A008592 -id:A008592 - cp's OEIS frontend

A008602 Multiples of 20.

0, 20, 40, 60, 80, 100, 120, 140, 160, 180, 200, 220, 240, 260, 280, 300, 320, 340, 360, 380, 400, 420, 440, 460, 480, 500, 520, 540, 560, 580, 600, 620, 640, 660, 680, 700, 720, 740, 760, 780, 800, 820, 840, 860, 880, 900, 920, 940, 960, 980, 1000

Offset: 0

N. J. A. Sloane

The multiples of 20 are exactly those integers which do not have a multiple whose decimal digits are of alternating parity. (International Mathematical Olympiad 2004, problem 6, see A110303) - Joseph Myers, Jul 13 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 332.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A005843, A008592, A008600, A008601, A110303 (complement), A215145, A317095.

Range[0, 1500, 20] (* Vladimir Joseph Stephan Orlovsky, Jun 01 2011 *)
CoefficientList[Series[20 x / (x - 1)^2, {x, 0, 60}], x] (* Vincenzo Librandi, Jun 10 2013 *)

a(n)=20*n \\ Charles R Greathouse IV, Sep 28 2015

(0 to 1000 by 20).toList // Alonso del Arte, Feb 20 2020

G.f.: 20*x/(x - 1)^2. - Vincenzo Librandi, Jun 10 2013
E.g.f.: 20*x*exp(x). - Stefano Spezia, Feb 20 2020
From Elmo R. Oliveira, Apr 10 2025: (Start)
a(n) = 20*n = 2*A008592(n) = 10*A005843(n) = A317095(n)/2.
a(n) = 2*a(n-1) - a(n-2). (End)

A017353 a(n) = 10*n + 7.

Original entry on oeis.org

7, 17, 27, 37, 47, 57, 67, 77, 87, 97, 107, 117, 127, 137, 147, 157, 167, 177, 187, 197, 207, 217, 227, 237, 247, 257, 267, 277, 287, 297, 307, 317, 327, 337, 347, 357, 367, 377, 387, 397, 407, 417, 427, 437, 447, 457, 467, 477, 487, 497, 507, 517, 527, 537

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008592, A016873, A017341.

[10*n+7: n in [0..60]]; // Vincenzo Librandi, May 28 2011

A017353:=n->10*n+7: seq(A017353(n), n=0..70); # Wesley Ivan Hurt, Mar 26 2015

Range[7, 497, 10] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)

vector(100,n,10*n-3) \\ Derek Orr, Mar 29 2015

From Vincenzo Librandi, May 28 2011: (Start)
a(n) = 10*n + 7.
a(n) = 2*a(n-1) - a(n-2). (End)
G.f.: (7+3*x)/(x-1)^2. - Wesley Ivan Hurt, Mar 26 2015
From Elmo R. Oliveira, Apr 05 2025: (Start)
E.g.f.: exp(x)*(7 + 10*x).
a(n) = A016873(2*n+1). (End)

A034048 Numbers with multiplicative digital root value 0.

Original entry on oeis.org

0, 10, 20, 25, 30, 40, 45, 50, 52, 54, 55, 56, 58, 59, 60, 65, 69, 70, 78, 80, 85, 87, 90, 95, 96, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 120, 125, 130, 140, 145, 150, 152, 154, 155, 156, 158, 159, 160, 165, 169, 170, 178, 180, 185, 187, 190, 195

Offset: 1

Patrick De Geest, Sep 15 1998

Numbers with root value 1 are the 'repunits' (see A000042).
This sequence has density 1. - Franklin T. Adams-Watters, Apr 01 2009
Any integer with at least one 0 among its base 10 digits is in this sequence. - Alonso del Arte, Aug 29 2014
This sequence is 10-automatic: it contains numbers with a 0, or with a 5 and any even digit. - Charles R Greathouse IV, Feb 13 2017

			20 is in the sequence because 2 * 0 = 0.
25 is in the sequence because 2 * 5 = 10 and 1 * 0 = 0.

Franklin T. Adams-Watters, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Multiplicative Digital Root
Index entries for 10-automatic sequences.

Cf. A031347, A034048, A002275, A034049, A034050, A034051, A034052, A034053, A034054, A034055, A034056 (numbers having multiplicative digital roots 0-9).

