A016134 -id:A016134 - cp's OEIS frontend

A016321 Expansion of 1/((1-2x)(1-9x)(1-10x)).

1, 21, 313, 4065, 49081, 566721, 6350473, 69654225, 751887961, 8016991521, 84652923433, 886876310385, 9231886792441, 95586981129921, 985282830165193, 10117545471478545, 103557909243290521, 1057021183189581921

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 (21,-128,180).

Cf. A016133, A016134.

[(175*10^n +2^n-2*9^(n+2))/14 : n in [0..20]]; // Vincenzo Librandi, Oct 09 2011

CoefficientList[Series[1/((1-2x)(1-9x)(1-10x)),{x,0,20}],x] (* or *) LinearRecurrence[{21,-128,180},{1,21,313},20] (* Harvey P. Dale, Aug 18 2014 *)

a(n) = (175*10^n+2^n-162*9^n)/14 \\ Charles R Greathouse IV, Sep 23 2012

[(10^n - 2^n)/8-(9^n - 2^n)/7 for n in range(2,20)] # Zerinvary Lajos, Jun 05 2009

From Zerinvary Lajos, Jun 05 2009 [corrected by R. J. Mathar, Mar 14 2011]: (Start)
a(n) = 2^(n-1)/7 - 9^(n+2)/7 + 25*10^n/2.
a(n) = A016134(n+1) - A016133(n+1). (End)
From Vincenzo Librandi, Oct 09 2011: (Start)
a(n) = (175*10^n + 2^n - 2*9^(n+2))/14.
a(n) = 19*a(n-1) - 90*a(n-2) + 2^n.
a(n) = 21*a(n-1) - 128*a(n-2) + 180*a(n-3), n >= 3. (End)

A016325 Expansion of 1/((1-2x)(1-10x)(1-11x)).

Original entry on oeis.org

1, 23, 377, 5395, 71841, 915243, 11317657, 136994195, 1631936081, 19201296763, 223714264137, 2585856904995, 29694425953921, 339138685491083, 3855525540397817, 43660780944367795, 492768590388029361

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 (23,-152,220)

Cf. A016134, A016135.

[(2*11^(n+2) +2^n-225*10^n)/18 : n in [0..20]]; // Vincenzo Librandi, Oct 09 2011

CoefficientList[Series[1/((1 - 2 x) (1 - 10 x) (1 - 11 x)), {x, 0, 16}], x] (* Michael De Vlieger, Jan 31 2018 *)

Vec(1/((1-2*x)*(1-10*x)*(1-11*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

[(11^n - 2^n)/9-(10^n - 2^n)/8 for n in range(2,19)] # Zerinvary Lajos, Jun 05 2009

From Zerinvary Lajos, Jun 05 2009 [corrected by R. J. Mathar, Mar 14 2011]: (Start)
a(n) = 11^(n+2)/9 + 2^(n-1)/9 - 25*10^n/2.
a(n) = A016135(n+1) - A016134(n+1). (End)
a(n) = 21*a(n-1) - 110*a(n-2) + 2^n. - Vincenzo Librandi, Oct 09 2011

A060458 Maximum value seen in the final n decimal digits of 2^j for all values of j.

Original entry on oeis.org

8, 96, 992, 9984, 99968, 999936, 9999872, 99999744, 999999488, 9999998976, 99999997952, 999999995904, 9999999991808, 99999999983616, 999999999967232, 9999999999934464, 99999999999868928, 999999999999737856, 9999999999999475712, 99999999999998951424, 999999999999997902848

Offset: 1

Labos Elemer, Apr 09 2001

Consider the final n decimal digits of 2^j for all values of j. They are periodic. Sequence gives maximal value seen in these n digits.

