A169604 -id:A169604 - cp's OEIS frontend

A000400 Powers of 6: a(n) = 6^n.

1, 6, 36, 216, 1296, 7776, 46656, 279936, 1679616, 10077696, 60466176, 362797056, 2176782336, 13060694016, 78364164096, 470184984576, 2821109907456, 16926659444736, 101559956668416, 609359740010496, 3656158440062976, 21936950640377856, 131621703842267136

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 6), L(1, 6), P(1, 6), T(1, 6). Essentially same as Pisot sequences E(6, 36), L(6, 36), P(6, 36), T(6, 36). See A008776 for definitions of Pisot sequences.
Central terms of the triangle in A036561. - Reinhard Zumkeller, May 14 2006
a(n) = A169604(n)/3; a(n+1) = 2*A169604(n). - Reinhard Zumkeller, May 02 2010
Number of pentagons contained within pentaflakes. - William A. Tedeschi, Sep 12 2010
Sum of coefficients of expansion of (1 + x + x^2 + x^3 + x^4 + x^5)^n.
a(n) is number of compositions of natural numbers into n parts less than 6. For example, a(2) = 36, and there are 36 compositions of natural numbers into 2 parts less than 6.
The compositions of n in which each part is colored by one of  p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 5-colored compositions of n such that no adjacent parts have the same color.
Number of words of length n over the alphabet of six letters. - Joerg Arndt, Sep 16 2014
The number of ordered triples (x, y, z) of binary words of length n such that D(x,z) = D(x, y) + D(y, z) where D(a, b) is the Hamming distance from a to b. - Geoffrey Critzer, Mar 06 2017
a(n) is the area of a triangle with vertices at (2^n, 3^n), (2^(n+1), 3^(n+1)), and (2^(n+2), 3^(n+2)); a(n) is also one fifth the area of a triangle with vertices at (2^n, 3^(n+2)), (2^(n+1), 3^(n+1)), and (2^(n+2), 3^n). - J. M. Bergot, May 07 2018
a(n) is the number of possible outcomes of n distinguishable 6-sided dice. - Stefano Spezia, Jul 06 2024

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 86.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 271
Tanya Khovanova, Recursive Sequences
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Yash Puri and Thomas Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Eric Weisstein's World of Mathematics, Pentaflake
Index entries for linear recurrences with constant coefficients, signature (6).

Column 3 of A225816.
Row 6 of A003992.
Row 3 of A329332.
Cf. A000005, A000290, A011577, A036561, A159991, A169604,

a000400 = (6 ^)
a000400_list = iterate (* 6) 1  -- Reinhard Zumkeller, Nov 21 2013

6^Range[0, 40] (* Harvey P. Dale, Jan 24 2013 *)

A000400(n):=6^n$
makelist(A000400(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

a(n)=6^n \\ Charles R Greathouse IV, Jun 16 2011

(List.fill(50)(6: BigInt)).scanLeft(1: BigInt)( * ) // Alonso del Arte, May 31 2019

a(n) = 6^n.
a(0) = 1; a(n) = 6*a(n-1).
G.f.: 1/(1-6*x). - Simon Plouffe in his 1992 dissertation.
E.g.f.: exp(6*x).
A000005(a(n)) = A000290(n+1). - Reinhard Zumkeller, Mar 04 2007
a(n) = A159991(n)/A011577(n). - Reinhard Zumkeller, May 02 2009
a(n) = det(|s(i+3,j)|, 1 <= i,j <= n), where s(n,k) are Stirling numbers of the first kind. - Mircea Merca, Apr 04 2013

A169634 a(n) = 3*7^n.

Original entry on oeis.org

3, 21, 147, 1029, 7203, 50421, 352947, 2470629, 17294403, 121060821, 847425747, 5931980229, 41523861603, 290667031221, 2034669218547, 14242684529829, 99698791708803, 697891541961621, 4885240793731347, 34196685556119429, 239376798892836003, 1675637592249852021

Offset: 0

Klaus Brockhaus, Apr 04 2010

Essentially first differences of A120741.
Binomial transform of A169604.
Second binomial transform of A005053 without initial term 1.
Inverse binomial transform of A103333 without initial term 1.
Second inverse binomial transform of A013708.
Except for first term 3, these are the integers that satisfy phi(n) = 4*n/7. - Michel Marcus, Jul 14 2015
Number of distinct quadratic residues (QR) over Z_7^n such that gcd(QR, 7^n) = 1 where n >= 1. - Param Mayurkumar Parekh, Feb 11 2023

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Shalosh B. Ekhad and Doron Zeilberger, A Bijective Proof of Richard Stanley's Observation that the sum of the cubes of the n-th row of Stern's Diatomic array equals 3 times 7 to the power n-1, arXiv:2103.12852 [math.CO], 2021.
Richard P. Stanley, Some Linear Recurrences Motivated by Stern's Diatomic Array, arXiv:1901.04647 [math.CO], 2019. Also American Mathematical Monthly 127.2 (2020): 99-111.
Index entries for linear recurrences with constant coefficients, signature (7).

