A000244 -id:A000244 - cp's OEIS frontend

A071521 Number of 3-smooth numbers <= n.

1, 2, 3, 4, 4, 5, 5, 6, 7, 7, 7, 8, 8, 8, 8, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18

Offset: 1

Benoit Cloitre, Jun 02 2002

A 3-smooth number is a number of the form 2^x * 3^y where x >= 0 and y >= 0.

Bruce C. Berndt and Robert A. Rankin, "Ramanujan : letters and commentary", History of Mathematics Volume 9, AMS-LMS, p. 23, p. 35.
G. H. Hardy, Ramanujan: Twelve lectures on subjects suggested by his life and work, AMS Chelsea Pub., 1999, pages 67-82.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Thierry Bousch, La Tour de Stockmeyer, Séminaire Lotharingien de Combinatoire 77 (2017), Article B77d.
M. Haussman and H. N. Shapiro, On Ramanujan right triangle conjecture, Comm. Pure Appl. Math. 42 (1989), 885-889.
A. M. Ostrowski, Bemerkungen zur Theorie der Diophantischen Approximationen, Abh. Math. Sem. Univ. Hamburg 1 (1922), 77-98; 250-251.
Raphael Schumacher, The Formulas for the Distribution of the 3-Smooth, 5-Smooth, 7-Smooth and all other Smooth Numbers, arXiv preprint arXiv:1608.06928 [math.NT], 2016.

Cf. A003586.

Cf. A000079, A000244, A022330, A022331, A065331, A322026.

a071521 n = length $ takeWhile (<= n) a003586_list
-- Reinhard Zumkeller, Aug 14 2011

N:= 10000: # to get a(1) to a(N)
V:= Vector(N):
for y from 0 to floor(log[3](N)) do
  for x from 0 to ilog2(N/3^y) do
    V[2^x*3^y]:= 1
od od:
convert(map(round,Statistics:-CumulativeSum(V)),list); # Robert Israel, Dec 16 2014

a[n_] := Sum[ MoebiusMu[6k]*Floor[n/k], {k, 1, n}]; Table[a[n], {n, 1, 75}] (* Jean-François Alcover, Oct 11 2011, after Benoit Cloitre *)
f[n_] := Sum[Floor@Log[3, n/2^i] + 1, {i, 0, Log[2, n]}]; Array[f, 75] (* faster, or *)
f[n_] := Sum[Floor@Log[2, n/3^i] + 1, {i, 0, Log[3, n]}]; Array[f, 75] (* Robert G. Wilson v, Aug 18 2012 *)
Accumulate[Table[If[Max[FactorInteger[n][[All,1]]]<4,1,0],{n,80}]] (* Harvey P. Dale, Jan 11 2017 *)

for(n=1,100,print1(sum(k=1,n,if(sum(i=3,n,if(k%prime(i),0,1)),0,1)),","))

a(n)=sum(k=1,n,moebius(2*3*k)*floor(n/k)) \\ Benoit Cloitre, Jun 14 2007

a(n)=my(t=1/3); sum(k=0,logint(n,3), t*=3; logint(n\t,2)+1) \\ Charles R Greathouse IV, Jan 08 2018

from sympy import integer_log
def A071521(n): return sum((n//3**i).bit_length() for i in range(integer_log(n,3)[0]+1)) # Chai Wah Wu, Sep 15 2024

a(n) = Card{ k | A003586(k) <= n }. Asymptotically: let a=1/(2*log(2)*log(3)), b=sqrt(6), then from Ramanujan a(n) ~ a*log(2*n)*log(3*n) or equivalently a(n) ~ a*log(b*n)^2.
A022331(n) = a(A000079(n)); A022330(n) = a(A000244(n)). - Reinhard Zumkeller, May 09 2006
a(n) = Sum_{k=1..n} mu(6k)*floor(n/k). - Benoit Cloitre, Jun 14 2007
a(n) = Sum_{k=1..n} (floor(6^k/k)-floor((6^k-1)/k)). - Anthony Browne, May 19 2016
From Ridouane Oudra, Jul 17 2020: (Start)
a(n) = Sum_{i=0..floor(log_2(n))} (floor(log_3(n/2^i)) + 1).
a(n) = Sum_{i=0..floor(log_3(n))} (floor(log_2(n/3^i)) + 1). (End)
A322026(n) = a(A065331(n)). - Antti Karttunen, Sep 08 2024

A166586 Totally multiplicative sequence with a(p) = p - 2 for prime p.

