A000400 -id:A000400 - cp's OEIS frontend

A013613 Triangle of coefficients in expansion of (1+6x)^n.

1, 1, 6, 1, 12, 36, 1, 18, 108, 216, 1, 24, 216, 864, 1296, 1, 30, 360, 2160, 6480, 7776, 1, 36, 540, 4320, 19440, 46656, 46656, 1, 42, 756, 7560, 45360, 163296, 326592, 279936, 1, 48, 1008, 12096, 90720, 435456, 1306368, 2239488, 1679616

Offset: 0

N. J. A. Sloane

T(n,k) equals the number of n-length words on {0,1,...,6} having n-k zeros. - Milan Janjic, Jul 24 2015

			Triangle begins:
1;
1, 6;
1, 12, 36;
1, 18, 108, 216;
1, 24, 216, 864, 1296;
...

Michael De Vlieger and Reinhard Zumkeller, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened, rows 0..125 from Reinhard Zumkeller)
Ömür Deveci and Anthony G. Shannon, Some aspects of Neyman triangles and Delannoy arrays, Mathematica Montisnigri (2021) Vol. L, 36-43.

Cf. A038255 (mirrored).

import Data.List (inits)
a013613 n k = a013613_tabl !! n !! k
a013613_row n = a013613_tabl !! n
a013613_tabl = zipWith (zipWith (*))
               (tail $ inits a000400_list) a007318_tabl
-- Reinhard Zumkeller, Nov 21 2013

G.f.: 1 / (1 - x(1+6y)).
T(n,k) = 6^k*C(n,k) = Sum_{i=n-k..n} C(i,n-k)*C(n,i)*5^(n-i). Row sums are 7^n = A000420. - Mircea Merca, Apr 28 2012
T(n,k) = A007318(n,k)*A000400(k), 0 <= k <= n. - Reinhard Zumkeller, Nov 21 2013

A038243 Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).

Original entry on oeis.org

1, 5, 1, 25, 10, 1, 125, 75, 15, 1, 625, 500, 150, 20, 1, 3125, 3125, 1250, 250, 25, 1, 15625, 18750, 9375, 2500, 375, 30, 1, 78125, 109375, 65625, 21875, 4375, 525, 35, 1, 390625, 625000, 437500, 175000, 43750, 7000, 700, 40, 1, 1953125, 3515625, 2812500, 1312500, 393750, 78750, 10500, 900, 45, 1

Offset: 0

N. J. A. Sloane

Mirror image of A013612. - Zerinvary Lajos, Nov 25 2007
T(i,j) is the number of i-permutations of 6 objects a,b,c,d,e,f, with repetition allowed, containing j a's. - Zerinvary Lajos, Dec 21 2007
Triangle of coefficients in expansion of (5+x)^n - N-E. Fahssi, Apr 13 2008
Also the convolution triangle of A000351. - Peter Luschny, Oct 09 2022

			Triangle begins as:
       1;
       5,      1;
      25,     10,      1;
     125,     75,     15,      1;
     625,    500,    150,     20,     1;
    3125,   3125,   1250,    250,    25,    1;
   15625,  18750,   9375,   2500,   375,   30,   1;
   78125, 109375,  65625,  21875,  4375,  525,  35,  1;
  390625, 625000, 437500, 175000, 43750, 7000, 700, 40, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
B. N. Cyvin, J. Brunvoll, and S. J. Cyvin, Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), 109-121.

Cf. A000351, A000400 (row sums), A036071, A050982, A053464, A081135, A081143.

Sequences of the form q^(n-k)*binomial(n, k): A007318 (q=1), A038207 (q=2), A027465 (q=3), A038231 (q=4), this sequence (q=5), A038255 (q=6), A027466 (q=7), A038279 (q=8), A038291 (q=9), A038303 (q=10), A038315 (q=11), A038327 (q=12), A133371 (q=13), A147716 (q=14), A027467 (q=15).

