A053469 -id:A053469 - cp's OEIS frontend

A016129 Expansion of 1/((1-2x)(1-6*x)).

1, 8, 52, 320, 1936, 11648, 69952, 419840, 2519296, 15116288, 90698752, 544194560, 3265171456, 19591036928, 117546237952, 705277460480, 4231664828416, 25389989101568, 152339934871552, 914039609753600, 5484237659570176, 32905425959518208, 197432555761303552

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (8,-12).

Row sums of A100851.
Cf. A071951, A089278, A089500, A129467, A144843.
Sequences with gf 1/((1-n*x)*(1-6*x)): A000400 (n=0), A003464 (n=1), this sequence (n=2), A016137 (n=3), A016149 (n=4), A005062 (n=5), A053469 (n=6), A016169 (n=7), A016170 (n=8), A016172 (n=9), A016173 (n=10), A016174 (n=11), A016175 (n=12).

[(6^(n+1)-2^(n+1))/4 : n in [0..30]]; // Vincenzo Librandi, Oct 09 2011

Table[(6^(n+1) -2^(n+1))/4, {n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Jan 19 2011 *)
CoefficientList[Series[1/((1-2x)(1-6x)),{x,0,30}],x] (* or *) LinearRecurrence[{8,-12},{1,8},30] (* Harvey P. Dale, Jan 15 2015 *)

Vec(1/(1-2*x)/(1-6*x)+O(x^30)) \\ Charles R Greathouse IV, Apr 17 2012

[lucas_number1(n,8,12) for n in range(1, 31)] # Zerinvary Lajos, Apr 23 2009

[(6^n - 2^n)/4 for n in range(1,31)] # Zerinvary Lajos, Jun 04 2009

a(n) = A071951(n+2, 2) = 9*(2*3)^(n-1) - (2*1)^(n-1) = (2^(n-1))*(3^(n+1)-1), n>=0. - Wolfdieter Lang, Nov 07 2003
From Lambert Klasen (lambert.klasen(AT)gmx.net), Feb 05 2005: (Start)
G.f.: 1/((1-2*x)*(1-6*x)).
E.g.f.: (-exp(2*x) + 3*exp(6*x))/2.
a(n) = (6^(n+1) - 2^(n+1))/4. (End)
a(n)^2 = A144843(n+1). - Philippe Deléham, Nov 26 2008
a(n) = 8*a(n-1) - 12*a(n-2). - Philippe Deléham, Jan 01 2009
a(n) = det(|ps(i+2,j+1)|, 1 <= i,j <= n), where ps(n,k) are Legendre-Stirling numbers of the first kind (A129467). - Mircea Merca, Apr 06 2013

A053541 a(n) = n*10^(n-1).

Original entry on oeis.org

1, 20, 300, 4000, 50000, 600000, 7000000, 80000000, 900000000, 10000000000, 110000000000, 1200000000000, 13000000000000, 140000000000000, 1500000000000000, 16000000000000000, 170000000000000000

Offset: 1

Barry E. Williams, Jan 15 2000

This sequence gives the number of 1's (or any other nonzero digit) required to write all integers from 0 up to 10^n-1. - Jason D. W. Taff (jtaff(AT)jburroughs.org), Dec 05 2004 (improved by Bernard Schott, Nov 17 2022)

The corresponding number of 0's required to write all these integers from 0 up to 10^n-1 is A033714(n). - Bernard Schott, Nov 17 2022

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Frank Ellermann, Illustration of binomial transforms.
Index entries for linear recurrences with constant coefficients, signature (20,-100).

Cf. A001787, A033714, A038303, A053464, A053469, A081045, A094798.