Cf. A120741, A169604 (3*6^n), A005053 (expand (1-2x)/(1-5x)), A103333 (expand (1-5x)/(1-8x)), A013708 (3^(2*n+1)), A007283 (3*2^n), A164346 (3*4^n).

Magma
```
[ 3*7^n: n in [0..19] ];
```

3*7^Range[0, 25] (* Paolo Xausa, Jan 17 2025 *)

a(n) = 7*a(n-1) for n > 0; a(0) = 3.

G.f.: 3/(1-7*x).

A332822 One part of a 3-way classification of the positive integers. Numbers n for which A048675(n) == 2 (mod 3).

Original entry on oeis.org

3, 4, 7, 10, 13, 18, 19, 22, 24, 25, 29, 32, 34, 37, 42, 43, 45, 46, 53, 55, 56, 60, 61, 62, 71, 78, 79, 80, 81, 82, 85, 89, 94, 98, 99, 101, 104, 105, 107, 108, 113, 114, 115, 118, 121, 131, 132, 134, 139, 140, 144, 146, 150, 151, 152, 153, 155, 163, 166, 173, 174, 176, 181, 182, 187, 189, 192, 193, 194, 195, 199, 200, 204

Offset: 1

Antti Karttunen and Peter Munn, Feb 25 2020

nonn

The positive integers are partitioned between A332820, A332821 and this sequence.
For each prime p, the terms include exactly one of p and p^2. The primes alternate between this sequence and A332821. This sequence has the primes with even indexes, those in A031215.
The terms are the even numbers in A332820 halved. The terms are also the numbers m such that 5m is in A332820, and so on for alternate primes: 11, 17, 23 etc. Likewise, the terms are the numbers m such that 3m is in A332821, and so on for alternate primes: 7, 13, 19, 29 etc.
If we take each odd term of this sequence and replace each prime in its factorization by the next smaller prime, we get the same set of numbers as we get from halving the even terms of this sequence, and A332821 consists exactly of those numbers. The numbers that are one third of the terms that are multiples of 3 are in A332820, which consists exactly of those numbers. The numbers that are one fifth of the terms that are multiples of 5 constitute A332821, and for larger primes, an alternating pattern applies as described in the previous paragraph.
The product of any 2 terms of this sequence is in A332821, the product of any 3 terms is in A332820, and the product of a term of A332820 and a term of this sequence is in this sequence. So if a number k is present, k^2 is in A332821, k^3 is in A332820, and k^4 is in this sequence.
If k is an even number, exactly one of {k/2, k, 2k} is in the sequence (cf. A191257 / A067368 / A213258); and generally if k is a multiple of a prime p, exactly one of {k/p, k, k*p} is in the sequence.

Positions of terms valued -1 in A332823; equivalently, numbers in row 3k-1 of A277905 for some k >= 1.
Cf. A048675, A332820, A332821.
Comparable 2 or 3-way classifications: A000379/A000028, A001969/A000069, A003159/A036554, A005843/A005408, A028260/A026424, A191257/A067368/A213258, A325431/A325432, A329609/A329604/A332812.
Subsequences: intersection of A026478 and A066207, A031215 (prime terms), A033430\{0}, A117642\{0}, A169604, A244727\{0}, A244729\{0}, A338910 (semiprime terms).

Select[Range@ 204, Mod[Total@ #, 3] == 2 &@ Map[#[[-1]]*2^(PrimePi@ #[[1]] - 1) &, FactorInteger[#]] &] (* Michael De Vlieger, Mar 15 2020 *)

isA332822(n) =  { my(f = factor(n)); (2==((sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2)%3)); };

{a(n) : n >= 1} = {2 * A332821(k) : k >= 1} U {A003961(A332821(k)) : k >= 1}.

{a(n) : n >= 1} = {A332821(k)^2 : k >= 1} U {A331590(2, A332821(k)) : k >= 1}.

A382491 a(n) is the numerator of the asymptotic density of the numbers whose number of 3-smooth divisors is n.