Original entry on oeis.org

1, 0, 1, 0, 3, 0, 5, 0, 1, 0, 9, 0, 11, 0, 3, 0, 15, 0, 17, 0, 5, 0, 21, 0, 9, 0, 1, 0, 27, 0, 29, 0, 9, 0, 15, 0, 35, 0, 11, 0, 39, 0, 41, 0, 3, 0, 45, 0, 25, 0, 15, 0, 51, 0, 27, 0, 17, 0, 57, 0, 59, 0, 5, 0, 33, 0, 65, 0, 21, 0, 69, 0, 71, 0, 9, 0

Offset: 1

Jaroslav Krizek, Oct 17 2009

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000244 (powers of 3).

f:= proc(n) local t;
    mul((t[1]-2)^t[2],t=ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, Jun 07 2016

a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 2)^fi[[All, 2]])); Table[a[n], {n, 1, 50}] (* G. C. Greubel, Jun 06 2016 *)

a(n) = my(f = factor(n)); for (i=1, #f~, f[i,1] -= 2); factorback(f); \\ Michel Marcus, Dec 13 2014

for(n=1, 100, print1(direuler(p=2, n, 1/(1-p*X+2*X))[n], ", ")) \\ Vaclav Kotesovec, Feb 10 2023

Multiplicative with a(p^e) = (p-2)^e. If n = Product p(k)^e(k) then a(n) = Product (p(k) - 2)^e(k). a(2k) = 0 for k >= 1.
a(A000244(n)) = 1. - Michel Marcus, Dec 13 2014
Dirichlet g.f.: 1 / Product_{p prime} (1 - p^(1 - s) + 2*p^(-s)). The Dirichlet inverse is multiplicative with b(p) = 2 - p, b(p^e) = 0, for e > 1. - Álvar Ibeas, Nov 24 2017 [corrected by Vaclav Kotesovec, Feb 10 2023]
Sum_{k=1..n} a(k) ~ c * n^2/2, where c = Product_{primes} (1 - 1/(1 + p*(p-1)/2)) = 0.3049173579282080265466051390930446635010608835584906520231313997... - Vaclav Kotesovec, Feb 10 2023

More terms from Alonso del Arte, Dec 10 2014
a(69) and a(75) corrected by G. C. Greubel, Jun 06 2016
Erroneous formula and program removed by G. C. Greubel, Jun 06 2016

A004656 Powers of 3 written in base 2.

Original entry on oeis.org

1, 11, 1001, 11011, 1010001, 11110011, 1011011001, 100010001011, 1100110100001, 100110011100011, 1110011010101001, 101011001111111011, 10000001101111110001, 110000101001111010011, 10010001111101101111001

Offset: 0

N. J. A. Sloane

S. Wolfram, A New Kind of Science, Wolfram Media, 2002, pp. 120 and 903.

Vincenzo Librandi, Table of n, a(n) for n = 0..640

Cf. A000079, A004643, ..., A004655: powers of 2 written in base 10, 4, 5, ..., 16

Cf. A000244, A004658, A004659, ... : powers of 3 written in base 10, 4, 5, ...

[Seqint(Intseq(3^n, 2)): n in [0..30]]; // G. C. Greubel, Sep 10 2018

Table[ FromDigits[ IntegerDigits[3^n, 2]], {n, 0, 14}]

a(n)=fromdigits(binary(3^n)) \\ M. F. Hasler, Jun 23 2018

A011754 Number of ones in the binary expansion of 3^n.

