A036290 -id:A036290 - cp's OEIS frontend

A120910 Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.

3, 6, 3, 12, 12, 3, 24, 36, 18, 3, 48, 96, 72, 24, 3, 96, 240, 240, 120, 30, 3, 192, 576, 720, 480, 180, 36, 3, 384, 1344, 2016, 1680, 840, 252, 42, 3, 768, 3072, 5376, 5376, 3360, 1344, 336, 48, 3, 1536, 6912, 13824, 16128, 12096, 6048, 2016, 432, 54, 3, 3072

Offset: 1

Emeric Deutsch, Jul 15 2006

Row sums are the powers of 3 (A000244).
T(n,k) = 3*A038207(n-1,k).
T(n,k) = A120909(n,n-k).
Sum_{k>=0} k*T(n,k) = (n-1)*3^(n-1) = A036290(n-1).

			T(3,1)=12 because we have 001,002,011,022,100,110,112,122,200,211,220 and 221.
Triangle starts:
   3;
   6,  3
  12, 12,  3;
  24, 36, 18,  3;
  48, 96, 72, 24,  3;

Cf. A000244, A038207, A120909, A036290.

T:=(n,k)->3*2^(n-k-1)*binomial(n-1,k): for n from 1 to 11 do seq(T(n,k),k=0..n-1) od; # yields sequence in triangular form

sol=Solve[{a==v(z^2+a z),b==v(z^2+b z),c==v(z^2+c z)},{a,b,c}];f[z_,u_]:=1/(1-3z-a-b-c)/.sol/.v->u-1;nn=10;Drop[Map[Select[#,#>0&]&,Level[CoefficientList[Series[f[z,u],{z,0,nn}],{z,u}],{2}]],1]//Grid (* Geoffrey Critzer, May 19 2014 *)

T(n,k) = 3*2^(n-k-1)*binomial(n-1,k).

G.f.: (1 - (y - 1)*x)/(1 - (y + 2)*x). Generally for the number of length n words with k levels on an m-ary alphabet (m>1): (1 - (y - 1)*x)/(1 - (y + m - 1)*x). - Geoffrey Critzer, May 19 2014

A178756 Rectangular array T(n,k) = binomial(n,2)kn^(k-1) read by antidiagonals.

Original entry on oeis.org

1, 4, 3, 12, 18, 6, 32, 81, 48, 10, 80, 324, 288, 100, 15, 192, 1215, 1536, 750, 180, 21, 448, 4374, 7680, 5000, 1620, 294, 28, 1024, 15309, 36864, 31250, 12960, 3087, 448, 36, 2304, 52488, 172032, 187500, 97200, 28812, 5376, 648, 45

Offset: 2

Geoffrey Critzer, Dec 26 2010

T(n,k) is the sum of the digits in all n-ary words of length k. That is, sequences of k digits taken on an alphabet of {0,1,2,...,n-1}.

Note the rectangle is indexed begining from n = 2 (binary sequences) which is A001787.

			1,4,12,32,80,192,448,1024
3,18,81,324,1215,4374,15309,52488
6,48,288,1536,7680,36864,172032,786432
10,100,750,5000,31250,187500,1093750,6250000
15,180,1620,12960,97200,699840,4898880,33592320

G. C. Greubel, Antidiagonals n=2..101, flattened

Cf. A036290 (ternary sequences), A034967 (decimal digits).