Cf. the subsets A011540 and A008592.

mdr0Q[n_]:=NestWhile[Times@@IntegerDigits[#]&,n,#>9&]==0; Select[Range[ 0,200],mdr0Q] (* Harvey P. Dale, Jul 21 2020 *)

is(n)=factorback(digits(n))==0 \\ Charles R Greathouse IV, Feb 13 2017

A195817 Multiples of 10 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 10, 3, 20, 5, 30, 7, 40, 9, 50, 11, 60, 13, 70, 15, 80, 17, 90, 19, 100, 21, 110, 23, 120, 25, 130, 27, 140, 29, 150, 31, 160, 33, 170, 35, 180, 37, 190, 39, 200, 41, 210, 43, 220, 45, 230, 47, 240, 49, 250, 51, 260, 53, 270, 55, 280, 57, 290, 59, 300

Offset: 0

Omar E. Pol, Sep 29 2011

A008592 and A005408 interleaved.
Partial sums give the generalized 14-gonal (or tetradecagonal) numbers A195818.
a(n) is also the length of the n-th line segment of a rectangular spiral on the infinite square grid. The vertices of the spiral are the generalized 14-gonal numbers. - Omar E. Pol, Jul 27 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Column 10 of A195151.
Sequences whose partial sums give the generalized n-gonal numbers, if n>=5: A026741, A001477, zero together with A080512, A022998, A195140, zero together with A165998, A195159, A195161, A195312, this sequence.
Cf. A091998, A195152.

[(2*(-1)^n+3)*n: n in [0..60]]; // Vincenzo Librandi, Sep 30 2011

With[{nn=30},Riffle[10*Range[0,nn],Range[1,2*nn+1,2]]] (* or *) LinearRecurrence[{0,2,0,-1},{0,1,10,3},70] (* Harvey P. Dale, Nov 24 2013 *)

a(n) = (2*(-1)^n+3)*n; \\ Andrew Howroyd, Jul 23 2018

a(n) = (2*(-1)^n+3)*n. - Vincenzo Librandi, Sep 30 2011
From Bruno Berselli, Sep 30 2011: (Start)
G.f.: x*(1+10*x+x^2)/((1-x)^2*(1+x)^2).
a(n) = -a(-n) = a(n-2)*n/(n-2) = 2*a(n-2)-a(n-4).
a(n) * a(n+1) = a(n(n+1)).
a(n) + a(n+1) = A091998(n+1). (End)
a(0)=0, a(1)=1, a(2)=10, a(3)=3, a(n)=2*a(n-2)-a(n-4). - Harvey P. Dale, Nov 24 2013
Multiplicative with a(2^e) = 5*2^e, a(p^e) = p^e for odd prime p. - Andrew Howroyd, Jul 23 2018
Dirichlet g.f.: zeta(s-1) * (1 + 2^(3-s)). - Amiram Eldar, Oct 25 2023

A168184 Characteristic function of numbers that are not multiples of 10.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1

Offset: 0

Reinhard Zumkeller, Nov 30 2009

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index entries for characteristic functions
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,1).

Cf. A168185, A145568, A168182, A168181, A109720, A097325, A011558, A166486, A011655, A000035, A010690, A033442.

a168184 = (1 -) . (0 ^) . (`mod` 10)
a168184_list = cycle [0,1,1,1,1,1,1,1,1,1]
-- Reinhard Zumkeller, Oct 10 2012

Table[If[Mod[n,10]==0,0,1],{n,0,110}] (* or *) PadRight[{},110,{0,1,1,1,1,1,1,1,1,1}] (* Harvey P. Dale, Jun 03 2023 *)

a(n)=n%10>0 \\ Charles R Greathouse IV, Sep 24 2015

a(n+10) = a(n);
a(n) = A000007(A010879(n));
a(A067251(n)) = 1; a(A008592(n)) = 0;
not the same as A168046: a(n)=A168046 for n<=100;
A033442(n) = Sum_{k=0..n} a(k)*(n-k).
Dirichlet g.f.: (1-1/10^s)*zeta(s). - R. J. Mathar, Feb 19 2011
For the general case: the characteristic function of numbers that are not multiples of m is a(n)=floor((n-1)/m)-floor(n/m)+1, m,n > 0. - Boris Putievskiy, May 08 2013

A017317 a(n) = 10*n + 4.

Original entry on oeis.org

4, 14, 24, 34, 44, 54, 64, 74, 84, 94, 104, 114, 124, 134, 144, 154, 164, 174, 184, 194, 204, 214, 224, 234, 244, 254, 264, 274, 284, 294, 304, 314, 324, 334, 344, 354, 364, 374, 384, 394, 404, 414, 424, 434, 444, 454, 464, 474, 484, 494, 504, 514, 524, 534

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0(59).

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences.
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N)).
William A. Stein, The modular forms database.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008592, A016873, A016897, A017305.