Original entry on oeis.org

1, 2, 2, 4, 3, 6, 6, 5, 6, 8, 9, 13, 10, 11, 14, 15, 11, 14, 14, 17, 17, 20, 19, 22, 16, 18, 24, 30, 25, 25, 25, 26, 26, 34, 29, 32, 27, 34, 36, 32, 28, 39, 38, 39, 34, 34, 45, 38, 41, 33, 41, 46, 42, 35, 39, 42, 39, 40, 42, 48, 56, 56, 49, 57, 56, 51, 45, 47, 55, 55, 64, 68, 58

Offset: 0

Allan C. Wechsler, Dec 11 1999

Conjecture: a(n)/n tends to log(3)/(2*log(2)) = 0.792481250... (A094148). - Ed Pegg Jr, Dec 05 2002
Senge & Straus prove that for every m, there is some N such that for all n > N, a(n) > m. Dimitrov & Howe make this effective, proving that for n > 25, a(n) > 22. - Charles R Greathouse IV, Aug 23 2021
Ed Pegg's conjecture means that about half of the bits of 3^n are nonzero. It appears that the same is true for 5^n (A000351, cf. A118738) and 7^n (A000420). - M. F. Hasler, Apr 17 2024

S. Wolfram, "A new kind of science", p. 903.

Hugo Pfoertner, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Vassil S. Dimitrov and Everett W. Howe, Powers of 3 with few nonzero bits and a conjecture of Erdős, arXiv:2105.06440 [math.NT], 2021.
Taylor Dupuy and David E. Weirich, Bits of 3^n in binary, Wieferich primes and a conjecture of Erdős, Journal of Number Theory, Volume 158, January 2016, Pages 268-280.
Hugo Pfoertner, Plot of a(n) - 0.79248*n, +-Pi*sqrt(n), n up to 10^6.
H. G. Senge and E. G. Straus, PV-numbers and sets of multiplicity, Periodica Mathematica Hungarica 3 (1973), pp. 93-100.

Cf. A007088, A000120 (Hamming weight), A000244 (3^n), A004656, A261009, A094148.

Cf. A118738 (same for 5^n).

a011754 = a000120 . a000244  -- Reinhard Zumkeller, Aug 14 2015

[&+Intseq(3^n, 2): n in [0..79]]; // Vincenzo Librandi, Nov 28 2018

f:= n -> convert(convert(3^n,base,2),`+`):
map(f, [$0..100]); # Robert Israel, Apr 17 2024

Table[DigitCount[3^n, 2][[1]], {n, 0, 100}] (* Stefan Steinerberger, Apr 03 2006 *)
DigitCount[3^Range[0,100],2,1] (* Harvey P. Dale, Apr 06 2012 *)

a(n)=hammingweight(3^n) \\ Charles R Greathouse IV, Feb 09 2017

A011754 = lambda n: (3**n).bit_count() # M. F. Hasler, Apr 17 2024

a(n) = A000120(3^n). - Benoit Cloitre, Dec 06 2002

a(n) = A000120(A000244(n)). - Reinhard Zumkeller, Aug 14 2015

More terms from Stefan Steinerberger, Apr 03 2006

A013610 Triangle of coefficients in expansion of (1+3*x)^n.

Original entry on oeis.org

1, 1, 3, 1, 6, 9, 1, 9, 27, 27, 1, 12, 54, 108, 81, 1, 15, 90, 270, 405, 243, 1, 18, 135, 540, 1215, 1458, 729, 1, 21, 189, 945, 2835, 5103, 5103, 2187, 1, 24, 252, 1512, 5670, 13608, 20412, 17496, 6561, 1, 27, 324, 2268, 10206, 30618, 61236, 78732, 59049, 19683

Offset: 0

N. J. A. Sloane

T(n,k) is the number of lattice paths from (0,0) to (n,k) with steps (1,0) and three kinds of steps (1,1). The number of paths with steps (1,0) and s kinds of steps (1,1) corresponds to the expansion of (1+s*x)^n. - Joerg Arndt, Jul 01 2011
Rows of A027465 reversed. - Michael Somos, Feb 14 2002
T(n,k) equals the number of n-length words on {0,1,2,3} having n-k zeros. - Milan Janjic, Jul 24 2015
T(n-1,k-1) is the number of 3-compositions of n with zeros having k positive parts; see Hopkins & Ouvry reference.  - Brian Hopkins, Aug 16 2020

			Triangle begins
  1;
  1,    3;
  1,    6,    9;
  1,    9,   27,   27;
  1,   12,   54,  108,   81;
  1,   15,   90,  270,  405,  243;
  1,   18,  135,  540, 1215, 1458,  729;
  1,   21,  189,  945, 2835, 5103, 5103, 2187;

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
J. Goldman and J. Haglund, Generalized rook polynomials, J. Combin. Theory A91 (2000), 509-530, 1-rook coefficients on the 3xn board (all heights 3) with k rooks
Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.

