A000400 -id:A000400 - cp's OEIS frontend

A248216 a(n) = 6^n - 2^n.

0, 4, 32, 208, 1280, 7744, 46592, 279808, 1679360, 10077184, 60465152, 362795008, 2176778240, 13060685824, 78364147712, 470184951808, 2821109841920, 16926659313664, 101559956406272, 609359739486208, 3656158439014400, 21936950638280704

Offset: 0

Vincenzo Librandi, Oct 04 2014

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-12).

Sequences of the form k^n - 2^n: A001047 (k=3), A020522 (k=4), A005057 (k=5), this sequence (k=6), A190540 (k=7), A248217 (k=8), A191465 (k=9), A060458 (k=10), A139740 (k=11).

Cf. A000079, A000400, A024023.

Magma
```
[6^n-2^n: n in [0..25]];
```

Table[6^n - 2^n, {n, 0, 25}] (* or *) CoefficientList[Series[4x/((1-2x)(1-6x)), {x, 0, 30}], x]
LinearRecurrence[{8,-12},{0,4},30] (* Harvey P. Dale, Dec 21 2019 *)

[2^n*(3^n -1) for n in (0..25)] # G. C. Greubel, Feb 09 2021

G.f.: 4*x/((1-2*x)*(1-6*x)).
a(n) = 8*a(n-1) - 12*a(n-2).
a(n) = 2^n*(3^n - 1) = A000079(n) * A024023(n).
E.g.f.: exp(6*x) - exp(2*x) = 2*exp(4*x)*sinh(2*x). - G. C. Greubel, Feb 09 2021
a(n) = 4*A016129(n-1). - R. J. Mathar, Mar 10 2022
a(n) = A000400(n) - A000079(n). - Bernard Schott, Mar 27 2022

A268472 T(n,k)=Number of nXk arrays containing k copies of 0..n-1 with column sums equal.

Original entry on oeis.org

1, 1, 2, 1, 4, 6, 1, 8, 36, 24, 1, 16, 216, 648, 120, 1, 32, 1296, 19584, 19800, 720, 1, 64, 7776, 643680, 4140000, 972000, 5040, 1, 128, 46656, 22374144, 1007460000, 1936704000, 69060600, 40320, 1, 256, 279936, 807480576, 269568000000

Offset: 1

R. H. Hardin, Feb 05 2016

Table starts
......1..........1.............1.............1............1..............1
......2..........4.............8............16...........32.............64
......6.........36...........216..........1296.........7776..........46656
.....24........648.........19584........643680.....22374144......807480576
....120......19800.......4140000....1007460000.269568000000.76711698000000
....720.....972000....1936704000.4757348160000
...5040...69060600.1610574336000
..40320.6756825600
.362880

			Some solutions for n=3 k=4
..2..2..2..2. .1..2..0..1. .0..1..2..2. .0..1..1..2. .0..2..1..0
..1..0..1..0. .0..0..2..2. .1..0..1..1. .1..2..2..1. .2..0..2..1
..0..1..0..1. .2..1..1..0. .2..2..0..0. .2..0..0..0. .1..1..0..2

R. H. Hardin, Table of n, a(n) for n = 1..50

Diagonal is A268362.
Column 1 is A000142.
Row 2 is A000079.
Row 3 is A000400.

A373282 Expansion of Sum_{k>=0} x^(6^k) / (1 - 6*x^(6^k)).

Original entry on oeis.org

1, 6, 36, 216, 1296, 7777, 46656, 279936, 1679616, 10077696, 60466176, 362797062, 2176782336, 13060694016, 78364164096, 470184984576, 2821109907456, 16926659444772, 101559956668416, 609359740010496, 3656158440062976, 21936950640377856

Offset: 1

Seiichi Manyama, May 30 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A187767, A373279, A373280, A373281, A373283.

Cf. A000400, A373216.

G.f. A(x) satisfies A(x) = x/(1 - 6*x) + A(x^6).