List([1..20], n-> n*10^(n-1)) # G. C. Greubel, May 16 2019

[n*10^(n-1): n in [1..30]]; // Vincenzo Librandi, Jun 06 2011

seq(n*10^(n-1), n = 1 .. 40); # Bernard Schott, Nov 17 2022

f[n_]:=n*10^(n-1);f[Range[40]] (* Vladimir Joseph Stephan Orlovsky, Feb 09 2011*)
LinearRecurrence[{20,-100},{1,20},20] (* Harvey P. Dale, Aug 08 2023 *)

a(n)=n*10^(n-1) \\ Charles R Greathouse IV, Dec 05 2011

[n*10^(n-1) for n in (1..20)] # G. C. Greubel, May 16 2019

a(n) = 20*a(n-1) - 100*a(n-2), with a(0)=0, a(1)=1, a(2)=20.
From Jason D. W. Taff (jtaff(AT)jburroughs.org), Dec 05 2004: (Start)
a(n) = 10*a(n-1) + 10*(n-1).
a(n) = Sum_{k=1..n} k*binomial(n,k)*9^(n-k).
a(n) = A094798(10^n - 1). (End)
From G. C. Greubel, May 16 2019: (Start)
G.f.: x/(1-10*x)^2.
E.g.f.: x*exp(10*x). (End)
From Amiram Eldar, Oct 28 2020: (Start)
Sum_{n>=1} 1/a(n) = 10*log(10/9).
Sum_{n>=1} (-1)^(n+1)/a(n) = 10*log(11/10). (End)
a(n) = Sum_{k=1..n} A081045(k-1). - Bernard Schott, Nov 17 2022

Offset changed from 0 to 1 by Vincenzo Librandi, Jun 06 2011

A212700 a(n) = 5n6^(n-1).

Original entry on oeis.org

5, 60, 540, 4320, 32400, 233280, 1632960, 11197440, 75582720, 503884800, 3325639680, 21767823360, 141490851840, 914248581120, 5877312307200, 37614798766080, 239794342133760, 1523399350026240, 9648195883499520, 60935974001049600, 383896636206612480, 2413064570441564160

Offset: 1

Stanislav Sykora, May 25 2012

Main transitions in systems of n particles with spin 5/2.
Refer to the general explanation in A212697.
This particular sequence is obtained for base b=6, corresponding to spin S=(b-1)/2=5/2.
Arithmetic derivative of 6^n: a(n) = A003415(6^n). - Bruno Berselli, Oct 22 2013

Stanislav Sykora, Table of n, a(n) for n = 1..100
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (12,-36).

Cf. A001787, A212697, A212698, A212699, A212701, A212702, A212703, A212704 (b = 2, 3, 4, 5, 7, 8, 9, 10).

Cf. A000400, A003415, A008587, A053469.

Rest@ CoefficientList[Series[5 x/(6 x - 1)^2, {x, 0, 18}], x] (* or *)
Array[5 # 6^(# - 1) &, 18] (* Michael De Vlieger, Nov 18 2019 *)

mtrans(n, b) = n*(b-1)*b^(n-1);
for (n=1, 100, write("b212700.txt", n, " ", mtrans(n, 6)))

a(n) = n*(b-1)*b^(n-1): for this sequence, set b=6.
From R. J. Mathar, Oct 15 2013: (Start)
G.f.: 5*x/(6*x-1)^2.
a(n) = 5*A053469(n). (End)
From Elmo R. Oliveira, May 14 2025: (Start)
E.g.f.: 5*x*exp(6*x).
a(n) = A008587(n)*A000400(n-1).
a(n) = 12*a(n-1) - 36*a(n-2) for n > 2. (End)

A027473 Second column of A027466.

Original entry on oeis.org

1, 14, 147, 1372, 12005, 100842, 823543, 6588344, 51883209, 403536070, 3107227739, 23727920916, 179936733613, 1356446145698, 10173346092735, 75960984159088, 564959819683217, 4187349251769726, 30939858360298531, 227977903707462860, 1675637592249852021, 12288009009832248154

Offset: 1

Olivier Gérard

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Vincenzo Librandi, Table of n, a(n) for n = 1..300
Frank Ellermann, Illustration of binomial transforms.
Index entries for linear recurrences with constant coefficients, signature (14,-49).