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

for i from 0 to 8 do seq(binomial(i, j)*5^(i-j), j = 0 .. i) od; # Zerinvary Lajos, Dec 21 2007
# Uses function PMatrix from A357368. Adds column 1, 0, 0, ... to the left.
PMatrix(10, n -> 5^(n-1)); # Peter Luschny, Oct 09 2022

With[{q=5}, Table[q^(n-k)*Binomial[n,k], {n,0,12}, {k,0,n}]//Flatten] (* G. C. Greubel, May 12 2021 *)

flatten([[5^(n-k)*binomial(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 12 2021

See A038207 and A027465 and replace 2 and 3 in analogous formulas with 5. - Tom Copeland, Oct 26 2012

A056308 Number of reversible strings with n beads using a maximum of six different colors.

Original entry on oeis.org

1, 6, 21, 126, 666, 3996, 23436, 140616, 840456, 5042736, 30236976, 181421856, 1088414496, 6530486976, 39182222016, 235093332096, 1410555793536, 8463334761216, 50779983373056, 304679900238336, 1828079250264576, 10968475501587456, 65810852102532096

Offset: 0

Marks R. Nester

A string and its reverse are considered to be equivalent. Thus aabc and cbaa are considered to be identical, but abca is a different string.

			For a(2)=21, the six achiral strings are AA, BB, CC, DD, EE, and FF; the 15 (equivalent) chiral pairs are AB-BA, AC-CA, AD-DA, AE-EA, AF-FA, BC-CB, BD-DB, BE-EB, BF-FB, CD-DC, CE-EC, CF-FC, DE-ED, DF-FD, and EF-FE.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,6,-36).

Column 6 of A277504.

Cf. A000400 (oriented), A320524 (chiral), A056452 (achiral).

I:=[1,6,21]; [n le 3 select I[n] else 6*Self(n-1) +6*Self(n-2) - 36*Self(n-3): n in [1..30]]; // G. C. Greubel, Nov 10 2018

k=6; Table[(k^n+k^Ceiling[n/2])/2,{n,0,30}] (* Robert A. Russell, Nov 25 2017 *)
a[ n_] := (6^n + 6^Quotient[n + 1, 2]) / 2; (* Michael Somos, Jul 10 2018 *)
LinearRecurrence[{6, 6, -36}, {1, 6, 21}, 31] (* Robert A. Russell, Nov 10 2018 *)

Vec((1-21*x^2) / ((1 - 6*x)*(1 - 6*x^2)) + O(x^30)) \\ Colin Barker, Mar 20 2017 [Adapted to offset 0 by Robert A. Russell, Nov 10 2018]

{a(n) = (6^n + 6^((n+1)\2)) / 2}; \\ Michael Somos, Jul 10 2018

a(n) = (6^n + 6^floor((n+1)/2))/2.
G.f.: (1-21*x^2) / ((1-6*x)*(1-6*x^2)). - R. J. Mathar, Jul 06 2011 [Adapted to offset 0 by Robert A. Russell, Nov 10 2018]
a(n) = 6*a(n-1) + 6*a(n-2) - 36*a(n-3) for n > 3. - Colin Barker, Mar 20 2017
a(n) = (A000400(n) + A056452(n)) / 2. - Robert A. Russell, Jun 19 2018
a(n) = 6^(n + floor((n-1)/2)) * a(1-n) for all n in Z. - Michael Somos, Jul 10 2018

a(0)=1 prepended by Robert A. Russell, Nov 10 2018

A061980 Square array A(n,k) = A(n-1,k) + A(n-1, floor(k/2)) + A(n-1, floor(k/3)), with A(0,0) = 1, read by antidiagonals.

