A052379 -id:A052379 - cp's OEIS frontend

A145729 Partial sums of A052379.

0, 8, 80, 664, 5344, 42792, 342384, 2739128, 21913088, 175304776, 1402438288, 11219506392, 89756051232, 718048409960, 5744387279792, 45955098238456, 367640785907776, 2941126287262344, 23529010298098896, 188232082384791320

Offset: 0

Vladimir Joseph Stephan Orlovsky, Oct 17 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (10,-17,8).

Cf. A052379.

[8*(8^(n+1)-7*n-8)/49 : n in [0..20]]; // Vincenzo Librandi, Jul 04 2011

lst={};s=0;Do[s+=(s+=(s+=(s+=n)));AppendTo[lst,s],{n,0,5!}];lst

concat(0, Vec(-8*x/((x-1)^2*(8*x-1)) + O(x^100))) \\ Colin Barker, Oct 27 2014

a(n) = sum_{i=0..n-1} A052379(i).
a(n) = 8*(8^(n+1)-7*n-8)/49 = 8*A014831(n) = 2*A145730(n). - R. J. Mathar, Oct 21 2008
a(n) = 10*a(n-1)-17*a(n-2)+8*a(n-3). G.f.: -8*x / ((x-1)^2*(8*x-1)). - Colin Barker, Oct 27 2014

Edited by R. J. Mathar, Oct 21 2008

A024101 a(n) = 9^n-1.

Original entry on oeis.org

0, 8, 80, 728, 6560, 59048, 531440, 4782968, 43046720, 387420488, 3486784400, 31381059608, 282429536480, 2541865828328, 22876792454960, 205891132094648, 1853020188851840, 16677181699666568, 150094635296999120

Offset: 0

N. J. A. Sloane

Number of integers from 0 to 10^(n+1)-1 that lack any particular digit other than 0. - Robert G. Wilson v, Apr 14 2003

These are the numbers 888...8 in base 9. - Zerinvary Lajos, Nov 21 2007

Index entries for linear recurrences with constant coefficients, signature (10,-9).

Cf. A001019, A024023, A034472, A052386, A052379, A248726.

Table[9^n - 1, {n, 0, 20}]
LinearRecurrence[{10, -9}, {0, 8}, 30] (* Harvey P. Dale, Apr 14 2015 *)

a(n)=9^n-1 \\ Charles R Greathouse IV, Jun 11 2015

G.f.: 1/(1-9*x)-1/(1-x). - Mohammad K. Azarian, Jan 14 2009
E.g.f.: e^(9*x)-e^x. - Mohammad K. Azarian, Jan 14 2009
a(n) = A024023(n)*A034472(n). - Reinhard Zumkeller, Feb 14 2009
a(n) = 9*a(n-1)+8 for n>0, a(0)=0. - Vincenzo Librandi, Nov 19 2010
a(0)=0, a(1)=8; for n>1, a(n) = 10*a(n-1)-9*a(n-2). - Harvey P. Dale, Apr 14 2015
a(n) = Sum_{i=1..n} 8^i*binomial(n,n-i) for n>0, a(0)=0. - Bruno Berselli, Nov 11 2015
a(n) = A001019(n) - 1. - Sean A. Irvine, Jun 19 2019
Sum_{n>=1} 1/a(n) = A248726. - Amiram Eldar, Nov 13 2020

A228275 A(n,k) = Sum_{i=1..k} n^i; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 6, 3, 0, 0, 4, 14, 12, 4, 0, 0, 5, 30, 39, 20, 5, 0, 0, 6, 62, 120, 84, 30, 6, 0, 0, 7, 126, 363, 340, 155, 42, 7, 0, 0, 8, 254, 1092, 1364, 780, 258, 56, 8, 0, 0, 9, 510, 3279, 5460, 3905, 1554, 399, 72, 9, 0

Offset: 0

Alois P. Heinz, Aug 19 2013

A(n,k) is the total sum of lengths of longest ending contiguous subsequences with the same value over all s in {1,...,n}^k:
A(4,1) = 4 = 1+1+1+1: [1], [2], [3], [4].
A(1,4) = 4: [1,1,1,1].
A(3,2) = 12 = 2+1+1+1+2+1+1+1+2: [1,1], [1,2], [1,3], [2,1], [2,2], [2,3], [3,1], [3,2], [3,3].
A(2,3) = 14 = 3+1+1+2+2+1+1+3: [1,1,1], [1,1,2], [1,2,1], [1,2,2], [2,1,1], [2,1,2], [2,2,1], [2,2,2].