Cf. A007318, A013609, A027465, etc.

Diagonals of the triangle: A000244 (k=n), A027471 (k=n-1), A027472 (k=n-2), A036216 (k=n-3), A036217 (k=n-4), A036219 (k=n-5), A036220 (k=n-6), A036221 (k=n-7), A036222 (k=n-8), A036223 (k=n-9), A172362 (k=n-10).

a013610 n k = a013610_tabl !! n !! k
a013610_row n = a013610_tabl !! n
a013610_tabl = iterate (\row ->
   zipWith (+) (map (* 1) (row ++ [0])) (map (* 3) ([0] ++ row))) [1]
-- Reinhard Zumkeller, May 26 2013

[3^k*Binomial(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 19 2021

T:= n-> (p-> seq(coeff(p, x, k), k=0..n))((1+3*x)^n):
seq(T(n), n=0..10);  # Alois P. Heinz, Jul 25 2015

t[n_, k_] := Binomial[n, k]*3^(n-k); Table[t[n, n-k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 05 2013 *)
BinomialROW[n_, k_, t_] := Sum[Binomial[n, k]*Binomial[k, j]*(-1)^(k - j)*t^j, {j, 0, k}]; Column[Table[BinomialROW[n, k, 4], {n, 0, 10}, {k, 0, n}], Center] (* Kolosov Petro, Jan 28 2019 *)
T[0, 0] := 1; T[n_, k_]/;0<=k<=n := T[n, k] = 3T[n-1, k-1]+T[n-1, k]; T[n_, k_] := 0; Flatten@Table[T[n, k], {n, 0, 7}, {k, 0, n}] (* Oliver Seipel, Jan 26 2025 *)

{T(n, k) = polcoeff((1 + 3*x)^n, k)}; /* Michael Somos, Feb 14 2002 */

/* same as in A092566 but use */
steps=[[1,0], [1,1], [1,1], [1,1]]; /* note triple [1,1] */
/* Joerg Arndt, Jul 01 2011 */

flatten([[3^k*binomial(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 19 2021

G.f.: 1 / (1 - x*(1+3*y)).
Row sums are 4^n. - Joerg Arndt, Jul 01 2011
T(n,k) = 3^k*C(n,k) = Sum_{i=n-k..n} C(i,n-k)*C(n,i)*2^(n-i). - Mircea Merca, Apr 28 2012
From Peter Bala, Dec 22 2014: (Start)
Riordan array ( 1/(1 - x), 3*x/(1 - x) ).
exp(3*x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(3*x)*(1 + 9*x + 27*x^2/2! + 27*x^3/3!) = 1 + 12*x + 90*x^2/2! + 540*x^3/3! + 2835*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), 3*x/(1 - x) ). (End)
T(n,k) = Sum_{j=0..k} (-1)^(k-j) * binomial(n,k) * binomial(k,j) * 4^j. - Kolosov Petro, Jan 28 2019
T(0,0)=1, T(n,k)=3*T(n-1,k-1)+T(n-1,k) for 0<=k<=n, T(n,k)=0 for k<0 or k>n. - Oliver Seipel, Feb 10 2025

A056576 Highest k with 2^k <= 3^n.

Original entry on oeis.org

0, 1, 3, 4, 6, 7, 9, 11, 12, 14, 15, 17, 19, 20, 22, 23, 25, 26, 28, 30, 31, 33, 34, 36, 38, 39, 41, 42, 44, 45, 47, 49, 50, 52, 53, 55, 57, 58, 60, 61, 63, 64, 66, 68, 69, 71, 72, 74, 76, 77, 79, 80, 82, 84, 85, 87, 88, 90, 91, 93, 95, 96, 98, 99, 101, 103, 104, 106, 107

Offset: 0

Henry Bottomley, Jun 29 2000

nonn

			a(3)=4 because 3^3=27 and 2^4=16 is power of 2 immediately below 27.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Mike Winkler, The algorithmic structure of the finite stopping time behavior of the 3x+ 1 function, arXiv:1709.03385 [math.GM], 2017.