Original entry on oeis.org

1, 5, 13, 71, 97, 1355, 793, 19163, 53473, 292355, 60073, 13102907, 535537, 78584915, 790859641, 3523099499, 43112257, 99646519235, 387682633, 2764285630427, 7604811750289, 7337148996275, 31385253913, 2226944658077771, 3656440886376673, 2341258386360995, 80539587570991081

Offset: 1

Amiram Eldar, Mar 29 2025

The denominator that corresponds to a(n) is 3*6^(n-1) = A169604(n-1) = A081341(n).

			Fractions begin with 1/3, 5/18, 13/108, 71/648, 97/3888, 1355/23328, 793/139968, 19163/839808, 53473/5038848, 292355/30233088, 60073/181398528, 13102907/1088391168, ...
a(1) = 1 since a(1)/A081341(1) = 1/3 is the asymptotic density of the numbers with a single 3-smooth divisor, 1, i.e., the numbers that are congruent to 1 or 5 mod 6 (A007310).
a(2) = 5 since a(2)/A081341(2) = 5/18 is the asymptotic density of the numbers with exactly two 3-smooth divisors, either 1 and 2 or 1 and 3, i.e., A171126.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A007310, A072078, A081341 (denominators), A169604, A171126.

a[n_] := DivisorSum[n, 2^(n-#) * 3^(n-n/#) &]; Array[a, 30]

PARI
```
a(n) = sumdiv(n, d, 2^(n-d)*3^(n-n/d));
```

a(n) = Sum_{d|n} 2^(n-d) * 3^(n-n/d).
a(p) = 2^(p-1) + 3^(p-1).
Let f(n) = a(n)/A081341(n). Then:
f(n) = (1/3) * Sum_{d|n} (1/2)^(d-1) * (1/3)^(n/d-1).
Sum_{n>=1} f(n) = 1.
Sum_{n>=1} n * f(n) = 3 (the asymptotic mean of A072078).
Sum_{n>=1} n^2 * f(n) = 18, and therefore, the asymptotic variance of A072078 is 18 - 3^2 = 9, and its asymptotic standard deviation is 3.

A174971 Periodic sequence: Repeat 3, -3.

Original entry on oeis.org

3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3

Offset: 0

Klaus Brockhaus, Apr 04 2010

Interleaving of A010701 and -A010701; signed version of A010701.
Essentially first differences of A010674.
Inverse binomial transform of 3 followed by A000004.
Second inverse binomial transform of A010701.
Third inverse binomial transform of A007283.
Fourth inverse binomial transform of A000244 without initial term 1.
Fifth inverse binomial transform of A164346.
Sixth inverse binomial transform of A005053 without initial term 1.
Seventh inverse binomial transform of A169604.
Eighth inverse binomial transform of A169634.
Ninth inverse binomial transform of A103333 without initial term 1.
Tenth inverse binomial transform of A013708.
Eleventh inverse binomial transform of A093138 without initial term 1.

Index entries for linear recurrences with constant coefficients, signature (-1).

Cf. A010701 (all 3's sequence), A000004 (all zeros sequence), A007283 (3*2^n), A000244 (powers of 3), A164346 (3*4^n), A005053 (expand (1-2x)/(1-5x)), A169604 (3*6^n), A169634 (3*7^n), A103333 (expand (1-5x)/(1-8x)), A013708 (3^(2n+1)), A093138 (expand (1-7x)/(1-10x)).

&cat[ [3, -3]: n in [0..41] ];
[ 3*(-1)^n: n in [0..83] ];

PadRight[{},120,{3,-3}] (* or *) NestList[-1#&,3,120] (* Harvey P. Dale, Dec 30 2023 *)

a(n)=3*(-1)^n \\ Charles R Greathouse IV, Jun 13 2013

a(n) = 3*(-1)^n.
a(n) = -a(n-1) for n > 0; a(0) = 3.
a(n) = a(n-2) for n > 1; a(0) = 3, a(1) = -3.
G.f.: 3/(1+x).

A352692 a(n) + a(n+1) = 2^n for n >= 0 with a(0) = 4.

Original entry on oeis.org

4, -3, 5, -1, 9, 7, 25, 39, 89, 167, 345, 679, 1369, 2727, 5465, 10919, 21849, 43687, 87385, 174759, 349529, 699047, 1398105, 2796199, 5592409, 11184807, 22369625, 44739239, 89478489, 178956967, 357913945, 715827879, 1431655769, 2863311527, 5726623065, 11453246119, 22906492249