If n == 0 (mod 6), a(n) = 6^n + a(n/6) otherwise a(n) = 6^n.

A009986 Powers of 42.

Original entry on oeis.org

1, 42, 1764, 74088, 3111696, 130691232, 5489031744, 230539333248, 9682651996416, 406671383849472, 17080198121677824, 717368321110468608, 30129469486639681536, 1265437718438866624512, 53148384174432398229504, 2232232135326160725639168, 93753749683698750476845056

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 42), L(1, 42), P(1, 42), T(1, 42). Essentially same as Pisot sequences E(42, 1764), L(42, 1764), P(42, 1764), T(42, 1764). See A008776 for definitions of Pisot sequences.

The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 42-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011

T. D. Noe, Table of n, a(n) for n = 0..100
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (42).

Cf. A000079, A000400, A000420, A008776, A009965.

[42^n: n in [0..20]]; // Vincenzo Librandi, Nov 21 2010

42^Range[0, 14] (* Michael De Vlieger, Jan 13 2018 *)

a(n) = 42^n; \\ Michel Marcus, Jan 14 2018

G.f.: 1/(1-42*x). - Philippe Deléham, Nov 24 2008
a(n) = 42^n; a(n) = 42*a(n-1), a(0)=1. - Vincenzo Librandi, Nov 21 2010
From Elmo R. Oliveira, Jul 10 2025: (Start)
E.g.f.: exp(42*x).
a(n) = A000079(n)*A009965(n) = A000400(n)*A000420(n). (End)

A038478 Sums of 2 distinct powers of 6.

Original entry on oeis.org

7, 37, 42, 217, 222, 252, 1297, 1302, 1332, 1512, 7777, 7782, 7812, 7992, 9072, 46657, 46662, 46692, 46872, 47952, 54432, 279937, 279942, 279972, 280152, 281232, 287712, 326592, 1679617, 1679622, 1679652, 1679832, 1680912, 1687392, 1726272, 1959552, 10077697, 10077702

Offset: 1

Olivier Gérard

Amiram Eldar, Table of n, a(n) for n = 1..10000

Base-6 interpretation of A038444.

Cf. A000400, A038479, A038480.

Total/@Subsets[6^Range[0,10],{2}]//Union (* Harvey P. Dale, Aug 11 2018 *)

from math import isqrt
def A038478(n): return 6**(m:=isqrt(n<<3)+1>>1)+6**(n-1-(m*(m-1)>>1)) # Chai Wah Wu, Apr 04 2025

Offset corrected by Amiram Eldar, Jul 14 2022

A069031 Powers of 6 with strictly increasing sum of digits.

Original entry on oeis.org

1, 6, 36, 1296, 7776, 279936, 362797056, 78364164096, 470184984576, 16926659444736, 21936950640377856, 4738381338321616896, 170581728179578208256, 36845653286788892983296, 2227915756473955677973140996096, 13367494538843734067838845976576

Offset: 1

Amarnath Murthy, Apr 02 2002

Harvey P. Dale, Table of n, a(n) for n = 1..51

Cf. A069027, A069028, A069029, A069030.

Subsequence of A000400.

DeleteDuplicates[Table[{6^n,Total[IntegerDigits[6^n]]},{n,0,40}],GreaterEqual[ #1[[2]],#2[[2]]]&][[All,1]] (* Harvey P. Dale, Jan 12 2023 *)

More terms from Larry Reeves (larryr(AT)acm.org), Jun 26 2002

More terms from Sean A. Irvine, Mar 29 2024

A089507 Second column of triangle A089504 and second column of array A078741 divided by 18.

Original entry on oeis.org

1, 30, 756, 18360, 441936, 10614240, 254788416, 6115201920, 146766525696, 3522406694400, 84537821131776, 2028908069959680, 48693795855814656, 1168651113600245760, 28047626804770062336, 673143043784666480640

Offset: 0

Wolfdieter Lang, Dec 01 2003

Convolution of A000400 (powers of 6) with A009968 (powers of 24).

