A228275 -id:A228275 - cp's OEIS frontend

A027444 a(n) = n^3 + n^2 + n.

0, 3, 14, 39, 84, 155, 258, 399, 584, 819, 1110, 1463, 1884, 2379, 2954, 3615, 4368, 5219, 6174, 7239, 8420, 9723, 11154, 12719, 14424, 16275, 18278, 20439, 22764, 25259, 27930, 30783, 33824, 37059, 40494, 44135, 47988, 52059, 56354, 60879, 65640, 70643, 75894

Offset: 0

Patrick De Geest and Mark Milhet (mm992395(AT)shellus.com)

For n>1, a(n) is the volume of a truncated square pyramid with height n and base lengths n+2 and n-1. - Wesley Ivan Hurt, Apr 05 2016

			For n = 4, 4^3 + 4^2 + 4 = 64 + 16 + 4 = 84.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. Applegate, M. LeBrun, and N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq. 14 (2011) # 11.9.8.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Eric Weisstein's World of Mathematics, Magic Circles.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Column k=3 of A228275.

Cf. A270109.

[n^3 + n^2 + n: n in [0..50]]; // Vincenzo Librandi, Jun 09 2011

A027444:=n->n^3+n^2+n: seq(A027444(n), n=0..100); # Wesley Ivan Hurt, Apr 05 2016

Table[n^3 + n^2 + n, {n, 0, 50}] (* Harvey P. Dale, Dec 13 2013 *)

O.g.f.: x*(3 + 2*x + x^2)/(1 - x)^4. - R. J. Mathar, Feb 04 2008

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3. - Wesley Ivan Hurt, Apr 05 2016

A080674 a(n) = (4/3)*(4^n - 1).

Original entry on oeis.org

0, 4, 20, 84, 340, 1364, 5460, 21844, 87380, 349524, 1398100, 5592404, 22369620, 89478484, 357913940, 1431655764, 5726623060, 22906492244, 91625968980, 366503875924, 1466015503700, 5864062014804, 23456248059220, 93824992236884, 375299968947540, 1501199875790164

Offset: 0

N. J. A. Sloane, Mar 02 2003

a(n) is the number of steps which are made when generating all n-step random walks that begin in a given point P on a two-dimensional square lattice. To make one step means to move along one edge on the lattice. - Pawel P. Mazur (Pawel.Mazur(AT)pwr.wroc.pl), Mar 10 2005
Conjectured to be the number of integers from 0 to (10^n)-1 that lack 0, 1, 2, 3, 4 and 5 as a digit. - Alexandre Wajnberg, Apr 25 2005
Gives the values of m such that binomial(4*m + 4,m) is odd. Cf. A002450, A020988 and A263132. - Peter Bala, Oct 11 2015
Also the partial sums of 4^n for n>0, cf. A000302. - Robert G. Wilson v, Sep 18 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..170
Peter Bala, A characterization of A002450, A020988 and A080674.
Mattia Fregola, Cellular Automata RULE13 generating OEIS sequence A080674
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Row n = 4 of A228275.

Cf. A000301, A000302, A002450, A020988, A024036, A080674, A263132.

[(4/3)*(4^n-1): n in [0..40] ]; // Vincenzo Librandi, Apr 28 2011

Table[4*(4^n-1)/3,{n,0,100}]  (* Vladimir Joseph Stephan Orlovsky, Jan 30 2012 *)
LinearRecurrence[{5,-4},{0,4},40] (* Harvey P. Dale, May 05 2018 *)

vector(100, n, n--; (4/3)*(4^n-1)) \\ Altug Alkan, Oct 11 2015

Vec(4*x/((x-1)*(4*x-1)) + O(x^40)) \\ Colin Barker, Oct 12 2015

a(n) = 2*A020988(n) = A002450(n+1) - 1 = 4*A002450(n).
a(n) = Sum_{i = 1..n} 4^i. - Adam McDougall (mcdougal(AT)stolaf.edu), Sep 29 2004
a(n) = 4*a(n-1) + 4. - Alexandre Wajnberg, Apr 25 2005
a(n) = 4^n + a(n-1) (with a(0) = 0). - Vincenzo Librandi, Aug 08 2010
From Colin Barker, Oct 12 2015: (Start)
a(n) = 5*a(n-1) - 4*a(n-2).
G.f.: 4*x / ((x-1)*(4*x-1)). (End)
E.g.f.: 4*exp(x)*(exp(3*x) - 1)/3. - Elmo R. Oliveira, Dec 17 2023

A060072 a(n) = (n^(n-1) - 1)/(n-1) for n>1, a(1) = 0.

Original entry on oeis.org

0, 1, 4, 21, 156, 1555, 19608, 299593, 5380840, 111111111, 2593742460, 67546215517, 1941507093540, 61054982558011, 2085209001813616, 76861433640456465, 3041324492229179280, 128583032925805678351, 5784852794328402307380, 275941052631578947368421

Offset: 1

Henry Bottomley, Feb 21 2001

nonn

(n-1)-digit repunits in base n written in decimal.

			a(10)=111111111; i.e., just nine 1's (converted from base 10 to decimal).