Cf. A000079 (powers of 2), A000244 (powers of 3), A020914, A022921.

Cf. A056850, A117630 (complement), A020857 (decimal expansion of log_2(3)), A076227, A100982.

a056576 = subtract 1 . a020914  -- Reinhard Zumkeller, May 17 2015

seq(ilog2(3^n), n= 0 .. 1000); # Robert Israel, Dec 11 2014

Table[Floor[Log[2, 3^n]], {n, 0, 69}] (* Robert G. Wilson v, Apr 06 2006 *)
Table[Floor[n*Log[2, 3]], {n, 0, 68}] (* L. Edson Jeffery, Dec 11 2014 *)

{a(n) = if( n<0, 0, logint(3^n, 2))}; /* Michael Somos, Dec 13 2014 */

def A056576(n): return (3**n).bit_length()-1 # Chai Wah Wu, Oct 09 2024

a(n) = floor(log_2(3^n)) = log_2(A000244(n)-A056576(n)) = a(n-1)+A022921(n-1).

a(n) = A020914(n) - 1. - L. Edson Jeffery, Dec 12 2014

A078125 Number of partitions of 3^n into powers of 3.

Original entry on oeis.org

1, 2, 5, 23, 239, 5828, 342383, 50110484, 18757984046, 18318289003448, 47398244089264547, 329030840161393127681, 6190927493941741957366100, 318447442589056401640929570896, 45106654667152833836835578059359839

Offset: 0

Paul D. Hanna, Nov 18 2002

nonn

a(n) = sum of the n-th row of lower triangular matrix of A078122.
From Valentin Bakoev, Feb 22 2009: (Start)
a(n) = the partitions of 3^n into powers of 3.
A125801(n) = a(n+1) - 1.
For given m, the general formula for t_m(n, k) and the corresponding tables T, computed as in the example, determine a family of related sequences (placed in the rows or in the columns of T). For example, the sequences from the 3rd, 4th, etc. rows of the given table are not represented in the OEIS till now. (End)

			Square of A078122 = A078123 as can be seen by 4 X 4 submatrix:
[1,_0,_0,0]^2=[_1,_0,_0,_0]
[1,_1,_0,0]___[_2,_1,_0,_0]
[1,_3,_1,0]___[_5,_6,_1,_0]
[1,12,_9,1]___[23,51,18,_1]
To obtain t_3(5,2) we use the table T, defined as T[i,j]= t_3(i,j), for i=1,2,...,5(=n), and j= 0,1,2,...,162(= k.m^{n-1}). It is: 1,2,3,4,5,6,7,8,...,162; 1,5,12,22,35,51,...,4510; (this row contains the first 55 members of A000326 - the pentagonal numbers) 1,23,93,238,485,...,29773; 1,239,1632,5827,15200,32856,62629; 1,5828,68457; Column 1 contains the first 5 members of this sequence. - _Valentin Bakoev_, Feb 22 2009

Alois P. Heinz, Table of n, a(n) for n = 0..40
V. Bakoev, Algorithmic approach to counting of certain types m-ary partitions, Discrete Mathematics, 275 (2004) pp. 17-41.

Cf. A078121, A078122 (matrix shift when cubed), A078123, A078124, A125801.
Column k=3 of A145515. - Alois P. Heinz, Sep 27 2011
Cf. A000244, A002577, A145513.

import Data.MemoCombinators (memo2, list, integral)
a078125 n = a078125_list !! n
a078125_list = f [1] where
   f xs = (p' xs $ last xs) : f (1 : map (* 3) xs)
   p' = memo2 (list integral) integral p
   p  0 = 1; p []  = 0
   p ks'@(k:ks) m = if m < k then 0 else p' ks' (m - k) + p' ks m
-- Reinhard Zumkeller, Nov 27 2015

m[i_, j_] := m[i, j]=If[j==0||i==j, 1, m3[i-1, j-1]]; m2[i_, j_] := m2[i, j]=Sum[m[i, k]m[k, j], {k, j, i}]; m3[i_, j_] := m3[i, j]=Sum[m[i, k]m2[k, j], {k, j, i}]; a[n_] := m2[n, 0]

