuuuu, uvuuuuuuu, uzuuuuuuu, uxuuuuuuu, vuuuuuuuu, zuuuuuuuu, xuuuuuuuu. - Zerinvary Lajos, Jun 23 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (27,-324,2268,-10206,30618,-61236,78732,-59049,19683).

Cf. A027465.

Sequences of the form 3^n*binomial(n+m, m): A000244 (m=0), A027471 (m=1), A027472 (m=2), A036216 (m=3), A036217 (m=4), A036219 (m=5), A036220 (m=6), A036221 (m=7), this sequence (m=8), A036223 (m=9), A172362 (m=10).

[3^n*Binomial(n+8, 8): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011

seq(3^n*binomial(n+8,8), n=0..18); # Zerinvary Lajos, Jun 23 2008

Table[3^n*Binomial[n+8, 8], {n, 0, 20}] (* Zerinvary Lajos, Jan 31 2010 *)
CoefficientList[Series[1/(1-3x)^9,{x,0,30}],x] (* or *) LinearRecurrence[{27,-324, 2268,-10206,30618,-61236,78732,-59049,19683}, {1,27,405,4455,40095,312741, 2189187,14073345,84440070}, 30] (* Harvey P. Dale, Jan 07 2016 *)

[3^n*binomial(n+8, 8) for n in range(30)] # Zerinvary Lajos, Mar 13 2009

a(n) = 3^n*binomial(n+8, 8).
a(n) = A027465(n+9, 9).
G.f.: 1/(1-3*x)^9.
a(0)=1, a(1)=27, a(2)=405, a(3)=4455, a(4)=40095, a(5)=312741, a(6)=2189187, a(7)=14073345, a(8)=84440070, a(n) = 27*a(n-1) - 324*a(n-2) + 2268*a(n-3) - 10206*a(n-4) + 30618*a(n-5) - 61236*a(n-6) + 78732*a(n-7) - 59049*a(n-8) + 19683*a(n-9). - Harvey P. Dale, Jan 07 2016
From Amiram Eldar, Sep 22 2022: (Start)
Sum_{n>=0} 1/a(n) = 43632/35 - 3072*log(3/2).
Sum_{n>=0} (-1)^n/a(n) = 393216*log(4/3) - 3959208/35. (End)

A036223 Expansion of 1/(1-3*x)^10; 10-fold convolution of A000244 (powers of 3).

Original entry on oeis.org

1, 30, 495, 5940, 57915, 486486, 3648645, 25019280, 159497910, 956987460, 5454828522, 29753610120, 156206453130, 793048146660, 3908594437110, 18761253298128, 87943374834975, 403504896301650, 1815772033357425

Offset: 0

Wolfdieter Lang

With a different offset, number of n-permutations (n >= 9) of 4 objects: u, v, z, x with repetition allowed, containing exactly nine (9) u's. - Zerinvary Lajos, Jul 02 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (30,-405,3240,-17010,61236,-153090,262440,-295245,196830,-59049).

Cf. A027465.

Sequences of the form 3^n*binomial(n+m, m): A000244 (m=0), A027471 (m=1), A027472 (m=2), A036216 (m=3), A036217 (m=4), A036219 (m=5), A036220 (m=6), A036221 (m=7), A036222 (m=8), this sequence (m=9), A172362 (m=10).

[3^n*Binomial(n+9, 9): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011

seq(3^n*binomial(n+9, 9), n=0..20); # Zerinvary Lajos, Jul 02 2008

Table[3^n*Binomial[n+9,9], {n,0,30}] (* G. C. Greubel, May 18 2021 *)
CoefficientList[Series[1/(1-3x)^10,{x,0,30}],x] (* or *) LinearRecurrence[ {30,-405,3240,-17010,61236,-153090,262440,-295245,196830,-59049},{1,30,495,5940,57915,486486,3648645,25019280,159497910,956987460},30] (* Harvey P. Dale, Jan 16 2022 *)