Cf. A001787, A003415, A053464, A053469.

[n*7^(n-1): n in [1..35]]; // Vincenzo Librandi, Jun 06 2011

Join[{a=1,b=14},Table[c=14*b-49*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)
LinearRecurrence[{14,-49},{1, 14},19] (* Stefano Spezia, May 05 2024 *)

a(n) = n*7^(n-1).
a(n) = 14*a(n-1) - 49*a(n-2) with a(1) = 1, a(2) = 14.
a(n) = A003415(7^n). - Bruno Berselli, Oct 22 2013
From Amiram Eldar, Oct 28 2020: (Start)
Sum_{n>=1} 1/a(n) = 7*log(7/6).
Sum_{n>=1} (-1)^(n+1)/a(n) = 7*log(8/7). (End)
From Stefano Spezia, May 05 2024: (Start)
G.f.: x/(1 - 7*x)^2.
E.g.f.: x*exp(7*x). (End)

More terms from Larry Reeves (larryr(AT)acm.org), May 29 2001

Offset changed from 2 to 1 by Vincenzo Librandi, Jun 06 2011

A053540 a(n) = n*9^(n-1).

Original entry on oeis.org

1, 18, 243, 2916, 32805, 354294, 3720087, 38263752, 387420489, 3874204890, 38354628411, 376572715308, 3671583974253, 35586121596606, 343151886824415, 3294258113514384, 31501343210481297, 300189270593998242, 2851798070642983299, 27017034353459841780

Offset: 1

Barry E. Williams, Jan 15 2000

Vincenzo Librandi, Table of n, a(n) for n = 1..300
Frank Ellermann, Illustration of binomial transforms.
Index entries for linear recurrences with constant coefficients, signature (18,-81).

Related to computing A023052.

Cf. A001787, A053464, A053469.

List([1..20], n-> n*9^(n-1)); # G. C. Greubel, May 16 2019

[n*9^(n-1): n in [1..20]]; // Vincenzo Librandi, Jun 11 2011

f[n_]:=n*9^(n-1); f[Range[40]] (* Vladimir Joseph Stephan Orlovsky, Feb 09 2011*)

a(n)=n*9^(n-1) \\ Charles R Greathouse IV, Oct 07 2015

[n*9^(n-1) for n in (1..20)] # G. C. Greubel, May 16 2019

From Colin Barker, Oct 17 2012: (Start)
a(n) = 18*a(n-1) - 81*a(n-2).
G.f.: x/(1-9*x)^2. (End)
E.g.f.: x*exp(9*x). - G. C. Greubel, May 16 2019
From Amiram Eldar, Oct 28 2020: (Start)
Sum_{n>=1} 1/a(n) = 9*log(9/8).
Sum_{n>=1} (-1)^(n+1)/a(n) = 9*log(10/9). (End)

More terms from Larry Reeves (larryr(AT)acm.org), May 29 2001

Edited by N. J. A. Sloane at the suggestion of Reinhard Zumkeller, Sep 16 2007

A053539 a(n) = n * 8^(n-1).

Original entry on oeis.org

0, 1, 16, 192, 2048, 20480, 196608, 1835008, 16777216, 150994944, 1342177280, 11811160064, 103079215104, 893353197568, 7696581394432, 65970697666560, 562949953421312, 4785074604081152, 40532396646334464, 342273571680157696, 2882303761517117440, 24211351596743786496

Offset: 0

Barry E. Williams, Jan 15 2000

The Szeged index of the hypercube Q_n (see the Ashrafi et al. reference, p. 45, last line). - Emeric Deutsch, Aug 06 2014

For n > 3, 2*a(n) is the number of spanning trees in a superprism on 2*n vertices (see Bogdanowicz). - Stefano Spezia, May 05 2024

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

G. C. Greubel, Table of n, a(n) for n = 0..1000
A. R. Ashrafi, B. Manoochehrian, and H. Yousefi-Azari, On Szeged polynomial of a graph, Bull. Iranian Math. Soc., 33, 2007, 37-46. - _Emeric Deutsch_, Aug 06 2014
Zbigniew R. Bogdanowicz, The number of spanning trees in a superprism, Discrete Math. Lett. 13 (2024) 66-73. See Theorem 3.1.
Frank Ellermann, Illustration of binomial transforms.
Index entries for linear recurrences with constant coefficients, signature (16,-64).