Vincenzo Librandi, Table of n, a(n) for n = 0..730
Index entries for linear recurrences with constant coefficients, signature (30,-144).

Cf. A089505, A089513, A089514.

[6^n*(4^(n+1)-1)/3: n in [0..15]]; // Vincenzo Librandi, Oct 18 2017

CoefficientList[Series[1/((1-6x)(1-24x)),{x,0,20}],x] (* or *) LinearRecurrence[{30,-144},{1,30},20] (* Harvey P. Dale, Sep 25 2017 *)

G.f.: 1/((1-3*2*1*x)*(1-4*3*2*x)).
a(n) = A089504(n+2, 2), n>=0.
a(n) = (4*(4*3*2)^n - (3*2*1)^n)/3 = (2^n)*(2^(2*(n+1))-1)*3^(n-1).
a(n) = 6^n*(4^(n+1)-1)/3. - Vincenzo Librandi, Oct 18 2017

A100851 Triangle read by rows: T(n,k) = 2^n * 3^k, 0 <= k <= n, n >= 0.

Original entry on oeis.org

1, 2, 6, 4, 12, 36, 8, 24, 72, 216, 16, 48, 144, 432, 1296, 32, 96, 288, 864, 2592, 7776, 64, 192, 576, 1728, 5184, 15552, 46656, 128, 384, 1152, 3456, 10368, 31104, 93312, 279936, 256, 768, 2304, 6912, 20736, 62208, 186624, 559872, 1679616, 512, 1536, 4608, 13824, 41472, 124416, 373248, 1119744, 3359232, 10077696

Offset: 0

Reinhard Zumkeller, Nov 20 2004

			From _Stefano Spezia_, Apr 28 2024: (Start)
Triangle begins:
   1;
   2,  6;
   4, 12,  36;
   8, 24,  72, 216;
  16, 48, 144, 432, 1296;
  32, 96, 288, 864, 2592, 7776;
  ...
(End)

G. C. Greubel, Rows n = 0..100 of the triangle, flattened
Eric Weisstein's World of Mathematics, Smooth Number.

Cf. A000079, A000400, A001021, A003586, A005010, A007283, A016129.
Cf. A016149, A051958, A053524, A100852, A248337.
Cf. A065333(T(n, k))=1.

A100851:= func< n,k | 2^n*3^k >;
[A100851(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 11 2024

A100851[n_, k_]= 2^n*3^k;
Table[A100851[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 11 2024 *)

def A100851(n,k): return 2^n*3^k
flatten([[A100851(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Nov 11 2024

T(n,0) = A000079(n).
T(n,1) = A007283(n) for n>0.
T(n,2) = A005010(n) for n>1.
T(n,n) = A000400(n) = A100852(n,n).
Sum_{k=0..n} T(n, k) = A016129(n).
T(2*n, n) = A001021(n). - Reinhard Zumkeller, Mar 04 2006
G.f.: 1/((1 - 2*x)*(1 - 6*x*y)). - Stefano Spezia, Apr 28 2024
From G. C. Greubel, Nov 11 2024: (Start)
Sum_{k=0..n} (-1)^k*T(n, k) = A053524(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = (1/2)*((1-(-1)^n)*A248337((n+1)/2) + (1 + (-1)^n)*A016149(n/2)).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (1/2)*(-1)^floor(n/2)*( (1+(-1)^n) *A051958((n+2)/2) + 2*(1-(-1)^n)*A051958((n+1)/2)). (End)
Sum_{n>=0, k=0..n} 1/T(n,k) = 12/5. - Amiram Eldar, May 12 2025

A139626 a(n) = binomial(n+4, 4)*6^n.