T:=Flat(List([3..10], n-> List([2..n-1], k-> Binomial(k,2)*(n-k)* k^(n-k-1) ))); # G. C. Greubel, Jan 24 2019

[[Binomial(k,2)*(n-k)*k^(n-k-1): k in [2..n-1]]: n in [3..10]]; // G. C. Greubel, Jan 24 2019

T:= (n, k)-> binomial(n, 2)*k*n^(k-1):
seq(seq(T(n, 1+d-n), n=2..d), d=2..14); # Alois P. Heinz, Jan 17 2013

Table[Range[8]! Rest[CoefficientList[Series[Binomial[n,2]x Exp[n x],{x,0,8}],x]],{n,2,10}]//Grid
T[n_, k_]:= Binomial[n, 2]*k*n^(k-1); Table[T[k,n-k], {n,2,10}, {k, 2, n-1}]//Flatten (* G. C. Greubel, Jan 24 2019 *)

{T(n,k) = binomial(n,2)*k*n^(k-1)};
for(n=2,10, for(k=2,n-1, print1(T(k,n-k), ", "))) \\ G. C. Greubel, Jan 24 2019

[[binomial(k,2)*(n-k)*k^(n-k-1) for k in (2..n-1)] for n in (3..10)] # G. C. Greubel, Jan 24 2019

E.g.f. for row n: binomial(n,2)*x*exp(n*x).

A265121 Integers k such that k*3^k + 2 is prime.

Original entry on oeis.org

0, 1, 3, 5, 11, 15, 17, 153, 169, 273, 317, 373, 923, 1403, 1969, 2349, 7809, 10313, 12291, 24865, 41289

Offset: 1

Altug Alkan, Dec 01 2015

Initial corresponding primes are 2, 5, 83 and 1217.
How do this sequence and A006552 compare asymptotically?
a(22) > 10^5. - Michael S. Branicky, Oct 08 2024

			a(3) = 3 because 3^3 * 3 + 2 = 83 is prime.

Cf. A006552, A036290, A182373.

[n: n in [0..400] | IsPrime(n*3^n+2)]; // Vincenzo Librandi, Dec 02 2015

Select[Range[0, 10000], PrimeQ[# 3^# + 2] &] (* Vincenzo Librandi, Dec 02 2015 *)

for(n=0, 1e6, if(ispseudoprime(3^n*n + 2), print1(n, ", ")));

a(1) = 0 added by Vincenzo Librandi, Dec 02 2015

a(21) from Michael S. Branicky, May 16 2023

A289399 Total path length of the complete ternary tree of height n.

Original entry on oeis.org

0, 3, 21, 102, 426, 1641, 6015, 21324, 73812, 250959, 841449, 2790066, 9167358, 29893557, 96855123, 312088728, 1000836264, 3196219035, 10169787837, 32252755710, 101988443730, 321655860993, 1012039172391, 3177332285412, 9955641160956, 31137856397031

Offset: 0

F. Skerman, Jul 05 2017

			The complete ternary tree of height two consists of one root node (at depth 0), three children of the root (at depth 1) and 9 leaf nodes (at depth 2). Thus a(2) = 0 + 3*1 + 9*2 = 21.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-15,9).

Partial sums of A036290.

concat(0, Vec(3*x / ((1 - x)*(1 - 3*x)^2) + O(x^30))) \\ Colin Barker, Jul 05 2017

From Colin Barker, Jul 05 2017: (Start)
G.f.: 3*x / ((1 - x)*(1 - 3*x)^2).
a(n) = 3*(1 - 3^n + 2*3^n*n) / 4.
a(n) = 7*a(n-1) - 15*a(n-2) + 9*a(n-3) for n>2.
(End)

A273893 Denominator of n/3^n.

Original entry on oeis.org

1, 3, 9, 9, 81, 243, 243, 2187, 6561, 2187, 59049, 177147, 177147, 1594323, 4782969, 4782969, 43046721, 129140163, 43046721, 1162261467, 3486784401, 3486784401, 31381059609, 94143178827, 94143178827, 847288609443, 2541865828329, 282429536481, 22876792454961