Original entry on oeis.org

1, 0, 3, 0, 2, 9, 0, 1, 8, 27, 0, 0, 6, 26, 81, 0, 0, 4, 23, 80, 243, 0, 0, 3, 20, 76, 242, 729, 0, 0, 3, 17, 72, 237, 728, 2187, 0, 0, 1, 17, 66, 232, 722, 2186, 6561, 0, 0, 1, 11, 66, 222, 716, 2179, 6560, 19683, 0, 0, 1, 11, 54, 222, 701, 2172, 6552, 19682, 59049

Offset: 0

Henry Bottomley, May 24 2001

			Array begins as:
    1,   0,   0,   0,   0,   0,   0, ...;
    3,   2,   1,   0,   0,   0,   0, ...;
    9,   8,   6,   4,   3,   3,   1, ...;
   27,  26,  23,  20,  17,  17,  11, ...;
   81,  80,  76,  72,  66,  66,  54, ...;
  243, 242, 237, 232, 222, 222, 202, ...;
  729, 728, 722, 716, 701, 701, 671, ...;
Antidiagonal rows begin as:
  1;
  0, 3;
  0, 2, 9;
  0, 1, 8, 27;
  0, 0, 6, 26, 81;
  0, 0, 4, 23, 80, 243;
  0, 0, 3, 20, 76, 242, 729;
  0, 0, 3, 17, 72, 237, 728, 2187;
  0, 0, 1, 17, 66, 232, 722, 2186, 6561;

G. C. Greubel, Antidiagonals n = 0..50, flattened

Row sums are 6^n: A000400.
Columns are A000244, A024023, A060188, A061981, A061982 twice, A061983 twice, etc.
Cf. A000244, A061290, A061930, A061979, A061984, A061987.

A[n_, k_]:= A[n, k]= If[n==0, Boole[k==0], A[n-1,k] +A[n-1,Floor[k/2]] +A[n-1, Floor[k/3]]];
T[n_, k_]:= A[k, n-k];
Table[A[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 18 2022 *)

@CachedFunction
def A(n,k):
    if (n==0): return 0^k
    else: return A(n-1, k) + A(n-1, (k//2)) + A(n-1, (k//3))
def T(n, k): return A(k, n-k)
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 18 2022

A(n,k) = A(n-1,k) + A(n-1, floor(k/2)) + A(n-1, floor(k/3)), with A(0,0) = 1.
T(n, k) = A(k, n-k).
Sum_{k=0..n} A(n, k) = A000400(n).
T(n, n) = A(n, 0) = A000244(n). - G. C. Greubel, Jun 18 2022

A067864 Numbers k such that k divides the sum of digits of 6^k.

Original entry on oeis.org

1, 3, 9, 18, 90

Offset: 1

Amarnath Murthy and Shyam Sunder Gupta, Feb 16 2002

Conjecture: the sequence is finite, since (sum of the digits of 6^k)/k -> log_10(6)*4.5 ~ 3.50168 as k->infinity (this is also a conjecture). - Robert Gerbicz, May 08 2008
Next term, if it exists, exceeds 100000. - Sean A. Irvine, Apr 05 2010
The keyword "hard" refers to the difficulty of finding, or disproving the existence of, the next term. - N. J. A. Sloane, Nov 09 2024

			a(1)=3, so 3 divides the sum of digits of 6^3 (i.e., 2 + 1 + 6 = 9).

Cf. A000400, A066002, A175457, A175169, A067862, A067863.

Select[Range[100],Mod[Total[IntegerDigits[6^#]],#]==0&] (* Harvey P. Dale, Nov 09 2024 *)

A169604 a(n) = 3*6^n.

Original entry on oeis.org

3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368, 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408

Offset: 0

Klaus Brockhaus, Apr 04 2010

a(n) = A081341(n+1).
Essentially first differences of A125682.
Binomial transform of A005053 without initial term 1.
Second binomial transform of A164346.
Inverse binomial transform of A169634.
Second inverse binomial transform of A103333 without initial term 1.
Contribution from Reinhard Zumkeller, May 02 2010: (Start)
a(n) = 3*A000400(n) = A000400(n+1)/2;
subsequence of A003586; a(n)=A003586(A014105(n)) for n<6. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (6).

Cf. A081341, A125682 ((6^n-1)*3/5), A005053 (expand (1-2x)/(1-5x)), A164346 (3*4^n), A169634 (3*7^n), A103333 (expand (1-5x)/(1-8x)).

