A027468 -id:A027468 - cp's OEIS frontend

A064222 a(0) = 0; a(n) = DecimalDigitsSortedDecreasing(a(n-1) + 1) for n > 0.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 21, 22, 32, 33, 43, 44, 54, 55, 65, 66, 76, 77, 87, 88, 98, 99, 100, 110, 111, 211, 221, 222, 322, 332, 333, 433, 443, 444, 544, 554, 555, 655, 665, 666, 766, 776, 777, 877, 887, 888, 988, 998, 999, 1000, 1100, 1110, 1111, 2111

Offset: 0

Reinhard Zumkeller, Sep 21 2001

a(n) = A004186(a(n-1) + 1). - Reinhard Zumkeller, Oct 31 2007

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A004186, A004216, A027468.

a064222 n = a064222_list !! n
a064222_list = iterate (a004186 . (+ 1)) 0
-- Reinhard Zumkeller, Apr 11 2012

NestList[FromDigits[Sort[IntegerDigits[#+1],Greater]]&,0,60] (* Harvey P. Dale, Sep 04 2011 *)

a(n+1) = (d+0^d)*10^floor(log_10(a(n)+1)) + (1-0^d)*floor(a(n)/10), where d = (a(n)+1) mod 10. - Reinhard Zumkeller, Oct 31 2007

a(n) = (ceiling( (n-G(D(n)-1))/D(n) )*(10^D(n) -1) - 10^( (G(D(n)-1)-n) mod (D(n)) ) + 1)/9, for n>0, where D(n) = floor( (sqrt(8n+1)+3)/6 ) is the number of digits in a(n), and G(k) = A027468(k) = 9*k*(k+1)/2. - Stefan Alexandru Avram, May 24 2023

A283394 a(n) = 3n(3*n + 7)/2 + 4.

Original entry on oeis.org

4, 19, 43, 76, 118, 169, 229, 298, 376, 463, 559, 664, 778, 901, 1033, 1174, 1324, 1483, 1651, 1828, 2014, 2209, 2413, 2626, 2848, 3079, 3319, 3568, 3826, 4093, 4369, 4654, 4948, 5251, 5563, 5884, 6214, 6553, 6901, 7258, 7624, 7999, 8383, 8776, 9178, 9589, 10009

Offset: 0

Bruno Berselli, Mar 23 2017

Sum_{k = 0..n} (3*k + r)^3 is divisible by 3*n*(3*n + 2*r + 3)/2 + r^2: the sequence corresponds to the case r = 2 of this formula (other cases are listed in Crossrefs section).
Also, Sum_{k = 0..n} (3*k + 2)^3 / a(n) gives 2, 7, 15, 26, 40, 57, 77, 100, 126, 155, 187, 222, ... (A005449).
a(n) is even if n belongs to A014601. No term is divisible by 3, 5, 7 and 11.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Nickolas Arustamyan, Christopher Cox, Erik Lundberg, Sean Perry, and Zvi Rosen, On the Number of Equilibria Balancing Newtonian Point Masses with a Central Force, arXiv:2106.11416 [math-ph], 2021.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005449, A034856, A095794, A101881, A165157, A186349.

Sequences with formula 3*n*(3*n + 2*r + 3)/2 + r^2: A038764 (r=-1), A027468 (r=0), A081271 (r=1), this sequence (r=2), A027468 (r=3; offset: -1), A080855 (r=4; offset: -2).

Magma
```
[3*n*(3*n+7)/2+4: n in [0..50]];
```

Table[3 n (3 n + 7)/2 + 4, {n, 0, 50}]
LinearRecurrence[{3,-3,1},{4,19,43},50] (* Harvey P. Dale, Mar 02 2019 *)

Maxima
```
makelist(3*n*(3*n+7)/2+4, n, 0, 50);
```

a(n) = 3*n*(3*n + 7)/2 + 4; \\ Indranil Ghosh, Mar 24 2017

Python
```
[3*n*(3*n+7)/2+4 for n in range(50)]
```
Sage
```
[3*n*(3*n+7)/2+4 for n in range(50)]
```

O.g.f.: (4 + 7*x - 2*x^2)/(1 - x)^3.
E.g.f.: (8 + 30*x + 9*x^2)*exp(x)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = A081271(-n-2).
a(n) = 3*A095794(n+1) + 1.
a(n) = A034856(3*n+2) = A101881(6*n+2) = A165157(6*n+3) = A186349(6*n+3).
The inverse binomial transform yields 4, 15, 9, 0 (0 continued), therefore:
a(n) = 4*binomial(n,0) + 15*binomial(n,1) + 9*binomial(n,2).