[10*n+4: n in [0..60]]; // Vincenzo Librandi, May 29 2011

Range[4, 1000, 10] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)

a(n)=10*n+4 \\ Charles R Greathouse IV, Jul 10 2016

a(n) = 10*n + 4; a(n) = 2*a(n-1) - a(n-2). - Vincenzo Librandi, May 29 2011
G.f.: 2*(2+3*x)/(x-1)^2. - R. J. Mathar, Mar 20 2018
From Elmo R. Oliveira, Apr 05 2025: (Start)
E.g.f.: 2*exp(x)*(2 + 5*x).
a(n) = 2*A016873(n) = A016897(2*n). (End)

A179051 Number of partitions of n into powers of 10 (cf. A011557).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 0

Reinhard Zumkeller, Jun 27 2010

nonn

A179052 and A008592 give record values and where they occur.

			a(19) = #{10 + 9x1, 19x1} = 2;
a(20) = #{10 + 10, 10 + 10x1, 20x1} = 3;
a(21) = #{10 + 10 + 1, 10 + 11x1, 21x1} = 3.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for related partition-counting sequences

Number of partitions of n into powers of b: A018819 (b=2), A062051 (b=3).
Cf. A206245, A000041, A179051.
Cf. A133880, A132272.
Cf. A008592, A179052.

a179051 = p 1 where
   p _ 0 = 1
   p k m = if m < k then 0 else p k (m - k) + p (k * 10) m
-- Reinhard Zumkeller, Feb 05 2012

terms = 10001;
CoefficientList[Product[1/(1 - x^(10^k)) + O[x]^terms,
     {k, 0, Log[10, terms] // Ceiling}], x]
(* Jean-François Alcover, Dec 12 2021, after Ilya Gutkovskiy *)

a(n) = A133880(n) for n < 90; a(n) = A132272(n) for n < 100.
a(10^n) = A145513(n).
a(10*n) = A179052(n).
A179052(n) = a(A008592(n));
a(n) = p(n,1) where p(n,k) = if k<=n then p(10*[(n-k)/10],k)+p(n,10*k) else 0^n.
G.f.: Product_{k>=0} 1/(1 - x^(10^k)). - Ilya Gutkovskiy, Jul 26 2017

A344330 Sides s of squares that can be tiled with squares of two different sizes so that the number of large or small squares is the same.

Original entry on oeis.org

10, 15, 20, 30, 40, 45, 50, 60, 65, 68, 70, 75, 78, 80, 90, 100, 105, 110, 120, 130, 135, 136, 140, 150, 156, 160, 165, 170, 175, 180, 190, 195, 200, 204, 210, 220, 222, 225, 230, 234, 240, 250, 255, 260, 270, 272, 280, 285, 290, 300, 310, 312, 315, 320, 325, 330, 340, 345, 350, 360, 369, 370

Offset: 1

Bernard Schott, May 15 2021

nonn

This sequence is a generalization of the 4th problem proposed for the pupils in grade 6 during the 19th Mathematical Festival at Moscow in 2008.
Some notations: s = side of the tiled square, a = side of small squares, b = side of large squares, and z = number of small squares = number of large squares.
Side s of such tiled squares must satisfy the Diophantine equation s^2 = z * (a^2+b^2).
There are two types of solutions. See A344331 for type 1 and A344332 for type 2.
If q is a term, k * q is another term for k > 1.

			-> Example of type 1:
Square 10 x 10 with a = 1, b = 2, s = 10, z = 20.
      ___ ___ _ ___ ___ _
     |   |   |_|   |   |_|
     |___|___|_|___|___|_|
     |   |   |_|   |   |_| with 10 elementary 2 x 5 rectangles
     |___|___|_|___|___|_|
     |   |   |_|   |   |_|              ___ ___ _
     |___|___|_|___|___|_|             |   |   |_|
     |   |   |_|   |   |_|             |___|___|_|
     |___|___|_|___|___|_|
     |   |   |_|   |   |_|
     |___|___|_|___|___|_|
.
-> Example of type 2:
Square 15 x 15 with a = 3, b = 4, s = 15, z = 9.
      ________ ________ ________ _____
     |        |        |        |     |
     |        |        |        |     |
     |        |        |        |_____|
     |_______ |________|________|     |
     |        |        |        |     |
     |        |        |        |_____|
     |        |        |        |     |
     |________|________|________|     |
     |        |        |        |_____|
     |        |        |        |     |
     |        |        |        |     |
     |_____ __|___ ____|_ ______|_____|
     |     |      |      |      |     |
     |     |      |      |      |     |
     |_____|______|______|______|_____|
Remarks:
- With terms as 10, 20, ... we only obtain sides of squares of type 1:
10 is a term of this type because the square 10 X 10 only can be tiled with 20 squares of size 1 X 1 and 20 squares of size 2 X 2 (see first example),
20 is another term of this type because the square 20 X 20 only can be tiled with 80 squares of size 1 x 1 and 80 squares of size 2 x 2.
- With terms as 15, 65, ... we only obtain sides of squares of type 2:
15 is a term of this type because the square 15 X 15 only can be tiled with 9 squares of size 3 X 3 and 9 squares of size 4 X 4 (see second example),
65 is another term of this type because the square 65 X 65 only can be tiled with 25 squares of size 5 X 5 and 25 squares of size 12 X 12.
- With terms as 30, 60, ... we obtain both sides of squares of type 1 and of type 2:
30 is a term of type 1 because the square 30 X 30 can be tiled with 180 squares of size 1 X 1 and 180 squares of size 2 X 2, but,
30 is also a term of type 2 because the square 30 X 30 can be tiled with 9 squares of size 6 X 6 and 9 squares of size 8 X 8.