Harry J. Smith, Table of n, a(n) for n=1..200

Cf. A000169, A037205, A055869, A060073, A125118, A228275.

Cf. other sequences of generalized repunits, such as A053696, A055129, A031973, A125598, A173468, A023037, A119598, A085104, and A162861.

[0] cat [ (n^(n-1) -1)/(n-1) : n in [2..25]]; // G. C. Greubel, Aug 15 2022

Join[{0},Array[(#^(#-1)-1)/(#-1)&,20,2]] (* Harvey P. Dale, Jun 04 2013 *)

a(n) = if (n==1, 0, (n^(n - 1) - 1)/(n - 1)); \\ Harry J. Smith, Jul 01 2009

[0]+[(n^(n-1) -1)/(n-1) for n in (2..25)] # G. C. Greubel, Aug 15 2022

a(n+1) = Sum_{k=1..n} n^(k-1)*C(n, k). - Olivier Gérard, Jun 26 2001 [Corrected by Mathew Englander, Dec 15 2020]
a(n) = Sum_{j=2..n} n^(n-j). - Zerinvary Lajos, Sep 11 2006
a(n+1) = A125118(n,n). - Reinhard Zumkeller, Nov 21 2006
a(n) = Integral_{x=1/n..1} 1/x^n dx. - Francesco Daddi, Aug 01 2011
a(n) = A037205(n-1)/(n-1) = A060073(n)*(n-1) = A023037(n) - A000169(n).
a(n) = [x^n] x^2/((1 - x)*(1 - n*x)). - Ilya Gutkovskiy, Oct 04 2017
a(n) = 1 + A228275(n, n-2) for n >= 2. - Mathew Englander, Dec 14 2020

Name edited by Michel Marcus, Dec 14 2020

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

A027445 a(n) = n^4 + n^3 + n^2 + n^1.

Original entry on oeis.org

0, 4, 30, 120, 340, 780, 1554, 2800, 4680, 7380, 11110, 16104, 22620, 30940, 41370, 54240, 69904, 88740, 111150, 137560, 168420, 204204, 245410, 292560, 346200, 406900, 475254, 551880, 637420, 732540, 837930, 954304, 1082400, 1222980, 1376830, 1544760, 1727604

Offset: 0

Patrick De Geest

a(A047203(n)) mod 10 = 0; a(A016861(n)) mod 10 = 4. - Reinhard Zumkeller, Oct 23 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Equals 2 * A071237(n).

Column k=4 of A228275.

List([0..50],n->n^4+n^3+n^2+n); # Muniru A Asiru, Jul 15 2018

[n^4 + n^3 + n^2 + n^1: n in [0..50]]; // Vincenzo Librandi, Jun 09 2011

seq(n^4+n^3+n^2+n,n=0..50); # Muniru A Asiru, Jul 15 2018

lst={};Do[AppendTo[lst, n^4+n^3+n^2+n], {n, 0, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 20 2008 *)
Table[Total[n^Range[4]],{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,4,30,120,340},40] (* Harvey P. Dale, Jul 01 2017 *)

a(n)=n^4+n^3+n^2+n^1 \\ Charles R Greathouse IV, Oct 21 2022

A031972 a(n) = Sum_{k=1..n} n^k.

Original entry on oeis.org

0, 1, 6, 39, 340, 3905, 55986, 960799, 19173960, 435848049, 11111111110, 313842837671, 9726655034460, 328114698808273, 11966776581370170, 469172025408063615, 19676527011956855056, 878942778254232811937, 41660902667961039785742, 2088331858752553232964199

Offset: 0

N. J. A. Sloane, Dec 11 1999

Sum of lengths of longest ending contiguous subsequences with the same value over all s in {1,...,n}^n: a(n) = Sum_{k=1..n} k*A228273(n,k). a(2) = 6 = 2+1+1+2: [1,1], [1,2], [2,1], [2,2]. - Alois P. Heinz, Aug 19 2013
a(n) is the expected wait time to see the contiguous subword 11...1 (n copies of 1) over all infinite sequences on alphabet {1,2,...,n}. - Geoffrey Critzer, May 19 2014
a(n) is the number of sequences of k elements from {1,2,...,n}, where 1<=k<=n. For example, a(2) = 6, counting the sequences, [1], [2], [1,1], [1,2], [2,1], [2,2]. Equivalently, a(n) is the number of bar graphs having a height and width of at most n. - Emeric Deutsch, Jan 24 2017.
In base n, a(n) has n+1 digits: n 1's followed by a 0. - Mathew Englander, Oct 20 2020

Alois P. Heinz, Table of n, a(n) for n = 0..386
A. Blecher, C. Brennan, A. Knopfmacher and H. Prodinger, The height and width of bargraphs, Discrete Applied Math. 180, (2015), 36-44.