[3^n*binomial(n+9,9) for n in range(30)] # Zerinvary Lajos, Mar 13 2009

a(n) = 3^n*binomial(n+9, 9).
a(n) = A027465(n+10, 10).
G.f.: 1/(1-3*x)^10.
E.g.f.: (4480 + 120960*x + 725760*x^2 + 1693440*x^3 + 1905120*x^4 + 1143072*x^5 + 381024*x^6 + 69984*x^7 + 6561*x^8 + 243*x^9)*exp(3*x)/4480. - G. C. Greubel, May 18 2021
From Amiram Eldar, Sep 22 2022: (Start)
Sum_{n>=0} 1/a(n) = 6912*log(3/2) - 784431/280.
Sum_{n>=0} (-1)^n/a(n) = 1769472*log(4/3) - 142532433/280. (End)

A038221 Triangle whose (i,j)-th entry is binomial(i,j)3^(i-j)3^j.

Original entry on oeis.org

1, 3, 3, 9, 18, 9, 27, 81, 81, 27, 81, 324, 486, 324, 81, 243, 1215, 2430, 2430, 1215, 243, 729, 4374, 10935, 14580, 10935, 4374, 729, 2187, 15309, 45927, 76545, 76545, 45927, 15309, 2187, 6561, 52488, 183708, 367416, 459270, 367416, 183708, 52488, 6561

Offset: 0

N. J. A. Sloane

Triangle of coefficients in expansion of (3 + 3x)^n = 3^n (1 +x)^n, where n is a nonnegative integer. (Coefficients in expansion of (1 +x)^n are given in A007318: Pascal's triangle). - Zagros Lalo, Jul 23 2018

			Triangle begins as:
     1;
     3,     3;
     9,    18,      9;
    27,    81,     81,     27;
    81,   324,    486,    324,     81;
   243,  1215,   2430,   2430,   1215,    243;
   729,  4374,  10935,  14580,  10935,   4374,    729;
  2187, 15309,  45927,  76545,  76545,  45927,  15309,  2187;
  6561, 52488, 183708, 367416, 459270, 367416, 183708, 52488, 6561;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 44, 48

Indranil Ghosh, Rows 0..100 of triangle, flattened
B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.

Cf. A007318, A304236, A304249.
Cf. A000007, A000400, A027465, A030195, A057083.
Columns k: A000244 (k=0), 3*A027471 (k=1), 3^2*A027472 (k=2), 3^3*A036216 (k=3), 3^4*A036217 (k=4), 3^5*A036219 (k=5), 3^6*A036220 (k=6), 3^7*A036221 (k=7), 3^8*A036222 (k=8), 3^9*A036223 (k=9), 3^10*A172362 (k=10).

Flat(List([0..8],i->List([0..i],j->Binomial(i,j)*3^(i-j)*3^j))); # Muniru A Asiru, Jul 23 2018

a038221 n = a038221_list !! n
a038221_list = concat $ iterate ([3,3] *) [1]
instance Num a => Num [a] where
   fromInteger k = [fromInteger k]
   (p:ps) + (q:qs) = p + q : ps + qs
   ps + qs         = ps ++ qs
   (p:ps) * qs'@(q:qs) = p * q : ps * qs' + [p] * qs
    *                = []
-- Reinhard Zumkeller, Apr 02 2011

[3^n*Binomial(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Oct 17 2022

(* programs from Zagros Lalo, Jul 23 2018 *)
t[0, 0]=1; t[n_, k_]:= t[n, k]= If[n<0 || k<0, 0, 3 t[n-1, k] + 3 t[n-1, k-1]]; Table[t[n, k], {n,0,10}, {k,0,n}]//Flatten
Table[CoefficientList[Expand[3^n *(1+x)^n], x], {n,0,10}]//Flatten
Table[3^n Binomial[n, k], {n,0,10}, {k,0,n}]//Flatten  (* End *)

def A038221(n,k): return 3^n*binomial(n,k)
flatten([[A038221(n,k) for k in range(n+1)] for n in range(10)]) # G. C. Greubel, Oct 17 2022