			Square array A(n,k) begins:
  0, 0,  0,   0,    0,     0,      0,      0, ...
  0, 1,  2,   3,    4,     5,      6,      7, ...
  0, 2,  6,  14,   30,    62,    126,    254, ...
  0, 3, 12,  39,  120,   363,   1092,   3279, ...
  0, 4, 20,  84,  340,  1364,   5460,  21844, ...
  0, 5, 30, 155,  780,  3905,  19530,  97655, ...
  0, 6, 42, 258, 1554,  9330,  55986, 335922, ...
  0, 7, 56, 399, 2800, 19607, 137256, 960799, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give: A000004, A001477, A002378, A027444, A027445, A152031, A228290, A228291, A228292, A228293, A228294.
Rows n=0-11 give: A000004, A001477, A000918(k+1), A029858(k+1), A080674, A104891, A105281, A104896, A052379(k-1), A052386, A105279, A105280.
Main diagonal gives A031972.
Lower diagonal gives A226238.
Cf. A228250.

A:= (n, k)-> `if`(n=1, k, (n/(n-1))*(n^k-1)):
seq(seq(A(n, d-n), n=0..d), d=0..12);

a[0, 0] = 0; a[1, k_] := k; a[n_, k_] := n*(n^k-1)/(n-1); Table[a[n-k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 16 2013 *)

A(1,k) = k, else A(n,k) = n/(n-1)*(n^k-1).
A(n,k) = Sum_{i=1..k} n^i.
A(n,k) = Sum_{i=1..k+1} binomial(k+1,i)*A(n-i,k)*(-1)^(i+1) for n>k, given values A(0,k), A(1,k),..., A(k,k). - Yosu Yurramendi, Sep 03 2013

A052386 Number of integers from 1 to 10^n-1 that lack 0 as a digit.

Original entry on oeis.org

0, 9, 90, 819, 7380, 66429, 597870, 5380839, 48427560, 435848049, 3922632450, 35303692059, 317733228540, 2859599056869, 25736391511830, 231627523606479, 2084647712458320, 18761829412124889, 168856464709124010, 1519708182382116099, 13677373641439044900

Offset: 0

Odimar Fabeny, Mar 10 2000

			For n=2, the numbers from 1 to 99 which *have* 0 as a digit are the 9 numbers 10, 20, 30, ..., 90. So a(1) = 99 - 9 = 90.

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Peter D. Loly and Ian D. Cameron, Frierson's 1907 Parameterization of Compound Magic Squares Extended to Orders 3^L, L = 1, 2, 3, ..., with Information Entropy, arXiv:2008.11020 [math.HO], 2020.
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Cf. A024101, A052379.

Row n=9 of A228275.

[9*(9^n-1)/8: n in [0..20]]; // Vincenzo Librandi, Jul 04 2011

Table[9(9^n - 1)/8, {n, 0, 20}]
LinearRecurrence[{10,-9},{0,9},30] (* Harvey P. Dale, Mar 22 2019 *)

a(n)=9^(n+1)\8 \\ Charles R Greathouse IV, Aug 25 2014

a(n) = 9*a(n-1) + 9.
a(n) = 9*(9^n-1)/8 = sum_{j=1..n} 9^j = a(n-1)+9^n = 9*A002452(n) = A002452(n+1)-1; write A000918(n+1) in base 2 and read as if written in base 9. - Henry Bottomley, Aug 30 2001
a(n) = 10*a(n-1)-9*a(n-2). G.f.: 9*x / ((x-1)*(9*x-1)). - Colin Barker, Sep 26 2013

More terms and revised description from James Sellers, Mar 13 2000
More terms and revised description from Robert G. Wilson v, Apr 14 2003
More terms from Colin Barker, Sep 26 2013

A105279 a(0)=0; a(n) = 10*a(n-1) + 10.