Main diagonal of A228275.

Cf. A031973, A228273, A023037, A226238.

a031972 n = sum $ take n $ iterate (* n) n
-- Reinhard Zumkeller, Nov 22 2014

[1] cat [(n^(n+1)-n)/(n-1): n in [2..20]]; // Vincenzo Librandi, Apr 16 2015

a:= n-> `if`(n<2, n, (n^(n+1)-n)/(n-1)):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 15 2013

f[n_]:=Sum[n^k,{k,n}];Array[f,30] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011*)

a(1)=1; for n!=1 a(n) = (n^(n+1)-1)/(n-1) - 1. - Benoit Cloitre, Aug 17 2002
a(n) = A031973(n)-1 for n>0. - Robert G. Wilson v, Apr 15 2015
a(n) = n*A023037(n) = n^n - 1 + A023037(n). - Mathew Englander, Oct 20 2020

a(0)=0 prepended by Alois P. Heinz, Oct 22 2019

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)

A226238 a(n) = (n^n - n)/(n - 1).

Original entry on oeis.org

2, 12, 84, 780, 9330, 137256, 2396744, 48427560, 1111111110, 28531167060, 810554586204, 25239592216020, 854769755812154, 31278135027204240, 1229782938247303440, 51702516367896047760, 2314494592664502210318, 109912203092239643840220

Offset: 2

Ralf Stephan, Aug 25 2013

a(n) expressed in base n is written with (n-1) ones followed by a zero. - Michel Marcus, Aug 25 2013

Michael De Vlieger, Table of n, a(n) for n = 2..387
Tanya Khovanova and Gregory Marton, Archive Labeling Sequences, arXiv:2305.10357 [math.HO], 2023. See p. 9.

A diagonal of A228275.

Array[(#^# - #)/(# - 1) &, 18, 2] (* Michael De Vlieger, May 24 2023 *)

PARI
```
a(n)=(n^n-n)/(n-1)
```

def A226238(n): return (n**n-n)//(n-1) # Chai Wah Wu, Sep 28 2023

a(n) = Sum_{k=1..n-1} n^k.

a(n) = A023037(n) - 1, for n>1. - Michel Marcus, Aug 25 2013

A152031 a(n) = n^5 + n^4 + n^3 + n^2 + n.

Original entry on oeis.org

0, 5, 62, 363, 1364, 3905, 9330, 19607, 37448, 66429, 111110, 177155, 271452, 402233, 579194, 813615, 1118480, 1508597, 2000718, 2613659, 3368420, 4288305, 5399042, 6728903, 8308824, 10172525, 12356630, 14900787, 17847788, 21243689, 25137930, 29583455

Offset: 0

Vladimir Joseph Stephan Orlovsky, Nov 20 2008

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6, -15, 20, -15, 6, -1).

Column k=5 of A228275.

Cf. A053699.

a:= n-> `if`(n=1, 5, (n^6-n)/(n-1)):
seq(a(n), n=0..35);  # Alois P. Heinz, Aug 20 2013

lst={};Do[AppendTo[lst,n^5+n^4+n^3+n^2+n],{n,0,5!}];lst
(* Other programs: *)
Table[Total[n^Range@ 5], {n, 0, 31}] (* or *)
CoefficientList[Series[x (5 + 32 x + 66 x^2 + 16 x^3 + x^4)/(x - 1)^6, {x, 0, 31}], x] (* Michael De Vlieger, Apr 05 2017 *)

a(n) = n^5 + n^4 + n^3 + n^2 + n; \\ Joerg Arndt, Sep 03 2013

def a(n): return n**5 + n**4 + n**2 + n # Indranil Ghosh, Apr 05 2017

a <- c(0, 5, 62, 363, 1364, 3905)
for(n in (length(a)+1):40) a[n] <- 6*a[n-1] -15*a[n-2] +20*a[n-3] -15*a[n-4] +6*a[n-5] -a[n-6]
a
[Yosu Yurramendi, Sep 03 2013]

a(n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6) n>5, a(0)=0, a(1)=5, a(2)=62, a(3)=363, a(4)=1364, a(5)=3905. [Yosu Yurramendi, Sep 03 2013]
From Indranil Ghosh, Apr 05 2017: (Start)
G.f.: x*(5 + 32x + 66x^2 + 16x^3 + x^4)/(x - 1)^6.
E.g.f.: exp(x)*x*(5 + 26x + 32x^2 + 11x^3 + x^4).
(End)
a(n) = n*A053699(n). - Michel Marcus, Apr 05 2017

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

cp's OEIS Frontend

A027444 a(n) = n^3 + n^2 + n.

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A080674 a(n) = (4/3)*(4^n - 1).

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A060072 a(n) = (n^(n-1) - 1)/(n-1) for n>1, a(1) = 0.

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

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A027445 a(n) = n^4 + n^3 + n^2 + n^1.

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A031972 a(n) = Sum_{k=1..n} n^k.

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