Binomial transform of A027473.

Cf. A001787, A053464, A053469, A053540.

List([0..20], n-> n*8^(n-1)); # G. C. Greubel, May 16 2019

[n*8^(n-1): n in [0..20]]; // Vincenzo Librandi, Feb 09 2011

a := proc(n) option remember; if n<2 then n else 16*a(n-1)-64*a(n-2) end if end proc: seq(a(n), n = 0 .. 20); # Emeric Deutsch, Aug 06 2014

Table[n 8^(n-1),{n,0,20}] (* or *) LinearRecurrence[{16,-64},{0,1},20] (* Harvey P. Dale, Feb 01 2017 *)

a(n) = n*8^(n-1); \\ Joerg Arndt, Aug 07 2014

[n*8^(n-1) for n in (0..20)] # G. C. Greubel, May 16 2019

a(n) = 16*a(n-1) - 64*a(n-2), with a(0)=0, a(1)=1. - Emeric Deutsch, Aug 06 2014
From G. C. Greubel, May 16 2019: (Start)
G.f.: x/(1-8*x)^2.
E.g.f.: x*exp(8*x). (End)
From Amiram Eldar, Oct 28 2020: (Start)
Sum_{n>=1} 1/a(n) = 8*log(8/7).
Sum_{n>=1} (-1)^(n+1)/a(n) = 8*log(9/8). (End)

Offset corrected and name edited by Emeric Deutsch, Aug 06 2014

A027271 a(n) = Sum_{k=0..2n} (k+1)*T(n,k), where T is given by A026536.

Original entry on oeis.org

1, 4, 18, 48, 180, 432, 1512, 3456, 11664, 25920, 85536, 186624, 606528, 1306368, 4199040, 8957952, 28553472, 60466176, 191476224, 403107840, 1269789696, 2660511744, 8344332288, 17414258688, 54419558400, 113192681472, 352638738432, 731398864896, 2272560758784

Offset: 0

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,12,0,-36).

Cf. A026536, A053469, A199299 (bisection).

[Round(6^(n/2)*( 3*((n+1) mod 2) + Sqrt(6)*(n mod 2) )*(n+1)/3): n in [0..40]]; // G. C. Greubel, Apr 12 2022

T[n_, k_]:= T[n, k]= If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[n/2], If[EvenQ[n], T[n-1, k-2] +T[n-1, k-1] +T[n-1, k], T[n-1, k-2] +T[n-1, k]] ]];
A027271[n_]:= A027271[n]= Sum[(k+1)*T[n,k], {k,0,2*n}];
Table[A027271[n], {n,0,40}] (* G. C. Greubel, Apr 12 2022 *)

A027271(n)=my(b(n)=if(!bittest(n,0),n\2*6^(n\2-1)));4*b(n+1)+b(n+2)+6*b(n) \\ could be made more efficient and explicit by simplifying the formula for n even and for n odd separately. - M. F. Hasler, Sep 29 2012

[6^(n/2)*( 3*((n+1)%2) + sqrt(6)*(n%2) )*(n+1)/3 for n in (0..40)] # G. C. Greubel, Apr 12 2022

From Paul Barry, Mar 03 2004: (Start)
G.f.: (1+4*x+6*x^2)/(1-6*x^2)^2 = (d/dx)((1+3*x)/(1-6*x^2)).
a(n) = 6^(n/2)*((3-sqrt(6))*(-1)^n + (3+sqrt(6)))*(n+1)/6. (End)
a(n) = 4*b(n) + b(n+1) + 6*b(n-1) with b(n)= 0, 1, 0, 12, 0, 108, 0, 864, ... (aerated A053469). - R. J. Mathar, Sep 29 2012
E.g.f.: (1 + 2*x)*cosh(sqrt(6)*x) + sqrt(2/3)*(1 + 3*x)*sinh(sqrt(6)*x). - Stefano Spezia, May 07 2023