Original entry on oeis.org

0, 10, 110, 1110, 11110, 111110, 1111110, 11111110, 111111110, 1111111110, 11111111110, 111111111110, 1111111111110, 11111111111110, 111111111111110, 1111111111111110, 11111111111111110, 111111111111111110, 1111111111111111110, 11111111111111111110, 111111111111111111110

Offset: 0

Alexandre Wajnberg, Apr 25 2005

a(n) is the smallest even number with digits in {0,1} having digit sum n; in other words, the base 10 reading of the binary string of A000918(n). Cf. A069532. - Jason Kimberley, Nov 02 2011

Also, except for a(0), the binary representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 645", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, Jul 19 2017

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Robert Price, Diagrams of first 20 stages of the Cellular Automata.
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton.
S. Wolfram, A New Kind of Science.
Wolfram Research, Wolfram Atlas of Simple Programs.
Index entries for sequences related to cellular automata.
Index to 2D 5-Neighbor Cellular Automata.
Index to Elementary Cellular Automata.
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A000918, A002275, A029858, A052379, A052386, A069532, A080674, A124166, A161770.
Row n=10 of A228275.
Partial sums of A178500.

a105279 n = a105279_list !! n
a105279_list = iterate ((* 10) . (+ 1)) 0
-- Reinhard Zumkeller, Feb 05 2012

[-10/9+(10/9)*10^n: n in [0..20]]; // Vincenzo Librandi, Jul 04 2011

NestList[10*(# + 1) &, 0, 25] (* Paolo Xausa, Jul 17 2024 *)

a(n) = (10/9)*(10^n - 1), with n>=0.
a(n) = Sum_{k=1..n} 10^k.
Repunits times 10: a(n) = 10 * A002275(n). - Reinhard Zumkeller, Feb 05 2012
From Stefano Spezia, Sep 15 2023: (Start)
O.g.f.: 10*x/((1 - x)*(1 - 10*x)).
E.g.f.: 10*exp(x)*(exp(9*x) - 1)/9. (End)
From Elmo R. Oliveira, Jun 18 2025: (Start)
a(n) = 11*a(n-1) - 10*a(n-2).
a(n) = A124166(n)/10.
a(n) = A161770(n)/100 for n >= 1. (End)

A104891 a(0) = 0; a(n) = 5*a(n-1) + 5.

Original entry on oeis.org

0, 5, 30, 155, 780, 3905, 19530, 97655, 488280, 2441405, 12207030, 61035155, 305175780, 1525878905, 7629394530, 38146972655, 190734863280, 953674316405, 4768371582030, 23841857910155, 119209289550780, 596046447753905, 2980232238769530, 14901161193847655

Offset: 0

Alexandre Wajnberg, Apr 24 2005

Number of integers from 0 to (10^n)-1 that lack 0, 1, 2, 3 and 4 as a digit.

Number of monic irreducible polynomials of degree 1 in GF(5)[x1,...,xn]. - Max Alekseyev, Jan 23 2006

			a(3) = 5*a(2) + 5 = 5*30 + 5 = 155.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A000225, A000918, A029858, A052379, A052386, A080674.

Row n=5 of A228275.