Original entry on oeis.org

1, 30, 540, 7560, 90720, 979776, 9797760, 92378880, 831409920, 7205552640, 60526642176, 495217981440, 3961743851520, 31084451758080, 239794342133760, 1822437000216576, 13668277501624320, 101306056776744960, 742911083029463040, 5395880497792942080

Offset: 0

Zerinvary Lajos, Jun 12 2008

With a different offset, number of n-permutations (n=5) of 7 objects t, u, v, w, z, x, y with repetition allowed, containing exactly four (4) u's. Example: a(1)=30 because we have
uuuut, uuutu, uutuu, utuuu, tuuuu,
uuuuv, uuuvu, uuvuu, uvuuu, vuuuu,
uuuuw, uuuwu, uuwuu, uwuuu, wuuuu,
uuuuz, uuuzu, uuzuu, uzuuu, zuuuu,
uuuux, uuuxu, uuxuu, uxuuu, xuuuu,
uuuuy, uuuyu, uuyuu, uyuuu, yuuuu.

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (30,-360,2160,-6480,7776).

Cf. A000332, A000400.

[6^n* Binomial(n+4, 4): n in [0..20]]; // Vincenzo Librandi, Oct 12 2011

Maple
```
seq(binomial(n+4,4)*6^n,n=0..22);
```

a(n)=binomial(n+4,4)*6^n \\ Charles R Greathouse IV, Sep 11 2013

a(n) = A000332(n+4) * A000400(n). - Michel Marcus, Sep 11 2013
G.f.: 1 / (1-6*x)^5. - Colin Barker, Sep 25 2013
From Amiram Eldar, Aug 28 2022: (Start)
Sum_{n>=0} 1/a(n) = 548 - 3000*log(6/5).
Sum_{n>=0} (-1)^n/a(n) = 8232*log(7/6) - 1268. (End)

More terms from Colin Barker, Sep 25 2013

A164532 a(n) = 6*a(n-2) for n > 2; a(1) = 1, a(2) = 4.

Original entry on oeis.org

1, 4, 6, 24, 36, 144, 216, 864, 1296, 5184, 7776, 31104, 46656, 186624, 279936, 1119744, 1679616, 6718464, 10077696, 40310784, 60466176, 241864704, 362797056, 1451188224, 2176782336, 8707129344, 13060694016, 52242776064, 78364164096

Offset: 1

Klaus Brockhaus, Aug 15 2009

nonn

Interleaving of A000400 and A067411 without initial term 1.

Binomial transform is apparently A123011. Fourth binomial transform is A154235.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,6).

Cf. A000400 (powers of 6), A067411, A123011, A154235.

[ n le 2 select 3*n-2 else 6*Self(n-2): n in [1..29] ];

LinearRecurrence[{0,6}, {1,4}, 40] (* G. C. Greubel, Jul 16 2021 *)

[((1 - (-1)^n)*sqrt(6)/2 + 2*(1 + (-1)^n))*6^(n/2 -1) for n in (1..40)] # G. C. Greubel, Jul 16 2021

a(n) = (5 - (-1)^n)*6^(1/4*(2*n - 5 + (-1)^n)).
G.f.: x*(1+4*x)/(1-6*x^2).
a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011
a(n) = ((1-(-1)^n)*sqrt(6)/2 + 2*(1+(-1)^n))*6^(n/2 -1). - G. C. Greubel, Jul 16 2021

cp's OEIS Frontend

A248216 a(n) = 6^n - 2^n.

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A268472 T(n,k)=Number of nXk arrays containing k copies of 0..n-1 with column sums equal.

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A373282 Expansion of Sum_{k>=0} x^(6^k) / (1 - 6*x^(6^k)).

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A009986 Powers of 42.

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A038478 Sums of 2 distinct powers of 6.

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A069031 Powers of 6 with strictly increasing sum of digits.

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A089507 Second column of triangle A089504 and second column of array A078741 divided by 18.

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A100851 Triangle read by rows: T(n,k) = 2^n * 3^k, 0 <= k <= n, n >= 0.

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