A104002 Triangle T(n,k) read by rows: number of permutations in S_n avoiding all k-length patterns that start with 1 except one fixed pattern and containing it exactly once.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 12, 6, 1, 5, 32, 27, 8, 1, 6, 80, 108, 48, 10, 1, 7, 192, 405, 256, 75, 12, 1, 8, 448, 1458, 1280, 500, 108, 14, 1, 9, 1024, 5103, 6144, 3125, 864, 147, 16, 1, 10, 2304, 17496, 28672, 18750, 6480, 1372, 192, 18, 1, 11, 5120, 59049, 131072

Offset: 2

Ralf Stephan, Feb 26 2005

T(n+k,k+1) = total number of occurrences of any given letter in all possible n-length words on a k-letter alphabet. For example, with the 2 letter alphabet {0,1} there are 4 possible 2-length words: {00,01,10,11}. The letter 0 occurs 4 times altogether, as does the letter 1. T(4,3) = 4. - Ross La Haye, Jan 03 2007

Table T(n,k) = k*n^(k-1) n,k > 0 read by antidiagonals. - Boris Putievskiy, Dec 17 2012

			Triangle begins:
  1;
  2,   1;
  3,   4,    1;
  4,  12,    6,    1;
  5,  32,   27,    8,   1;
  6,  80,  108,   48,  10,   1;
  7, 192,  405,  256,  75,  12,  1;
  8, 448, 1458, 1280, 500, 108, 14, 1;

Michael De Vlieger, Table of n, a(n) for n = 2..11176 (rows 2 <= n <= 150).
T. Mansour, Permutations containing and avoiding certain patterns, arXiv:math/9911243 [math.CO], 1999-2000.
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.

Cf. Left-edge columns include A001787, A027471, A002697, A053464, A053469, A027473, A053539, A053540, A053541, A081127, A081128.

Table[(n - k + 1) (k - 1)^(n - k), {n, 2, 12}, {k, 2, n}] // Flatten (* Michael De Vlieger, Aug 22 2018 *)

T(n, k) = (n-k+1) * (k-1)^(n-k), k<=n.

As a linear array, the sequence is a(n) = A004736(n)*A002260(n)^(A004736(n)-1) or a(n) = ((t*t+3*t+4)/2-n)*(n-(t*(t+1)/2))^((t*t+3*t+4)/2-n-1), where t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 17 2012

A121124 Unbranched a-4-catapolynonagons (see Brunvoll reference for precise definition).

Original entry on oeis.org

1, 4, 21, 138, 864, 5526, 34992, 221724, 1399680, 8818632, 55427328, 347684400, 2176782336, 13604912928, 84894511104, 528958247616, 3291294892032, 20453047668864, 126949945835520, 787089669219072, 4874877920083968, 30163307160752640, 186464080443211776, 1151689908801235968

Offset: 2

N. J. A. Sloane, Aug 13 2006

J. Brunvoll, S. J. Cyvin and B. N. Cyvin, Isomer enumeration of polygonal systems..., J. Molec. Struct. (Theochem), 364 (1996), 1-13, Table 12, q=9, alpha=1.
Index entries for linear recurrences with constant coefficients, signature (12,-30,-72,216).