[5*(5^n -1)/4: n in [0..30]]; // G. C. Greubel, Jun 15 2021

a:=n->add(5^j,j=1..n): seq(a(n),n=0..30); # Zerinvary Lajos, Jun 27 2007

RecurrenceTable[{a[n]==5*a[n-1]+5, a[0]==0}, a, {n, 0, 30}] (* Vaclav Kotesovec, Jul 25 2014 *)
NestList[5#+5&,0,30] (* Harvey P. Dale, Oct 04 2019 *)

concat(0, Vec(5*x/((x-1)*(5*x-1)) + O(x^30))) \\ Colin Barker, Jul 25 2014

[5*(5^n -1)/4 for n in (0..30)] # G. C. Greubel, Jun 15 2021

a(n) = 5*(5^n - 1)/4. - Max Alekseyev, Jan 23 2006
a(n) = a(n-1) + 5^n with a(0)=0. - Vincenzo Librandi, Nov 13 2010
From Colin Barker, Jul 25 2014: (Start)
a(n) = 6*a(n-1) - 5*a(n-2).
G.f.: 5*x / ((1-x)*(1-5*x)). (End)
E.g.f.: (5/4)*(exp(5*x) - exp(x)). - G. C. Greubel, Jun 15 2021

A104896 a(0) = 0; a(n) = 7*a(n-1) + 7.

Original entry on oeis.org

0, 7, 56, 399, 2800, 19607, 137256, 960799, 6725600, 47079207, 329554456, 2306881199, 16148168400, 113037178807, 791260251656, 5538821761599, 38771752331200, 271402266318407, 1899815864228856, 13298711049601999, 93090977347214000, 651636841430498007

Offset: 0

Alexandre Wajnberg, Apr 24 2005

Conjecture: this is also the number of integers from 0 to 10^n - 1 that lack 0, 1 and 2 as a digit.

Number of monic irreducible polynomials of degree 1 in GF(7)[x1,...,xn]. - Max Alekseyev, Jan 23 2006

Alois P. Heinz, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (8,-7).

Cf. A000918, A029858, A052379, A052386, A080674.

Row n=7 of A228275.

[(7/6)*(7^n -1): n in [0..30]]; // G. C. Greubel, Jun 09 2021

a:=n->sum (7^j,j=1..n): seq(a(n), n=0..30); # Zerinvary Lajos, Oct 03 2007

RecurrenceTable[{a[n]==7*a[n-1]+7,a[0]==0},a,{n,0,30}] (* Vaclav Kotesovec, Jul 25 2014 *)

concat(0, Vec(7*x/((x-1)*(7*x-1)) + O(x^30))) \\ Colin Barker, Jul 25 2014

[(7/6)*(7^n -1) for n in (0..30)] # G. C. Greubel, Jun 09 2021

a(n) = (7^(n+1) - 7) / 6. - Max Alekseyev, Jan 23 2006
a(n) = a(n-1) + 7^n with a(0)=0. - Vincenzo Librandi, Nov 13 2010
From Colin Barker, Jul 25 2014: (Start)
a(n) = 8*a(n-1) - 7*a(n-2).
G.f.: 7*x / ((x-1)*(7*x-1)). (End)
E.g.f.: (7/6)*(exp(7*x) - exp(x)). - G. C. Greubel, Jun 09 2021

A105281 a(0)=0; a(n) = 6*a(n-1) + 6.

Original entry on oeis.org

0, 6, 42, 258, 1554, 9330, 55986, 335922, 2015538, 12093234, 72559410, 435356466, 2612138802, 15672832818, 94036996914, 564221981490, 3385331888946, 20311991333682, 121871948002098, 731231688012594, 4387390128075570, 26324340768453426, 157946044610720562

Offset: 0

Alexandre Wajnberg, Apr 25 2005

Number of integers from 0 to (10^n) - 1 that lack 0, 1, 2 and 3 as a digit.

a(n) is the expected number of tosses of a single die needed to obtain for the first time a string of n consecutive 6's. - Jean M. Morales, Aug 04 2012

Index entries for linear recurrences with constant coefficients, signature (7,-6).

Cf. A000918, A003464, A029858, A052379, A052386, A080674.