A102741 a(n) = 3^4 * binomial(n+3, 4).

Original entry on oeis.org

81, 405, 1215, 2835, 5670, 10206, 17010, 26730, 40095, 57915, 81081, 110565, 147420, 192780, 247860, 313956, 392445, 484785, 592515, 717255, 860706, 1024650, 1210950, 1421550, 1658475, 1923831, 2219805, 2548665, 2912760, 3314520, 3756456, 4241160, 4771305, 5349645

Offset: 1

Zerinvary Lajos, Aug 06 2008

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A027465.

Sequences of the form 3^m*binomial(n+m-1, m): A008585 (m=1), A027468 (m=2), A134171 (m=3), this sequence (m=4), A113335 (m=5).

[3^4*Binomial(n+3,4): n in [1..30]]; // G. C. Greubel, May 17 2021

Maple
```
seq(binomial(n+3,4)*3^4, n=1..27);
```

With[{c=3^4},Table[c Binomial[n+3,4],{n,40}]]  (* Harvey P. Dale, Mar 12 2011 *)

[3^4*binomial(n+3,4) for n in (1..30)] # G. C. Greubel, May 17 2021

G.f.: 81*x/(1-x)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009
E.g.f.: (27/8)*x*(24 + 36*x + 12*x^2 + x^3)*exp(x). - G. C. Greubel, May 17 2021
From Amiram Eldar, Aug 28 2022: (Start)
Sum_{n>=1} 1/a(n) = 4/243.
Sum_{n>=1} (-1)^(n+1)/a(n) = 32*log(2)/81 - 64/243. (End)

A134171 a(n) = (9/2)(n-1)(n-2)*(n-3).

Original entry on oeis.org

0, 0, 0, 27, 108, 270, 540, 945, 1512, 2268, 3240, 4455, 5940, 7722, 9828, 12285, 15120, 18360, 22032, 26163, 30780, 35910, 41580, 47817, 54648, 62100, 70200, 78975, 88452, 98658, 109620, 121365, 133920, 147312, 161568, 176715, 192780, 209790, 227772, 246753

Offset: 1

N. J. A. Sloane, Jan 30 2008

Number of n permutations (n>=3) of 4 objects u, v, z, x with repetition allowed, containing n-3=0 u's. Example: if n=3 then n-3 =zero u, a()=27 because we have vzx, vxz, zvx, zxv, xvz, xzv, vvv, zzz, xxx, vvx, vxv, xvv, xxv, xvx, vxx, vvz, vzv, zvv, zzv, zvz, vzz, xzz, zxz, zzx, xxz, xzx, zxx. A027465 formatted as a triangular array: diagonal: 27, 108, 270, 540, 945, 1512. - Zerinvary Lajos, Aug 06 2008

G. C. Greubel, Table of n, a(n) for n = 1..1000
D. Zvonkine, Home Page.
D. Zvonkine, Counting ramified coverings and intersection theory on Hurwitz spaces II (local structure of Hurwitz spaces and combinatorial results), Moscow Mathematical Journal, Vol. 7, No. 1 (2007), 135-162.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A008585, A027465, A027468. - Zerinvary Lajos, Aug 06 2008

Cf. A000292, A102741, A113335.