# Exhibit 1
Hra := proc(r::integer,a::integer,q::integer)
    binomial(r-1,a-1)*(q-3)+binomial(r-1,a) ;
    %*(q-3)^(r-a-1) ;
end proc:
Jra := proc(r::integer,a::integer,q::integer)
    binomial(r-2,a-2)*(q-3)^2 +2*binomial(r-2,a-1)*(q-3) +binomial(r-2,a) ;
    %*(q-3)^(r-a-2) ;
end proc:
# Exhibit 2
A121124 := proc(r::integer)
    q := 9 ;
    a := 1 ;
    Jra(r,a,q)+binomial(2,r-a)+( 1 +(-1)^(r+a) +(1+(-1)^a)*(1-(-1)^r)*floor((q-3)/2)/2)*Hra(floor(r/2),floor(a/2),q) ;
    %/4 ;
end proc:
seq(A121124(n),n=2..30)  # R. J. Mathar, Aug 01 2019

Join[{1, 4}, LinearRecurrence[{12, -30, -72, 216}, {21, 138, 864, 5526}, 22]] (* Jean-François Alcover, Apr 04 2020 *)

From R. J. Mathar, Aug 01 2019: (Start)
G.f.: x^2 +4*x^3 -3*x^4*(7-38*x-54*x^2+270*x^3) / ( (6*x^2-1)*(-1+6*x)^2 ).
a(n) = A000400((n-1)/2)/12 +6^(n-1)/16 +A053469(n+1)/864, where Axxxxx(.) is zero for fractional indices, n>3. (End)

A304255 Triangle read by rows: T(0,0) = 1; T(n,k) = 6*T(n-1,k) + T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 6, 36, 1, 216, 12, 1296, 108, 1, 7776, 864, 18, 46656, 6480, 216, 1, 279936, 46656, 2160, 24, 1679616, 326592, 19440, 360, 1, 10077696, 2239488, 163296, 4320, 30, 60466176, 15116544, 1306368, 45360, 540, 1, 362797056, 100776960, 10077696, 435456, 7560, 36

Offset: 0

Zagros Lalo, May 09 2018

The numbers in rows of the triangle are along skew diagonals pointing top-left in center-justified triangle given in A013613 ((1+6*x)^n).
The coefficients in the expansion of 1/(1-6x-x^2) are given by the sequence generated by the row sums.
The row sums are Denominators of continued fraction convergent to sqrt(10), see A005668.
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 6.162277660..., a metallic mean (see A176398), when n approaches infinity.

			Triangle begins:
1;
6;
36, 1;
216, 12;
1296, 108, 1;
7776, 864, 18;
46656, 6480, 216, 1;
279936, 46656, 2160, 24;
1679616, 326592, 19440, 360, 1;
10077696, 2239488, 163296, 4320, 30;
60466176, 15116544, 1306368, 45360, 540, 1;
362797056, 100776960, 10077696, 435456, 7560, 36;
2176782336, 665127936, 75582720, 3919104, 90720, 756, 1;
13060694016, 4353564672, 554273280, 33592320, 979776, 12096, 42;
78364164096, 28298170368, 3990767616, 277136640, 9797760, 163296, 1008, 1;
470184984576, 182849716224, 28298170368, 2217093120, 92378880, 1959552, 18144, 48;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 72, 94.

Zagros Lalo, Left-justified triangle
Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (1+6x)^n

Row sums give A005668.
Cf. A000400 (column 0), A053469 (column 1), A081136 (column 2), A081144 (column 3).
Cf. A013613.
Cf. A176398.

t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, 6 t[n - 1, k] + t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 11}, {k, 0, Floor[n/2]}] // Flatten

T(n, k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, 6*T(n-1, k) + T(n-2, k-1)));
tabf(nn) = for (n=0, nn, for (k=0, n\2, print1(T(n,k), ", ")); print); \\ Michel Marcus, May 26 2018

cp's OEIS Frontend

A016129 Expansion of 1/((1-2*x)*(1-6*x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Sage

Formula

A053541 a(n) = n*10^(n-1).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A212700 a(n) = 5*n*6^(n-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A027473 Second column of A027466.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A053540 a(n) = n*9^(n-1).

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

Extensions

A053539 a(n) = n * 8^(n-1).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

GAP

Magma

Maple

A016129 Expansion of 1/((1-2x)(1-6*x)).

A212700 a(n) = 5n6^(n-1).