Denote the sum m^n + m^n + ... + m^n, k times, by k*m^n (m > 1, n > 0 and k are natural numbers). The general formula for the number of all partitions of the sum k*m^n into powers of m is t_m(n, k)= k+1 if n=1, t_m(n, k)= 1 if k=0, and t_m(n, k)= t_m(n, k-1) + t_m(n-1, k*m) if n > 1 and k > 0. a(n) is obtained for m=3 and n=1,2,3,... - Valentin Bakoev, Feb 22 2009

a(n) = [x^(3^n)] 1/Product_{j>=0} (1-x^(3^j)). - Alois P. Heinz, Sep 27 2011

A108716 a(n) = tan(Pi/14)^(-2n) + tan(3Pi/14)^(-2n) + tan(5Pi/14)^(-2n).

Original entry on oeis.org

3, 21, 371, 7077, 135779, 2606261, 50028755, 960335173, 18434276035, 353858266965, 6792546291251, 130387472704741, 2502874814474531, 48044357383337973, 922243598852422035, 17703083191185355397

Offset: 0

Philippe Deléham, Jun 20 2005

The Berndt-type sequence number 11 for the argument 2*Pi/7 defined by the relation a(n) = t(1)^(2*n) + t(2)^(2*n) + t(4)^(2*n) = (-sqrt(7) + 4*s(1))^(2*n) + (-sqrt(7) + 4*s(2))^(2*n) + (-sqrt(7) + 4*s(4))^(2*n), where t(j) = tan(2*Pi*j/7) and s(j) = sin(2*Pi*j/7) (the respective sum with odd powers are discussed in A215794). See also A215007, A215008, A215143, A215493, A215494, A215510, A215512, A215694, A215695, A215828 and especially A215575, where a(n) = B(2n) for the function B(n) defined in the comments. - Roman Witula, Aug 23 2012
The sequence a(n+1)/a(n) is increasing and convergent to (t(2))^2 = 19,195669... (we note that the sequence A215794(n+1)/A215794(n) is decreasing and converges to the same limit). - Roman Witula, Aug 24 2012
Let L(p) be the total length of all sides and diagonals of a regular p-sided polygon inscribed in a unit circle. Then (L(p)/p)^2 = cot(Pi/(2p))^2 is the largest root of the equation: C(p,k)-C(p,2+k)*x+C(p,4+k)*x^2-C(p,6+k)*x^3+ ... +(-1)^q*x^q = 0, where k=1 if p is odd, k=0 if p is even, q = floor(p/2), and where C denotes the binomial coefficient. The complete set of roots is: x(i) = cot((2*i-1)*Pi/(2p))^2, i=1,2,...,q. Then a(n) = x(1)^n+x(2)^n+...x(q)^n for p=7. - Seppo Mustonen, Mar 25 2014
Sum_{k=1..(m-1)/2} tan^(2n) (k*Pi/m) is an integer when m >= 3 is an odd integer (see AMM link and formula); this sequence is the particular case m = 7. All terms are odd. - Bernard Schott, Apr 22 2022

Robert Israel, Table of n, a(n) for n = 0..700
Michel Bataille and Li Zhou, A Combinatorial Sum Goes on Tangent, The American Mathematical Monthly, Vol. 112, No. 7 (Aug. - Sep., 2005), Problem 11044, pp. 657-659.
Seppo Mustonen, Lengths of edges and diagonals and sums of them in regular polygons as roots of algebraic equations (2013).
Seppo Mustonen, Lengths of edges and diagonals and sums of them in regular polygons as roots of algebraic equations [Local copy]
Roman Witula, Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2Pi/7, J. Integer Seq., 12 (2009), Article 09.8.5.
Roman Witula and Damian Slota, New Ramanujan-Type Formulas and Quasi-Fibonacci Numbers of Order 7, Journal of Integer Sequences, Vol. 10 (2007), Article 07.5.6
Index entries for linear recurrences with constant coefficients, signature (21,-35,7).

Similar to: A000244 (m=3), 2*A165225 (m=5), this sequence (m=7), A353410 (m=9), A275546 (m=11), A353411 (m=13).