[(9/2)*(n-1)*(n-2)*(n-3) : n in [1..50]]; // Wesley Ivan Hurt, May 29 2016

seq(27*binomial(n-1, 3), n=1..30); # Zerinvary Lajos, May 18 2008

LinearRecurrence[{4,-6,4,-1}, {0,0,0,27}, 50] (* G. C. Greubel, May 29 2016 *)

a(n) = 27 * binomial(n-1,3). - Zerinvary Lajos, Aug 06 2008
From Chai Wah Wu, May 29 2016: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>4.
G.f.: 27*x^4/(1-x)^4. (End)
E.g.f.: 27 + (9/2*(x^3-3*x^2+6*x-6))*exp(x). - G. C. Greubel, May 17 2021
a(n) = 27 * A000292(n-3) for n >= 3. - Alois P. Heinz, May 17 2021
From Amiram Eldar, Sep 24 2022: (Start)
Sum_{n>=4} 1/a(n) = 1/18.
Sum_{n>=4} (-1)^n/a(n) = 4*log(2)/9 - 5/18. (End)

A113335 a(n) = 3^5 * binomial(n+4, 5).

Original entry on oeis.org

243, 1458, 5103, 13608, 30618, 61236, 112266, 192456, 312741, 486486, 729729, 1061424, 1503684, 2082024, 2825604, 3767472, 4944807, 6399162, 8176707, 10328472, 12910590, 15984540, 19617390, 23882040, 28857465, 34628958, 41288373, 48934368, 57672648, 67616208

Offset: 1

Zerinvary Lajos, Aug 06 2008

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A027465.

Sequences of the form 3^m*binomial(n+m-1, m): A008585 (m=1), A027468 (m=2), A134171 (m=3), A102741 (m=4), this sequence (m=5).

[3^5*Binomial(n+4,5): n in [1..30]]; // G. C. Greubel, May 17 2021

Maple
```
seq(binomial(n+4,5)*3^5, n=1..27);
```

With[{c=3^5},Table[c Binomial[n+4,5],{n,30}]]  (* Harvey P. Dale, Apr 11 2011 *)

[3^5*binomial(n+4,5) for n in (1..30)] # G. C. Greubel, May 17 2021

a(n) = 3^5 * binomial(n+4, 5), n >= 1.
From G. C. Greubel, May 17 2021: (Start)
G.f.: 243*x/(1-x)^6.
E.g.f.: (81/40)*x*(120 + 240*x + 120*x^2 + 20*x^3 + x^4)*exp(x). (End)
From Amiram Eldar, Aug 29 2022: (Start)
Sum_{n>=1} 1/a(n) = 5/972.
Sum_{n>=1} (-1)^(n+1)/a(n) = 80*log(2)/243 - 655/2916. (End)

A198392 a(n) = (6n(3n+7)+(2n+13)*(-1)^n+3)/16 + 1.

Original entry on oeis.org

2, 4, 12, 18, 31, 41, 59, 73, 96, 114, 142, 164, 197, 223, 261, 291, 334, 368, 416, 454, 507, 549, 607, 653, 716, 766, 834, 888, 961, 1019, 1097, 1159, 1242, 1308, 1396, 1466, 1559, 1633, 1731, 1809, 1912, 1994, 2102, 2188, 2301, 2391, 2509, 2603, 2726, 2824, 2952

Offset: 0

Bruno Berselli, Oct 25 2011

For an origin of this sequence, see the triangular spiral illustrated in the Links section.

First bisection gives A117625 (without the initial term).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A152832 (by Superseeker).

Cf. sequences related to the triangular spiral: A022266, A022267, A027468, A038764, A045946, A051682, A062708, A062725, A062728, A062741, A064225, A064226, A081266-A081268, A081270-A081272, A081275 [incomplete list].

[(6*n*(3*n+7)+(2*n+13)*(-1)^n+3)/16+1: n in [0..50]];

LinearRecurrence[{1,2,-2,-1,1},{2,4,12,18,31},60] (* Harvey P. Dale, Jun 15 2022 *)

for(n=0, 50, print1((6*n*(3*n+7)+(2*n+13)*(-1)^n+3)/16+1", "));

G.f.: (2+2*x+4*x^2+2*x^3-x^4)/((1+x)^2*(1-x)^3).
a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5).
a(n)-a(-n-1) = A168329(n+1).
a(n)+a(n-1) = A102214(n).
a(2n)-a(2n-1) = A016885(n).
a(2n+1)-a(2n) = A016825(n).

A275414 Triangle read by rows: T(n,k) is the number of multisets of k ternary words with a total of n letters.