Row n=6 of A228275.

a:=n->add(6^j,j=1..n): seq(a(n),n=0..30); # Zerinvary Lajos, Oct 03 2007

NestList[6#+6&,0,30] (* Harvey P. Dale, Jul 24 2012 *)

PARI
```
a(n)=if(n<0,0, (6^n-1)*6/5)
```

a(n) = 6^n + a(n-1) (with a(0)=0). - Vincenzo Librandi, Nov 13 2010
From Colin Barker, Jan 28 2013: (Start)
a(n) = 7*a(n-1) - 6*a(n-2).
G.f.: 6*x/((x-1)*(6*x-1)). (End)
From Elmo R. Oliveira, Mar 16 2025: (Start)
E.g.f.: 6*exp(x)*(exp(5*x) - 1)/5.
a(n) = 6*(6^n - 1)/5.
a(n) = 6*A003464(n). (End)

More terms from Harvey P. Dale, Jul 24 2012

A105280 a(0)=0; a(n) = 11*a(n-1) + 11.

Original entry on oeis.org

0, 11, 132, 1463, 16104, 177155, 1948716, 21435887, 235794768, 2593742459, 28531167060, 313842837671, 3452271214392, 37974983358323, 417724816941564, 4594972986357215, 50544702849929376, 555991731349223147, 6115909044841454628, 67274999493256000919, 740024994425816010120

Offset: 0

Alexandre Wajnberg, Apr 25 2005

Harvey P. Dale, Table of n, a(n) for n = 0..960
Index entries for linear recurrences with constant coefficients, signature (12,-11).

Cf. A000918, A016123, A029858, A052379, A052386, A080674.

Row n=11 of A228275.

a:=n-> add(11^j,j=1..n): seq(a(n),n=0..12); # Zerinvary Lajos, Oct 03 2007

NestList[11#+11&,0,20] (* or *) LinearRecurrence[{12,-11},{0,11},20] (* Harvey P. Dale, Dec 02 2023 *)

a(n) = 11^n + a(n-1) (with a(0)=0). - Vincenzo Librandi, Nov 13 2010
From Elmo R. Oliveira, May 24 2025: (Start)
G.f.: 11*x/((x-1)*(11*x-1)).
E.g.f.: 11*exp(x)*(exp(10*x) - 1)/10.
a(n) = 11*(11^n - 1)/10.
a(n) = 12*a(n-1) - 11*a(n-2).
a(n) = A016123(n) - 1. (End)

Corrected by T. D. Noe, Nov 07 2006

A247841 a(n) = Sum_{k=2..n} 8^k.

Original entry on oeis.org

0, 64, 576, 4672, 37440, 299584, 2396736, 19173952, 153391680, 1227133504, 9817068096, 78536544832, 628292358720, 5026338869824, 40210710958656, 321685687669312, 2573485501354560, 20587884010836544, 164703072086692416, 1317624576693539392

Offset: 1

Vincenzo Librandi, Sep 25 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (9,-8).

Cf. similar sequences listed in A247817.

Cf. A052379.

[0] cat [&+[8^k: k in [2..n]]: n in [2..30]];

Magma
```
[(8^(n+1)-64)/7: n in [1..30]];
```

RecurrenceTable[{a[1] == 0, a[n] == a[n-1] + 8^n}, a, {n, 30}] (* or *) CoefficientList[Series[64 x / ((1 - x) (1 - 8 x)), {x, 0, 30}], x]
LinearRecurrence[{9,-8},{0,64},30] (* Harvey P. Dale, May 01 2018 *)

G.f.: 64*x^2/((1-x)*(1-8*x)).
a(n) = a(n-1) + 8^n.
a(n) = (8^(n+1) - 64)/7.
a(n) = 9*a(n-1) - 8*a(n-2).
a(n) = A052379(n) - 8. - Michel Marcus, Sep 25 2014

cp's OEIS Frontend

A145729 Partial sums of A052379.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A024101 a(n) = 9^n-1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A228275 A(n,k) = Sum_{i=1..k} n^i; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A052386 Number of integers from 1 to 10^n-1 that lack 0 as a digit.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A105279 a(0)=0; a(n) = 10*a(n-1) + 10.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

Formula

A104891 a(0) = 0; a(n) = 5*a(n-1) + 5.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A104896 a(0) = 0; a(n) = 7*a(n-1) + 7.

Views

Author