Magma
```
[ 3*6^n: n in [0..19] ];
```

3*6^Range[0, 25] (* Paolo Xausa, Jan 17 2025 *)

a(n)=3*6^n \\ Charles R Greathouse IV, Oct 16 2015

a(n) = 6*a(n-1) for n > 0; a(0) = 3.

G.f.: 3/(1-6*x).

A210436 Number of digits in 6^n.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 5, 6, 7, 8, 8, 9, 10, 11, 11, 12, 13, 14, 15, 15, 16, 17, 18, 18, 19, 20, 21, 22, 22, 23, 24, 25, 25, 26, 27, 28, 29, 29, 30, 31, 32, 32, 33, 34, 35, 36, 36, 37, 38, 39, 39, 40, 41, 42, 43, 43, 44, 45, 46, 46, 47, 48, 49, 50, 50, 51, 52, 53

Offset: 0

Luc Comeau-Montasse, Mar 21 2012

			a(4) = 4 because 6^4 = 1296, which has 4 digits.
a(5) = 4 because 6^5 = 7776, which has 4 digits.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000400, A034887, A034888, A210062, A210434, A210435.

[#Intseq(6^n): n in [0..67]]; // Bruno Berselli, Mar 22 2012

a:= n-> length(6^n): seq (a(n), n=0..100); # Alois P. Heinz, Mar 22 2012

Table[Length[IntegerDigits[6^n]], {n, 0, 99}] (* Alonso del Arte, Mar 22 2012 *)

a(n) = A055642(A000400(n)) = A055642(6^n) = floor(log_10(10*(6^n))). - Jonathan Vos Post, Mar 23 2012

A239011 Exponents m such that the decimal expansion of 6^m exhibits its first zero from the right later than any previous exponent.

Original entry on oeis.org

0, 2, 3, 4, 6, 7, 8, 12, 17, 24, 29, 42, 44, 101, 104, 128, 1015, 1108, 2629, 9683, 676076, 917474, 34882222, 53229360, 58230015, 90064345, 309000041, 319582553, 342860474, 382090917, 2770253437, 4380407969, 4407585753, 6966554399, 21235488251, 99404304146

Offset: 1

Charles R Greathouse IV and Robert G. Wilson v, Mar 14 2014

Assume that a zero precedes all decimal expansions. This will take care of those cases in A030702.

Inspired by the seqfan list discussion Re: "possible sequence", beginning with David Wilson 7:57 PM Mar 06 2014 and continued by M. F. Hasler, Allan C. Wechsler and Franklin T. Adams-Watters.

Cf. A000400, A030702, A020665, A031142, A239008, A239009, A239010, A239012, A239013, A239014, A239015.

f[n_] := Position[ Reverse@ Join[{0}, IntegerDigits[ PowerMod[6, n, 10^500]]], 0, 1, 1][[1, 1]]; k = mx = 0; lst = {}; While[k < 10000001, c = f[k]; If[c > mx, mx = c; AppendTo[ lst, k]; Print@ k]; k++]; lst

a(27)-a(34) from Bert Dobbelaere, Jan 21 2019

a(35)-a(36) from Chai Wah Wu, Jan 23 2020

A288211 Irregular triangle read by rows of normalized Girard-Waring formula (cf. A210258), for m=6 data values.

Original entry on oeis.org

1, 6, -5, 36, -45, 10, 216, -360, 80, 75, -10, 1296, -2700, 600, 1125, -250, -75, 5, 7776, -19440, 4320, 12150, -3600, -1125, -540, 225, 200, 36, -1, 46656, -136080, 30240, 113400, -37800, -23625, 2800, 5250, -3780, 3150, -350, 252, -105, -7