Original entry on oeis.org

3, 9, 6, 27, 27, 10, 81, 126, 54, 15, 243, 486, 297, 90, 21, 729, 1836, 1380, 540, 135, 28, 2187, 6561, 5994, 2763, 855, 189, 36, 6561, 23004, 24543, 13212, 4635, 1242, 252, 45, 19683, 78732, 96723, 59130, 23490, 6996, 1701, 324, 55, 59049, 265842, 368874, 253719

Offset: 1

R. J. Mathar, Jul 27 2016

Ternary analog of A209406. Multiset transformation of A000244.

			      3
      9       6
     27      27      10
     81     126      54      15
    243     486     297      90      21
    729    1836    1380     540     135      28
   2187    6561    5994    2763     855     189      36
   6561   23004   24543   13212    4635    1242     252      45
  19683   78732   96723   59130   23490    6996    1701     324      55
  59049  265842  368874  253719  111609   36828    9846    2232     405      66

Alois P. Heinz, Rows n = 1..141, flattened
Index entries for triangles generated by the Multiset Transformation

Cf. A144067 (row sums), A000244 (column 1), A027468 (subdiagonal ?).

b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
      `if`(min(i, p)<1, 0, add(b(n-i*j, i-1, p-j)*
       binomial(3^i+j-1, j), j=0..min(n/i, p)))))
    end:
T:= (n, k)-> b(n$2, k):
seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Apr 13 2017

b[n_, i_, p_] := b[n, i, p] = If[p > n, 0, If[n == 0, 1, If[Min[i, p] < 1, 0, Sum[b[n - i*j, i-1, p - j]*Binomial[3^i + j - 1, j], {j, 0, Min[n/i, p]}]]]];
T[n_, k_] := b[n, n, k];
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 19 2018, after Alois P. Heinz *)

T(n,1) = A000244(n).

T(n,k) = Sum_{c_i*N_i=n,i=1..k} binomial(T(N_i,1)+c_i-1,c_i) for 1

G.f.: Product_{j>=1} (1-y*x^j)^(-3^j). - Alois P. Heinz, Apr 13 2017

A370347 Number T(n,k) of partitions of [3n] into n sets of size 3 having exactly k sets {3j-2,3j-1,3j} (1<=j<=n); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 9, 0, 1, 252, 27, 0, 1, 14337, 1008, 54, 0, 1, 1327104, 71685, 2520, 90, 0, 1, 182407545, 7962624, 215055, 5040, 135, 0, 1, 34906943196, 1276852815, 27869184, 501795, 8820, 189, 0, 1, 8877242235393, 279255545568, 5107411260, 74317824, 1003590, 14112, 252, 0, 1

Offset: 0

Alois P. Heinz, Feb 15 2024

			T(2,0) = 9: 124|356, 125|346, 126|345, 134|256, 135|246, 136|245, 145|236, 146|235, 156|234.
T(2,2) = 1: 123|456.
Triangle T(n,k) begins:
            1;
            0,          1;
            9,          0,        1;
          252,         27,        0,      1;
        14337,       1008,       54,      0,    1;
      1327104,      71685,     2520,     90,    0,   1;
    182407545,    7962624,   215055,   5040,  135,   0, 1;
  34906943196, 1276852815, 27869184, 501795, 8820, 189, 0, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Partition of a set

Row sums give A025035.
Column k=0 gives A370357.
T(n+1,n-1) gives A027468.
T(n+2,n-1) gives 252*A000292.
Cf. A023531, A055140, A370358.

b:= proc(n) option remember; `if`(n<3, [1, 0, 9][n+1],
      9*(n*(n-1)/2*b(n-1)+(n-1)^2*b(n-2)+(n-1)*(n-2)/2*b(n-3)))
    end:
T:= (n, k)-> b(n-k)*binomial(n, k):
seq(seq(T(n, k), k=0..n), n=0..10);

T(n,k) = binomial(n,k) * A370357(n-k).
Sum_{k=1..n} T(n,k) = A370358(n).
T(n,k) mod 9 = A023531(n,k).

A124110 Primes of the form A124080 (10 times triangular numbers) +- 1.