A:= gfun:-rectoproc({-a(n+3)+21*a(n+2)-35*a(n+1)+7*a(n), a(0) = 3, a(1) = 21, a(2) = 371},a(n), remember):
seq(A(n),n=0..20); # Robert Israel, Aug 23 2015

Table[ Round[ Cot[Pi/14]^(2n) + Cot[3Pi/14]^(2n) + Cot[5Pi/14]^(2n)], {n, 0, 12}] (* Robert G. Wilson v, Jun 21 2005 *)
RecurrenceTable[{a[0]== 3, a[1]== 21, a[2]==371, a[n]== 21*a[n-1] - 35*a[n-2] + 7*a[n-3]}, a, {n,30}] (* G. C. Greubel, Aug 22 2015 *)

a(n)=round(tan(Pi/14)^(-2*n) + tan(3*Pi/14)^(-2*n) + tan(5*Pi/14)^(-2*n)); \\ Anders Hellström, Aug 22 2015

a(n) = 7^n*A(2n), where A(n) := A(n-1) + A(n-2) + A(n-3)/7, with A(0)=3, A(1)=1, and A(2)=3. - see Witula-Slota's (Section 6) and Witula's (Remark 11) papers for the proofs and details. In these papers A(n) denotes the value of the big omega function with index n for the argument 2*i/sqrt(7) (see also A215512). - Roman Witula, Aug 23 2012
Conjecture: a(n) = 21*a(n-1)-35*a(n-2)+7*a(n-3). G.f.: -(35*x^2-42*x+3) / (7*x^3-35*x^2+21*x-1). - Colin Barker, Jun 01 2013
To verify conjecture, note that the roots of 7*x^3-35*x^2+21*x-1 are tan(Pi/14)^2, tan(3*Pi/14)^2 and tan(5*Pi/14)^2. - Robert Israel, Aug 23 2015
E.g.f.: exp((tan(Pi/7))^2*x) + exp((cot(Pi/14))^2*x) + exp((cot(3*Pi/14))^2*x). - G. C. Greubel, Aug 22 2015
a(n) = A275195(2*n)/(7^n). - Kai Wang, Aug 02 2016
a(n) = (tan(1*Pi/7))^(2*n) + (tan(2*Pi/7))^(2*n) + (tan(3*Pi/7))^(2*n). - Bernard Schott, Apr 22 2022

More terms from Robert G. Wilson v, Jun 21 2005

A163632 Triple and reverse digits.

Original entry on oeis.org

1, 3, 9, 72, 612, 6381, 34191, 375201, 3065211, 3365919, 75779001, 300733722, 661102209, 7266033891, 37610189712, 631965038211, 3364115985981, 34975974329001, 300789229729401, 302881986763209, 726982069546809

Offset: 1

Dmitry Kamenetsky, Aug 02 2009

Vaclav Kotesovec, Table of n, a(n) for n = 1..1000
Vaclav Kotesovec, Plot of a(n)^(1/n) for n = 1..10000

The following are parallel families: A000079 (2^n), A004094 (2^n reversed), A028909 (2^n sorted up), A028910 (2^n sorted down), A036447 (double and reverse), A057615 (double and sort up), A263451 (double and sort down); A000244 (3^n), A004167 (3^n reversed), A321540 (3^n sorted up), A321539 (3^n sorted down), A163632 (triple and reverse), A321542 (triple and sort up), A321541 (triple and sort down).

Cf. A132064, A045539, A132078, A132114, A132113, A133361.

a[n_] := a[n] = If[n==1, 1, IntegerReverse[3a[n-1]]];
Array[a, 40] (* Jean-François Alcover, Jan 01 2021 *)

Offset changed from 0 to 1 by Vaclav Kotesovec, Jan 03 2020

A183111 Magnetic Tower of Hanoi, number of moves of disk number k, optimally solving the [RED ; BLUE ; NEUTRAL] or [NEUTRAL ; RED ; BLUE] pre-colored puzzle.