With f(n) = a(n+1) - a(n), the difference f(n) - a(n) is always 8*10^n meaning that a(n) becomes its own "first differences" sequence when each term is prefixed a digit '8'. For higher order differences, the prefix 8 becomes: 8*10^n*Sum_{k=0..m-1} 9^k where m is the order. - R. J. Cano, May 11 2014

			Maximum of the last 4 digits of powers of 2 is 9984=10000-16. It occurs at 2^254. 2^254 = 289480223.....01978282409984 (with 77 digits, last 4 ones are ...9984). The period length of the last-4-digit segment is A005054(4)=500. For n=4 period: amplitude=9984, phase=254.

Index entries for sequences related to final digits of numbers.
Index entries for linear recurrences with constant coefficients, signature (12,-20).

Cf. A000079, A003463, A005054, A060460, A016134.

[10^n-2^n : n in [1..20]]; // Wesley Ivan Hurt, Sep 25 2014

A060458:=n->10^n-2^n: seq(A060458(n), n=1..20); # Wesley Ivan Hurt, Sep 25 2014

RecurrenceTable[{a[n] == 12 a[n - 1] - 20 a[n - 2], a[0] == 0, a[1] == 8}, a[n], {n, 1, 20}]  (* Geoffrey Critzer, Dec 15 2011*)

a(n)=sum(j=0,n-1,2^(3*n-2*j)*binomial(n,j)) \\ R. J. Cano, May 15 2014

A060458(n)=(5^n-1)<M. F. Hasler, Oct 31 2014

[10^n - 2^n for n in range(1,19)] # Zerinvary Lajos, Jun 05 2009

a(n) = 10^n - 2^n = 2^n*(5^n - 1).
From Geoffrey Critzer, Dec 15 2011: (Start)
a(n) = 12*a(n-1) - 20*a(n-2).
O.g.f.: 1/(1-10*x) - 1/(1-2*x). (End)
a(n) = f(n,0) where f(x,y) = Sum_{j=0..x+y-1} (2^(3*x-2*j)*binomial(x,j)). - R. J. Cano, May 15 2014
a(n) = 2^(n+2)*A003463(n). - R. J. Cano, Sep 25 2014
a(n) = 8*A016134(n-1). - R. J. Mathar, Mar 10 2022
E.g.f.: exp(2*x)*(exp(8*x) - 1). - Elmo R. Oliveira, Mar 26 2025

Edited by M. F. Hasler, Oct 31 2014

More terms from Elmo R. Oliveira, Mar 26 2025

A332690 Sum of all numbers in bijective base-9 numeration with digit sum n.

Original entry on oeis.org

0, 1, 12, 124, 1248, 12496, 124992, 1249984, 12499968, 124999936, 1249999862, 12499999623, 124999998144, 1249999984364, 12499999840480, 124999998308464, 1249999981991936, 12499999808733888, 124999997974967808, 1249999978624935680, 12499999774999871588

Offset: 0

Alois P. Heinz, Feb 19 2020

Different from A016134.

			a(2) = 12 = 2 + 10 = 2_bij9 + 11_bij9.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Bijective numeration
Index entries for linear recurrences with constant coefficients, signature (10,1,-8,-17,-26,-35,-44,-53,-62,-81,-72,-63,-54,-45,-36,-27,-18,-9).

Cf. A007953, A016134, A028904, A052382, A211072, A214676, A332691.

b:= proc(n) option remember; `if`(n=0, [1, 0], add((p->
      [p[1], p[2]*9+p[1]*d])(b(n-d)), d=1..min(n, 9)))
    end:
a:= n-> b(n)[2]:
seq(a(n), n=0..23);

G.f.: (Sum_{j=1..9} j*x^j) / ((B(x) - 1) * (9*B(x) - 1)) with B(x) = Sum_{j=1..9} x^j.
a(n) = A028904(A332691(n)).
a(n) = A016134(n-1) for n = 1..9.

A332691 Bijective base-9 representation of the sum of all numbers in bijective base-9 numeration with digit sum n.

Original entry on oeis.org

1, 13, 147, 1636, 18124, 199399, 2314581, 25461653, 281178597, 3192976395, 35233852789, 387573484456, 4374418444135, 48228613881184, 541525753635894, 5956784387951128, 66635738355523786, 743994232656361639, 8285146556418623572, 92246623188575957748

Offset: 1

Alois P. Heinz, Feb 19 2020

			a(2) = 13_bij9 = 12 = 2 + 10 = 2_bij9 + 11_bij9.