Original entry on oeis.org

11, 29, 31, 59, 61, 101, 149, 151, 211, 281, 359, 449, 659, 661, 911, 1049, 1051, 1201, 1361, 1531, 1709, 1901, 2099, 2309, 2311, 2531, 2999, 3001, 3251, 3511, 3779, 4349, 4649, 4651, 5279, 5281, 6299, 6301, 6659, 6661, 7411, 8609, 9029, 9461, 9901, 11279

Offset: 1

Jonathan Vos Post, Nov 26 2006

Numbers j such that A124080(j)-1 is prime or A124080(j)+1 is prime, where repetition means a twin prime, are 1, 2, 2, 3, 3, 4, 5, 5, 6, 7, 8, 9, 11, 11, 13, 14, 14, 15, 16, 17, 18, 19, 20, 21, 21, 22, 24, 24, 25, ..., . - Robert G. Wilson v, Nov 29 2006

			a(1) = A124080(1)+1 = (10*T(1)) - 1 = 10*(1*(1+1)/2) + 1 = 10+1 = 11 is prime.
a(2) = A124080(2)-1 = (10*T(2))-1 = 10*(2*(2+1)/2) - 1 = 30-1 = 29 is prime.
a(3) = A124080(2)+1 = (10*T(2))+1 = 10*(2*(2+1)/2) + 1 = 30+1 = 31 is prime.

Cf. A000040, A000217, A028895, A046092, A045943, A002378, A028896, A024966, A033996, A027468.

s = {}; Do[t = 5n(n + 1); If[PrimeQ[t - 1], AppendTo[s, t - 1]]; If[PrimeQ[t + 1], AppendTo[s, t + 1]], {n, 47}]; s (* Robert G. Wilson v *)

{A124080(j)-1 when prime} U {A124080(j)+1 when prime} = {i = 10*T(j)-1 such that i is prime} U {i = 10*T(j)+1 such that i is prime} where T(j) = A000217(j) = j*(j+1)/2.

More terms from Robert G. Wilson v, Nov 29 2006

A124007 Number of permutations of n distinct letters (ABCD...) each of which appears thrice with n-3 fixed points.

Original entry on oeis.org

0, 0, 54, 216, 540, 1080, 1890, 3024, 4536, 6480, 8910, 11880, 15444, 19656, 24570

Offset: 0

Zerinvary Lajos, Nov 01 2006

nonn

			Maple produces the following triangle - the entries in quotes give the sequence:
1
"0", 0, 0, 1
1, 0, 9, "0", 9, 0, 1
56, 216, 378, 435, 324, 189, "54", 27, 0, 1
13833, 49464, 84510, 90944, 69039, 38448, 16476, 5184, 1431, "216", 54, 0, 1
6699824, 23123880, 38358540, 40563765, 30573900, 17399178, 7723640, 2729295, 776520, 180100, 33372, 5355, "540", 90, 0, 1
etc...

Cf. A059058, A027468, A059073, A000459.

p := (x, k)->k!^2*sum(x^j/((k-j)!^2*j!), j=0..k); R := (x, n, k)->p(x, k)^n; f := (t, n, k)->sum(coeff(R(x, n, k), x, j)*(t-1)^j*(n*k-j)!, j=0..n*k); for n from 0 to 6 do seq(coeff(f(t, n, 3), t, m)/3!^n, m=0..3*n); od;

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cp's OEIS Frontend

A064222 a(0) = 0; a(n) = DecimalDigitsSortedDecreasing(a(n-1) + 1) for n > 0.

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A283394 a(n) = 3*n*(3*n + 7)/2 + 4.

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A102741 a(n) = 3^4 * binomial(n+3, 4).

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A134171 a(n) = (9/2)*(n-1)*(n-2)*(n-3).

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Magma

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A113335 a(n) = 3^5 * binomial(n+4, 5).

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Magma

Maple

Mathematica

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A198392 a(n) = (6*n*(3*n+7)+(2*n+13)*(-1)^n+3)/16 + 1.

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Magma

Mathematica

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Formula

A275414 Triangle read by rows: T(n,k) is the number of multisets of k ternary words with a total of n letters.

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A283394 a(n) = 3n(3*n + 7)/2 + 4.

A134171 a(n) = (9/2)(n-1)(n-2)*(n-3).

A198392 a(n) = (6n(3n+7)+(2n+13)*(-1)^n+3)/16 + 1.