Offset: 1

Gregory Gerard Wojnar, Jun 06 2017

Let SM_k = Sum( d_(t_1, t_2, ..., t_6)* eM_1^t_1 * eM_2^t_2 * ... * eM_6^t_6) summed over all length 6 integer partitions of k, i.e., 1*t_1 + 2*t_2 + 3*t_3 + ... + 6*t_6 = k, where SM_k are the averaged k-th power sum symmetric polynomials in 6 data (i.e., SM_k = S_k/6 where S_k are the k-th power sum symmetric polynomials, and where eM_k are the averaged k-th elementary symmetric polynomials, eM_k = e_k/binomial(6,k) with e_k being the k-th elementary symmetric polynomials. The data d_(t_1, t_2, t_3, ..., t_6) form a triangle, with one row for each k value starting with k=1; the number of terms in successive rows is nondecreasing.

Row sums of positive entries give 1,6,46,371,3026,24707,201748. Row sums of negative entries are always 1 less than corresponding row sums of positive entries.

			Triangle begins:
 1;
 6,-5;
 36,-45,10;
 216,-360,80,75,-10;
 1296,-2700,600,1125,-250,-75,5;
 7776,-19440,4320,12150,-3600,-1125,-540,225,200,36,-1;
 ...
Above represents:
 SM_1 = eM_1;
 SM_2 = 6*(eM_1)^2 - 5*eM_2;
 SM_3 = 36*(eM_1)^3 - 45*eM_1*eM_2 + 10*eM_3;
 SM_4 = 216*(eM_1)^4 - 360*(eM_1)^2*eM_2 + 80*eM_1*eM_3 + 75*(eM_2)^2 - 10*eM_4;
 SM_5 = 1296*(eM_1)^5 - 2700*(eM_1)^3*eM_2 + 600*(eM_1)^2*eM_3 + 1125*eM_1*(eM_2)^2 - 250*eM_2*eM_3 - 75*eM_1*eM_4 + 5*eM_5;
 ...

Gregory Gerard Wojnar, Java program
G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, Table GW.n=6, p.24, arXiv:1706.08381 [math.GM], 2017.

Cf. A028297 (m=2), A287768 (m=3), A288199 (m=4), A288207 (m=5), A288245 (m=7), A288188 (m=8). Also see A210258 Girard-Waring.

First column of triangle is powers of m=6, A000400.

Java
```
// See link.
```

A367246 a(n) is the number of n-digit numbers whose difference between the largest and smallest digits is equal to 5.

Original entry on oeis.org

0, 9, 139, 1419, 12079, 92859, 669319, 4617699, 30878959, 201792939, 1295974999, 8212422579, 51499341439, 320287850619, 1978857202279, 12161478061059, 74421280021519, 453832688077899, 2759692966903159, 16742329188365139, 101377580843991199, 612894508749226779, 3700556151386869639

Offset: 1

Stefano Spezia, Nov 11 2023

a(n) is the number of n-digit numbers in A366962.

Index entries for linear recurrences with constant coefficients, signature (15,-74,120).

Cf. A366962.
Cf. A000302, A000351, A000400.
Cf. A367243, A367244, A367245, A367247, A367248, A367249, A367250.

LinearRecurrence[{15,-74,120},{0,9,139},23]

a(n) = 29*6^(n-1) - 49*5^(n-1) + 5*4^n.
a(n) = 15*a(n-1) - 74*a(n-2) + 120*a(n-3) for n > 3.
O.g.f.: x^2*(9 + 4*x)/((1 - 4*x)*(1 - 5*x)*(1 - 6*x)).
E.g.f.: (145*exp(6*x) - 294*exp(5*x) + 150*exp(4*x) - 1)/30.

cp's OEIS Frontend

A013613 Triangle of coefficients in expansion of (1+6x)^n.

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A038243 Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).

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A056308 Number of reversible strings with n beads using a maximum of six different colors.

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A061980 Square array A(n,k) = A(n-1,k) + A(n-1, floor(k/2)) + A(n-1, floor(k/3)), with A(0,0) = 1, read by antidiagonals.

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A067864 Numbers k such that k divides the sum of digits of 6^k.

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A169604 a(n) = 3*6^n.

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A210436 Number of digits in 6^n.

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