Offset: 0

Paul Curtz, Mar 29 2022

Difference table D(n,k) = D(n-1,k+1) - D(n-1,k), D(0,k) = a(k):
    4,  -3,   5,  -1,   9,   7, 25, ...
   -7,   8,  -6,  10,  -2,  18, 14,  50, ...
   15, -14,  16, -12,  20,  -4, 36,  28, 100, ...
  -29,  30, -28,  32, -24,  40, -8,  72,  56, 200, ...
   59, -58,  60, -56,  64, -48, 80, -16, 144, 112, 400, ...
  ...
The diagonals are given by D(n,n+k) = a(k)*2^n.
D(n,1) = -(-1)^n* A340627(n).
a(n)   - a(n) =  0,  0,  0,  0,   0, ...       (trivially)
a(n+1) + a(n) =  1,  2,  4,  8,  16, ... = 2^n (by definition)
a(n+2) - a(n) =  1,  2,  4,  8,  16, ... = 2^n
a(n+3) + a(n) =  3,  6, 12, 24,  48, ... = 2^n*3
a(n+4) - a(n) =  5, 10, 20, 40,  80, ... = 2^n*5
a(n+5) + a(n) = 11, 22, 44, 88, 176, ... = 2^n*11
     (...)
This table is given by T(r,n) = A001045(r)*2^n with r, n >= 0.
Sums of antidiagonals are A045883(n).
Main diagonal: A192382(n).
First upper diagonal: A054881(n+1).
First subdiagonal: A003683(n+1).
Second subdiagonal: A246036(n).
Now consider the array from c(n) = (-1)^n*a(n) with its difference table:
   4,   3,   5,    1,    9,   -7,   25,   -39, ...  = c(n)
  -1,   2,  -4,    8,  -16,   32,  -64,   128, ...  = -A122803(n)
   3,  -6,  12,  -24,   48,  -96,  192,  -384, ...  =
  -9,  18, -36,   72, -144,  288, -576,  1152, ...
  27, -54, 108, -216,  432, -864, 1728, -3456, ...
...
The first subdiagonal is -A000400(n). The second is A169604(n).

Index entries for linear recurrences with constant coefficients, signature (1,2).

If a(0) = k then A001045 (k=0), A078008 (k=1), A140966 (k=2), A154879 (k=3), this sequence (k=4).
Essentially the same as A115335.
Cf. A000079, A002450, A340627.
Cf. A020714, A175805.
Cf. A045883, A192382, A003683, A246036.
Cf. A054881, A000400, A169604, A122803.
Cf. A132805, A082311, A082365.
Cf. A024495, A132804, A163868.

a := proc(n) option remember; ifelse(n = 0, 4, 2^(n-1) - a(n-1)) end: # Peter Luschny, Mar 29 2022
A352691 := proc(n)
    (11*(-1)^n + 2^n)/3
end proc: # R. J. Mathar, Apr 26 2022

LinearRecurrence[{1, 2}, {4, -3}, 40] (* Amiram Eldar, Mar 29 2022 *)

a(n) = (11*(-1)^n + 2^n)/3; \\ Thomas Scheuerle, Mar 29 2022

abs(a(n)) = A115335(n-1) for n >= 1.
a(3*n) - (-1)^n*4 = A132805(n).
a(3*n+1) + (-1)^n*4 = A082311(n).
a(3*n+2) - (-1)^n*4 = A082365(n).
From Thomas Scheuerle, Mar 29 2022: (Start)
G.f.: (-4 + 7*x)/(-1 + x + 2*x^2).
Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*a(m + 2*n-k) = a(m)*2^n.
Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*a(1 + n-k) = -(-1)^n*A340627(n).
a(n) = (11*(-1)^n + 2^n)/3.
a(n + 2*m) = a(n) + A002450(m)*2^n.
a(2*n) = A192382(n+1) + (-1)^n*a(n).
a(n) = ( A045883(n) - Sum_{k=0..n-1}(-1)^k*a(k) )/n, for n > 0. (End)
a(n) = A001045(n) + 4*(-1)^n.
a(n+1) = 2*a(n) -11*(-1)^n.
a(n+2) = a(n) + 2^n.
a(n+4) = a(n) + A020714(n).
a(n+6) = a(n) + A175805(n).
a(2*n) = A163868(n).
a(2*n+1) = (2^(2*n+1) - 11)/3.

Warning: The DATA is correct, but there may be errors in the COMMENTS, which should be rechecked. - Editors of OEIS, Apr 26 2022

Edited by M. F. Hasler, Apr 26 2022.

cp's OEIS Frontend

A000400 Powers of 6: a(n) = 6^n.

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A169634 a(n) = 3*7^n.

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A332822 One part of a 3-way classification of the positive integers. Numbers n for which A048675(n) == 2 (mod 3).

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A382491 a(n) is the numerator of the asymptotic density of the numbers whose number of 3-smooth divisors is n.

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A174971 Periodic sequence: Repeat 3, -3.

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A352692 a(n) + a(n+1) = 2^n for n >= 0 with a(0) = 4.

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