A053464 -id:A053464 - cp's OEIS frontend

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

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

A053469 a(n) = n*6^(n-1).

Original entry on oeis.org

1, 12, 108, 864, 6480, 46656, 326592, 2239488, 15116544, 100776960, 665127936, 4353564672, 28298170368, 182849716224, 1175462461440, 7522959753216, 47958868426752, 304679870005248, 1929639176699904, 12187194800209920, 76779327241322496, 482612914088312832

Offset: 1

Barry E. Williams, Jan 13 2000

Binomial transform of A053464. - R. J. Mathar, Oct 26 2011

			G.f. = x + 12*x^2 + 108*x^3 + 864*x^4 + 6480*x^5 + 46656*x^6 + ... - _Michael Somos_, Dec 16 2019

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..400
Frank Ellermann, Illustration of binomial transforms.
Index entries for linear recurrences with constant coefficients, signature (12,-36).

Cf. A002697, A027471.

[n*(6^(n-1)): n in [1..30]]; // Vincenzo Librandi, Jun 09 2011

f[n_]:=n*6^(n-1);f[Range[40]] (* Vladimir Joseph Stephan Orlovsky, Feb 09 2011 *)
LinearRecurrence[{12,-36},{1,12},20] (* Harvey P. Dale, Apr 28 2015 *)

a(n)=n*6^(n-1) \\ Charles R Greathouse IV, Oct 07 2015

[lucas_number1(n,12,36) for n in range(1, 21)] # Zerinvary Lajos, Apr 28 2009

a(n) = 12*a(n-1) - 36*a(n-2), n>=3.
G.f.: x/(6x-1)^2. - Zerinvary Lajos, Apr 28 2009
E.g.f.: x*exp(6*x). - Michael Somos, Dec 16 2019
From Amiram Eldar, Oct 28 2020: (Start)
Sum_{n>=1} 1/a(n) = 6*log(6/5).
Sum_{n>=1} (-1)^(n+1)/a(n) = 6*log(7/6). (End)

More terms from James Sellers, Feb 02 2000

More terms from Zerinvary Lajos, Oct 02 2007

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

A036291 a(n) = n*5^n.

Original entry on oeis.org

0, 5, 50, 375, 2500, 15625, 93750, 546875, 3125000, 17578125, 97656250, 537109375, 2929687500, 15869140625, 85449218750, 457763671875, 2441406250000, 12969970703125, 68664550781250, 362396240234375, 1907348632812500, 10013580322265625, 52452087402343750

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (10,-25).

Cf. A000351, A053464.

[n*5^n: n in [0..20]]; // Vincenzo Librandi, Sep 09 2014

g:=1/(1-5*z): gser:=series(g, z=0, 43): seq(coeff(gser, z, n)*n, n=0..31); # Zerinvary Lajos, Jan 09 2009

Table[n 5^n, {n, 0, 20}] (* Vincenzo Librandi, Sep 09 2014 *)

G.f.: 5*x/(1 - 5*x)^2. - Vincenzo Librandi, Sep 09 2014
From Amiram Eldar, Jul 20 2020: (Start)
Sum_{n>=1} 1/a(n) = log(5/4).
Sum_{n>=1} (-1)^(n+1)/a(n) = log(6/5). (End)
From Elmo R. Oliveira, Sep 09 2024: (Start)
E.g.f.: 5*x*exp(5*x).
a(n) = n*A000351(n) = 5*A053464(n).
a(n) = 10*a(n-1) - 25*a(n-2) for n > 1. (End)

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

A212699 Main transitions in systems of n particles with spin 2.

Original entry on oeis.org

4, 40, 300, 2000, 12500, 75000, 437500, 2500000, 14062500, 78125000, 429687500, 2343750000, 12695312500, 68359375000, 366210937500, 1953125000000, 10375976562500, 54931640625000, 289916992187500, 1525878906250000, 8010864257812500, 41961669921875000, 219345092773437500

Offset: 1

Stanislav Sykora, May 25 2012

Please, refer to the general explanation in A212697.

This particular sequence is obtained for base b=5, corresponding to spin S=(b-1)/2=2.

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 (10,-25).

Cf. A001787, A212697, A212698, A212700, A212701, A212702, A212703, A212704 (b = 2, 3, 4, 6, 7, 8, 9, 10).

Cf. A000351, A008586, A053464.

Join[{4},Table[4n*5^(n-1),{n,20}]] (* or *) Join[{4},LinearRecurrence[{10,-25},{4,40},20]] (* Harvey P. Dale, Aug 19 2014 *)

mtrans(n, b) = n*(b-1)*b^(n-1);
for (n=1, 100, write("b212699.txt", n, " ", mtrans(n, 5)))

a(n) = n*(b-1)*b^(n-1) where b=5.
a(n) = 10*a(n-1) - 25*a(n-2), a(0)=a(1)=4, a(2)=40. - Harvey P. Dale, Aug 19 2014
From Elmo R. Oliveira, May 13 2025: (Start)
G.f.: 4*x/(5*x-1)^2.
E.g.f.: 4*x*exp(5*x).
a(n) = 4*A053464(n) = A008586(n)*A000351(n-1). (End)

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

A218016 Triangle, read by rows, where T(n,k) = k!C(n, k)5^(n-k) for n>=0, k=0..n.

Original entry on oeis.org

1, 5, 1, 25, 10, 2, 125, 75, 30, 6, 625, 500, 300, 120, 24, 3125, 3125, 2500, 1500, 600, 120, 15625, 18750, 18750, 15000, 9000, 3600, 720, 78125, 109375, 131250, 131250, 105000, 63000, 25200, 5040, 390625, 625000, 875000, 1050000, 1050000, 840000, 504000, 201600, 40320

Offset: 0

Vincenzo Librandi, Nov 10 2012

Triangle formed by the derivatives of x^n evaluated at x=5.
Sum(T(n,k), k=0..n) = A080954(n) (see the Formula section of A080954). . Also:
first column:    A000351;
second column:   A053464;
third column:  2*A084902;
fourth column: 6*A081143.

			Triangle begins:
1;
5,      1;
25,     10,     2;
125,    75,     30,     6;
625,    500,    300,    120,     24;
3125,   3125,   2500,   1500,    600,     120;
15625,  18750,  18750,  15000,   9000,    3600,   720;
78125,  109375, 131250, 131250,  105000,  63000,  25200,  5040;
390625, 625000, 875000, 1050000, 1050000, 840000, 504000, 201600, 40320; etc.

Vincenzo Librandi, Rows n = 0..100, flattened

Cf. A000351, A053464, A080954, A081143, A084902, A090802, A217629, A218017.

[Factorial(n)/Factorial(n-k)*5^(n-k): k in [0..n], n in [0..10]];

Flatten[Table[n!/(n-k)!*5^(n-k), {n, 0, 10}, {k, 0, n}]]

T(n,k) = 5^(n-k)*n!/(n-k)! for n>=0, k=0..n.

E.g.f. (by columns): exp(5x)*x^k.

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

cp's OEIS Frontend

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

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A053469 a(n) = n*6^(n-1).

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A027473 Second column of A027466.

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A036291 a(n) = n*5^n.

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

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

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A218016 Triangle, read by rows, where T(n,k) = k!C(n, k)5^(n-k) for n>=0, k=0..n.