Alois P. Heinz, Table of n, a(n) for n = 1..955
Wikipedia, Bijective numeration

Cf. A007953, A016134, A052382, A211072, A214676, A332690.

b:= proc(n) option remember; `if`(n=0, [1, 0], add((p->
      [p[1], p[2]*9+p[1]*d])(b(n-d)), d=1..min(n, 9)))
    end:
g:= proc(n) local d, l, m; m, l:= n, "";
      while m>0 do d:= irem(m, 9, 'm');
        if d=0 then d:=9; m:= m-1 fi; l:= d, l
      od; parse(cat(l))
    end:
a:= n-> g(b(n)[2]):
seq(a(n), n=1..23);

a(n) = A052382(A332690(n)).

A096043 Triangle read by rows: T(n,k) = (n+1,k)-th element of (M^9-M)/8, where M is the infinite lower Pascal's triangle matrix, 1<=k<=n.

Original entry on oeis.org

1, 10, 2, 91, 30, 3, 820, 364, 60, 4, 7381, 4100, 910, 100, 5, 66430, 44286, 12300, 1820, 150, 6, 597871, 465010, 155001, 28700, 3185, 210, 7, 5380840, 4782968, 1860040, 413336, 57400, 5096, 280, 8, 48427561, 48427560, 21523356, 5580120, 930006

Offset: 1

Gary W. Adamson, Jun 17 2004

			Triangle begins:
1
10 2
91 30 3
820 364 60 4
7381 4100 910 100 5
66430 44286 12300 1820 150 6

Cf. A007318. First column gives A002452. Row sums give A016134.

P:= proc(n) option remember; local M; M:= Matrix(n, (i, j)-> binomial(i-1, j-1)); (M^9-M)/8 end: T:= (n, k)-> P(n+1)[n+1, k]: seq(seq(T(n, k), k=1..n), n=1..11); # Alois P. Heinz, Oct 07 2009

P[n_] := P[n] = With[{M = Array[Binomial[#1-1, #2-1]&, {n, n}]}, (MatrixPower[M, 9] - M)/8]; T[n_, k_] := P[n+1][[n+1, k]]; Table[ Table[T[n, k], {k, 1, n}], {n, 1, 11}] // Flatten (* Jean-François Alcover, Jan 28 2015, after Alois P. Heinz *)

Edited with more terms by Alois P. Heinz, Oct 07 2009

A279035 Left-concatenate zeros to 2^(n-1) such that it has n digits. In the regular array formed by listing the found powers, a(n) is the sum of (nonzero) digits in column n.

Original entry on oeis.org

1, 2, 4, 9, 9, 9, 8, 19, 9, 8, 17, 27, 27, 27, 28, 17, 26, 35, 45, 45, 46, 37, 25, 44, 53, 65, 42, 72, 74, 52, 70, 90, 92, 74, 53, 62, 72, 70, 93, 61, 81, 80, 89, 100, 91, 80, 91, 79, 99, 99, 99, 98, 107, 117, 118, 106, 130, 86, 123, 155, 137, 117, 118, 105, 136

Offset: 1

David A. Corneth, Dec 03 2016

After carries, this is the decimal expansion of Sum_{i>=0} 0.2^i = 1.25. For n > 2, the 10^0's digit of a(n) + the 10^1's digit of a(n+1) + ... + the 10^m's digit of a(n+m) = 9 for some finite m.
Conjecture: a(n) ~ c*n where c ~= 1.93.
Conjecture: lim_{n->infinity} a(n)/n = (9/2)*log_5(2) =
1.93804... - Jon E. Schoenfield, Dec 09 2016