Original entry on oeis.org

0, 1, 3, 9, 25, 75, 223, 665, 1993, 5971, 17903, 53697, 161065, 483163, 1449439, 4348233, 13044585, 39133571, 117400431, 352200881, 1056601993, 3169805003, 9509413535, 28528238329, 85584711561, 256754129459, 770262380399, 2310787129121, 6932361368937

Offset: 0

Uri Levy, Dec 25 2010

A. The Magnetic Tower of Hanoi puzzle is described in link 1 listed below. The Magnetic Tower is pre-colored. Pre-coloring is [RED ; BLUE ; NEUTRAL] or [NEUTRAL ; RED ; BLUE], given in [Source ; Intermediate ; Destination] order. The solution algorithm producing the sequence is optimal (the sequence presented gives the minimum number of moves to solve the puzzle for the given pre-coloring configurations). Optimal solutions are discussed and their optimality is proved in link 2 listed below.
B. Disk numbering is from largest disk (k = 1) to smallest disk (k = N)
C. The above-listed "original" sequence generates a "partial-sums" sequence - describing the total number of moves required to solve the puzzle.
D. The number of moves of disk k, for large k, is close to (10/11)*3^(k-1) ~ 0.909*3^(k-1). Series designation: P909(k).

Uri Levy, The Magnetic Tower of Hanoi, Journal of Recreational Mathematics, Volume 35 Number 3 (2006), 2010, pp 173; arXiv:1003.0225 [math.CO], 2010.
Uri Levy, Magnetic Towers of Hanoi and their Optimal Solutions, arXiv:1011.3843 [math.CO], 2010.
Web applet to play The Magnetic Tower of Hanoi
Index entries for linear recurrences with constant coefficients, signature (3,1,-1,-6).

A000244 "Powers of 3" is the sequence (also) describing the number of moves of the k-th disk solving [RED ; BLUE ; BLUE] or [RED ; RED ; BLUE] pre-colored Magnetic Tower of Hanoi.

Cf. A183111 - A183125.

LinearRecurrence[{3,1,-1,-6},{0,1,3,9,25},30] (* Harvey P. Dale, Apr 30 2018 *)

G.f.: -x*(-1+4*x^3+x^2) / ( (3*x-1)*(2*x^3+x^2-1) ).
Recurrence Relations (a(n)=P909(n) as in referenced paper):
a(n) = a(n-2) + a(n-3) + 2*3^(n-2) + 2*3^(n-4) ; n >= 4
Closed-Form Expression:
Define:
λ1 = [1+sqrt(26/27)]^(1/3) +  [1-sqrt(26/27)]^(1/3)
λ2 = -0.5* λ1 + 0.5*i*{[sqrt(27)+sqrt(26)]^(1/3)- [sqrt(27)-sqrt(26)]^(1/3)}
λ3 = -0.5* λ1 - 0.5*i*{[sqrt(27)+sqrt(26)]^(1/3)- [sqrt(27)-sqrt(26)]^(1/3)}
AP = [(1/11)* λ2* λ3 - (3/11)*(λ2 + λ3) + (9/11)]/[( λ2 - λ1)*( λ3 - λ1)]
BP = [(1/11)* λ1* λ3 - (3/11)*(λ1 + λ3) + (9/11)]/[( λ1 - λ2)*( λ3 - λ2)]
CP = [(1/11)* λ1* λ2 - (3/11)*(λ1 + λ2) + (9/11)]/[( λ2 - λ3)*( λ1 - λ3)]
For any n > 0:
a(n) = (10/11)*3^(n-1) + AP* λ1^(n-1) + BP* λ2^(n-1) + CP* λ3^(n-1)
33*a(n) = 10*3^n -3*( A052947(n-2) -A052947(n-1) -4*A052947(n) ). - R. J. Mathar, Feb 05 2020

More terms from Harvey P. Dale, Apr 30 2018

cp's OEIS Frontend

A071521 Number of 3-smooth numbers <= n.

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A166586 Totally multiplicative sequence with a(p) = p - 2 for prime p.

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A004656 Powers of 3 written in base 2.

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A011754 Number of ones in the binary expansion of 3^n.

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A013610 Triangle of coefficients in expansion of (1+3*x)^n.

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A056576 Highest k with 2^k <= 3^n.

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A108716 a(n) = tan(Pi/14)^(-2n) + tan(3Pi/14)^(-2n) + tan(5Pi/14)^(-2n).