G.f.: 1/(1 - 3*x - 3*x*y). - Ilya Gutkovskiy, Apr 21 2017
T(0,0) = 1; T(n,k) = 3 T(n-1,k) + 3 T(n-1,k-1) for k = 0...n; T(n,k)=0 for n or k < 0. - Zagros Lalo, Jul 23 2018
From G. C. Greubel, Oct 17 2022: (Start)
T(n, k) = T(n, n-k).
T(n, n) = A000244(n).
T(n, n-1) = 3*A027471(n).
T(n, n-2) = 9*A027472(n+1).
T(n, n-3) = 27*A036216(n-3).
T(n, n-4) = 81*A036217(n-4).
T(n, n-5) = 243*A036219(n-5).
Sum_{k=0..n} T(n, k) = A000400(n).
Sum_{k=0..n} (-1)^k * T(n, k) = A000007(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A030195(n+1), n >= 0.
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A057083(n).
T(n, k) = 3^k * A027465(n, k). (End)

A053469 a(n) = n*6^(n-1).

Original entry on oeis.org

1, 12, 108, 864, 6480, 46656, 326592, 2239488, 15116544, 100776960, 665127936, 4353564672, 28298170368, 182849716224, 1175462461440, 7522959753216, 47958868426752, 304679870005248, 1929639176699904, 12187194800209920, 76779327241322496, 482612914088312832

Offset: 1

Barry E. Williams, Jan 13 2000

Binomial transform of A053464. - R. J. Mathar, Oct 26 2011

			G.f. = x + 12*x^2 + 108*x^3 + 864*x^4 + 6480*x^5 + 46656*x^6 + ... - _Michael Somos_, Dec 16 2019

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Vincenzo Librandi, Table of n, a(n) for n = 1..400
Frank Ellermann, Illustration of binomial transforms.
Index entries for linear recurrences with constant coefficients, signature (12,-36).

Cf. A002697, A027471.

[n*(6^(n-1)): n in [1..30]]; // Vincenzo Librandi, Jun 09 2011

f[n_]:=n*6^(n-1);f[Range[40]] (* Vladimir Joseph Stephan Orlovsky, Feb 09 2011 *)
LinearRecurrence[{12,-36},{1,12},20] (* Harvey P. Dale, Apr 28 2015 *)

a(n)=n*6^(n-1) \\ Charles R Greathouse IV, Oct 07 2015

[lucas_number1(n,12,36) for n in range(1, 21)] # Zerinvary Lajos, Apr 28 2009

a(n) = 12*a(n-1) - 36*a(n-2), n>=3.
G.f.: x/(6x-1)^2. - Zerinvary Lajos, Apr 28 2009
E.g.f.: x*exp(6*x). - Michael Somos, Dec 16 2019
From Amiram Eldar, Oct 28 2020: (Start)
Sum_{n>=1} 1/a(n) = 6*log(6/5).
Sum_{n>=1} (-1)^(n+1)/a(n) = 6*log(7/6). (End)

More terms from James Sellers, Feb 02 2000

More terms from Zerinvary Lajos, Oct 02 2007

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

cp's OEIS Frontend

A014915 a(1)=1, a(n) = n*3^(n-1) + a(n-1).

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A036219 Expansion of 1/(1-3*x)^6; 6-fold convolution of A000244 (powers of 3).

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A036220 Expansion of 1/(1-3*x)^7; 7-fold convolution of A000244 (powers of 3).

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A061593 Number of ways to place 2n nonattacking kings on a 4 X 2n chessboard.

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A036221 Expansion of 1/(1-3*x)^8; 8-fold convolution of A000244 (powers of 3).

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A036222 Expansion of 1/(1-3*x)^9; 9-fold convolution of A000244 (powers of 3).

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A036223 Expansion of 1/(1-3*x)^10; 10-fold convolution of A000244 (powers of 3).

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A038221 Triangle whose (i,j)-th entry is binomial(i,j)3^(i-j)3^j.

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