Offset: 0

Paul Curtz, Jun 02 2016

The reduced values are Ms(n) = 0, 1/3, 2/9, 1/9, 4/81, 5/243, 2/243, 7/2187, 8/6561, 1/2187, ... .
Numerators: 0, 1, 2, 1, 4, ... = A038502(n).
Ms(-n) = 0, -3, -18, ... = - A036290(n).
Difference table of Ms(n):
0,          1/3,    2/9,    1/9,   4/81, 5/243, 2/243, ...
1/3,       -1/9,   -1/9,  -5/81, -7/243, -1/81, ...
-4/9,         0,   4/81,  8/243,  4/243, ...
4/9,       4/81, -4/243, -4/243, ...
-32/81, -16/243,      0, ...
80/243,  16/243, ...
-64/243, ...
etc.
The difference table of O(n) = n/2^n (Oresme numbers) has its 0's on the main diagonal. Here the 0's appear every two rows. For n/4^n,they appear every three rows. (The denominators of O(n) are 2^A093048(n)).
All terms are powers of 3 (A000244).

Cf. A007949, A000244, A013732, A013733, A036290, A038502, A075101.

Table[Denominator[n/3^n], {n, 0, 28}] (* Michael De Vlieger, Jun 03 2016 *)

a(n) = denominator(n/3^n) \\ Felix Fröhlich, Jun 07 2016

[1] + [3^(n-n.valuation(3)) for n in [1..30]] # Tom Edgar, Jun 02 2016

For n>0, a(n) = 3^(n - valuation(n,3)) = 3^(n - A007949(n)). - Tom Edgar, Jun 02 2016
a(3n+1) = 3^(3n+1), a(3n+2) = 3^(3n+2).
a(3n+6) = 27*(3n+3).
From Peter Bala, Feb 25 2019: (Start)
a(n) = 3^n/gcd(n,3^n).
O.g.f.: 1 + F(3*x) - (2/3)*F((3*x)^3) - (2/9)*F((3*x)^9) - (2/27)*F((3*x)^27) - ..., where F(x) = x/(1 - x).
O.g.f. for reciprocals: Sum_{n >= 0} x^n/a(n) = 1 +  F((x/3)) + 2*( F((x/3)^3) + 3*F((x/3)^9) + 9*F((x/3)^27) + ... ). Cf. A038502. (End)

A352081 Numbers of the form k*p^k, where k>1 and p is a prime.

Original entry on oeis.org

8, 18, 24, 50, 64, 81, 98, 160, 242, 324, 338, 375, 384, 578, 722, 896, 1029, 1058, 1215, 1682, 1922, 2048, 2500, 2738, 3362, 3698, 3993, 4374, 4418, 4608, 5618, 6591, 6962, 7442, 8978, 9604, 10082, 10240, 10658, 12482, 13778, 14739, 15309, 15625, 15842, 18818

Offset: 1

Amiram Eldar, Apr 16 2022

nonn

Each term in this sequence has a single presentation in the form k*p^k.

			8 is a term since 8 = 2*2^2.
18 is a term since 18 = 2*3^2.
24 is a term since 24 = 3*2^3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Peter Lindqvist and Jaak Peetre, On the remainder in a series of Mertens, Expositiones Mathematicae, Vol. 15 (1997), pp. 467-478. See eq. (3).
Eric Weisstein's World of Mathematics, Mertens Constant. See eq. (5).

Subsequences: A036289 \ {0, 2}, A036290 \ {0, 3}, A036291 \ {0, 5}, A036293 \ {0, 7}, A073113 \ {2}, A079704, A100042, A104126.

Cf. A001620, A077761, A143524.

addP[p_, n_] := Module[{k = 2, s = {}, m}, While[(m = k*p^k) <= n, k++; AppendTo[s, m]]; s]; seq[max_] := Module[{m = Floor[Sqrt[max/2]], s = {}, ps}, ps = Select[Range[m], PrimeQ]; Do[s = Join[s, addP[p, max]], {p, ps}]; Sort[s]]; seq[2*10^4]

Sum_{n>=1} 1/a(n) = -A143524 = gamma - B_1, where gamma is Euler's constant (A001620), and B_1 is Mertens's constant (A077761).

A004142 a(n) = n*(3^n - 2^n).