			1
.2
. 4
. .8
. .16
. . 32
. . .64
. . .128
. . . 256
. . . .512
. . . .1024
The sum of digits of the first column is 1. Therefore, a(1) = 1.
The sum of digits in column 4 is 8 + 1 = 9. Therefore, a(4) = 9.
With the powers of 2 listed above, we can find n up to n = 7. For n > 8, some digits from 2^m compose a(n) for m > 10.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Cf. A000079, A016134.

f[n_, b_] := Block[{k = n}, While[k < n + Floor[ k*Log10[b]], k++]; Plus @@ Mod[ Quotient[ Table[ b^j*10^(k - j), {j, n -1, k}], 10^(k - n +1)], 10]]; Table[f[n, 2], {n, 65}]
 (* Robert G. Wilson v, Dec 03 2016 *)

A016205 Expansion of g.f. 1/((1-x)(1-2x)(1-10x)).

Original entry on oeis.org

1, 13, 137, 1385, 13881, 138873, 1388857, 13888825, 138888761, 1388888633, 13888888377, 138888887865, 1388888886841, 13888888884793, 138888888880697, 1388888888872505, 13888888888856121, 138888888888823353, 1388888888888757817, 13888888888888626745, 138888888888888364601

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (13,-32,20).

Cf. A002275, A016134.

Vec(1/(1-13*x+32*x^2-20*x^3) + O(x^21)) \\ Elmo R. Oliveira, Mar 26 2025

a(n) = (25*10^n - 9*2^n + 2)/18. - Bruno Berselli, Feb 09 2011
a(n) = 10*a(n-1) + 2^(n+1) - 1 if n > 0; a(0)=1. - Vincenzo Librandi, Feb 09 2011
From Elmo R. Oliveira, Mar 26 2025: (Start)
E.g.f.: exp(x)*(25*exp(9*x) - 9*exp(x) + 2)/18.
a(n) = 13*a(n-1) - 32*a(n-2) + 20*a(n-3).
a(n) = A016134(n+1) - A002275(n+2). (End)

More terms from Elmo R. Oliveira, Mar 26 2025

A350592 Integers m such that b(m) := 20^m*(5^(m+1) - 1)/4 + (20^m - 1)/19 is a prime.

Original entry on oeis.org

2, 4, 5, 7, 9, 13, 85, 222, 249, 1843

Offset: 1

Ya-Ping Lu, Jan 07 2022

b(m) = Sum_{i=0..2m} 2^(m - |m - i|)*10^i.
a(11) > 5000. - Michael S. Branicky, Jun 07 2022
a(11) > 50000. - Michael S. Branicky, Dec 21 2024

			m            b(m)          n    a(n)
--   -------------------   --   ----
0             1
1            121
2           12421          1     2
3          1248421
4         124968421        2     4
5        12499368421       3     5
6       1249987368421
7      124999747368421     4     7
8     12499994947368421
9    1249999898947368421   5     9

Cf. A002477, A016134, A260802.

Select[Range[250], PrimeQ[20^# * (5^(# + 1) - 1)/4 + (20^# - 1)/19] &] (* Amiram Eldar, Jan 08 2022 *)

from sympy import isprime; {print(m, end = ', ') for m in range(2000) if isprime(20**m*(5**(m+1) - 1)//4 + (20**m - 1)//19)}

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A016321 Expansion of 1/((1-2x)(1-9x)(1-10x)).

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A016325 Expansion of 1/((1-2x)(1-10x)(1-11x)).

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A060458 Maximum value seen in the final n decimal digits of 2^j for all values of j.

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A332690 Sum of all numbers in bijective base-9 numeration with digit sum n.

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A332691 Bijective base-9 representation of the sum of all numbers in bijective base-9 numeration with digit sum n.

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A096043 Triangle read by rows: T(n,k) = (n+1,k)-th element of (M^9-M)/8, where M is the infinite lower Pascal's triangle matrix, 1<=k<=n.

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A279035 Left-concatenate zeros to 2^(n-1) such that it has n digits. In the regular array formed by listing the found powers, a(n) is the sum of (nonzero) digits in column n.

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A016205 Expansion of g.f. 1/((1-x)(1-2x)(1-10x)).