Original entry on oeis.org

0, 1, 10, 57, 260, 1055, 3990, 14413, 50440, 172539, 580250, 1926089, 6328140, 20619703, 66732190, 214742085, 687698960, 2193154547, 6968850210, 22073006401, 69714716500, 219623377071, 690291036710

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (10,-37,60,-36).

Equals n * A001047(n).

Table[n(3^n-2^n),{n,0,30}] (* or *) LinearRecurrence[{10,-37,60,-36},{0,1,10,57},30] (* Harvey P. Dale, Sep 04 2023 *)

From R. J. Mathar, Feb 10 2022: (Start)
G.f.: -x*(-1+6*x^2) / ( (3*x-1)^2*(2*x-1)^2 ).
a(n) = A036290(n) - A036289(n). (End)

A134574 Array, a(n,k) = total size of all n-length words on a k-letter alphabet, read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 3, 8, 3, 4, 24, 18, 4, 5, 64, 81, 32, 5, 6, 160, 324, 192, 50, 6, 7, 384, 1215, 1024, 375, 72, 7, 8, 896, 4374, 5120, 2500, 648, 98, 8, 9, 2048, 15309, 24576, 15625, 5184, 1029, 128, 9, 10, 4608, 52488, 114688, 93750, 38880, 9604, 1536, 162, 10

Offset: 1

Ross La Haye, Jan 22 2008

			a(2,2) = 8 because there are 2^2 = 4 2-length words on a 2 letter alphabet, each of size 2 and 2*4 = 8.
Array begins:
==================================================================
n\k|  1     2       3        4         5         6          7  ...
---|--------------------------------------------------------------
1  |  1     2       3        4         5         6          7  ...
2  |  2     8      18       32        50        72         98  ...
3  |  3    24      81      192       375       648       1029  ...
4  |  4    64     324     1024      2500      5184       9604  ...
5  |  5   160    1215     5120     15625     38880      84035  ...
6  |  6   384    4374    24576     93750    279936     705894  ...
7  |  7   896   15309   114688    546875   1959552    5764801  ...
8  |  8  2048   52488   524288   3125000  13436928   46118408  ...
9  |  9  4608  177147  2359296  17578125  90699264  363182463  ...
... - _Franck Maminirina Ramaharo_, Aug 07 2018

Cf. a(n, 1) = a(1, k) = A000027(n); a(n, 2) = A036289(n); a(n, 3) = A036290(n); a(n, 4) = A018215(n); a(n, 5) = A036291(n); a(n, 6) = A036292(n); a(n, 7) = A036293(n); a(n, 8) = A036294(n); a(2, k) = A001105(k); a(3, k) = A117642(k); a(n, n) = A007778(n); a(n, n+1) = A066274(n+1): sum[a(i-1, n-i+1), {i, 1, n}] = A062807(n).

t[n_, k_] := Sum[k^n, {j, n}]; Table[ t[n - k + 1, k], {n, 10}, {k, n}] // Flatten (* Robert G. Wilson v, Aug 07 2018 *)

a(n,k) = n*k^n.
O.g.f. (by columns): (k*x)/(-1+k*x)^2.
E.g.f. (by columns): k*x*exp(k*x).
a(n,k) = Sum[k^n,{j,1,n}] = n*Sum[C(n,m)*(k-1)^m,{m,0,n}]. - Ross La Haye, Jan 26 2008

cp's OEIS Frontend

A120910 Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.

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A178756 Rectangular array T(n,k) = binomial(n,2)*k*n^(k-1) read by antidiagonals.

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A265121 Integers k such that k*3^k + 2 is prime.

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A289399 Total path length of the complete ternary tree of height n.

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A273893 Denominator of n/3^n.

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A352081 Numbers of the form k*p^k, where k>1 and p is a prime.

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Mathematica

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A004142 a(n) = n*(3^n - 2^n).

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A178756 Rectangular array T(n,k) = binomial(n,2)kn^(k-1) read by antidiagonals.