Ivan Yashchenko, Invitation to a Mathematical Festival, pp. 10 and 102, MSRI, Mathematical Circles Library, 2013.

Index to sequences related to Olympiads.

Subsequences: A008592 \ {0}, A008597 \ {0}, A034262 \ {0,1}.

Cf. A344331, A344332, A344333, A344334.

pts(lim) = my(v=List(), m2, s2, h2, h); for(middle=4, lim-1, m2=middle^2; for(small=1, middle, s2=small^2; if(issquare(h2=m2+s2, &h), if(h>lim, break); listput(v, [small, middle, h])))); vecsort(Vec(v)); \\ A009000
isdp4(s) = my(k=1, x); while(((x=k^4 - (k-1)^4) <= s), if (x == s, return (1)); k++); return(0);
isokp2(s) = {if (!isdp4(s), return(0)); if (s % 2, my(vp = pts(s)); for (i=1, #vp, my(vpi = vp[i], a = vpi[1], b = vpi[2], c = vpi[3]); if (a*c/(c-b) == s, return(1)); ); ); }
isok2(s) = {if (isokp2(s), return (1)); fordiv(s, d, if ((d>1) || (dx*y*(x^2+y^2), [1..m]), s);}
isok1(s) = {if (isokp1(s), return (1)); fordiv(s, d, if ((d>1) || (dMichel Marcus, Jun 04 2021

Corrected by Michel Marcus, May 18 2021

Incorrect term 145 removed by Michel Marcus, Jun 04 2021

A094053 Triangle read by rows: T(n,k) = k*(n-k), 1 <= k <= n.

Original entry on oeis.org

0, 1, 0, 2, 2, 0, 3, 4, 3, 0, 4, 6, 6, 4, 0, 5, 8, 9, 8, 5, 0, 6, 10, 12, 12, 10, 6, 0, 7, 12, 15, 16, 15, 12, 7, 0, 8, 14, 18, 20, 20, 18, 14, 8, 0, 9, 16, 21, 24, 25, 24, 21, 16, 9, 0, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 0, 11, 20, 27, 32, 35, 36, 35, 32, 27, 20, 11, 0, 12

Offset: 1

Reinhard Zumkeller, May 31 2004

T(n,k) = A003991(n-1,k) for 1 <= k < n;
T(n,k) = T(n,n-1-k) for k < n;
T(n,1) = n-1; T(n,n) = 0;          T(n,2) = A005843(n-2) for n > 1;
T(n,3) = A008585(n-3) for n>2;     T(n,4) = A008586(n-4) for n > 3;
T(n,5) = A008587(n-5) for n>4;     T(n,6) = A008588(n-6) for n > 5;
T(n,7) = A008589(n-7) for n>6;     T(n,8) = A008590(n-8) for n > 7;
T(n,9) = A008591(n-9) for n>8;     T(n,10) = A008592(n-10) for n > 9;
T(n,11) = A008593(n-11) for n>10;  T(n,12) = A008594(n-12) for n > 11;
T(n,13) = A008595(n-13) for n>12;  T(n,14) = A008596(n-14) for n > 13;
T(n,15) = A008597(n-15) for n>14;  T(n,16) = A008598(n-16) for n > 15;
T(n,17) = A008599(n-17) for n>16;  T(n,18) = A008600(n-18) for n > 17;
T(n,19) = A008601(n-19) for n>18;  T(n,20) = A008602(n-20) for n > 19;
Row sums give A000292; triangle sums give A000332;
All numbers m > 0 occur A000005(m) times;
A002378(n) = T(A005408(n),n+1) = n*(n+1).
k-th columns are arithmetic progressions with step k, starting with 0. If a zero is prefixed to the sequence, then we get a new table where the columns are again arithmetic progressions with step k, but starting with k, k=0,1,2,...: 1st column = (0,0,0,...), 2nd column = (1,2,3,...), 3rd column = (2,4,6,8,...), etc. - M. F. Hasler, Feb 02 2013
Construct the infinite-dimensional matrix representation of angular momentum operators (J_1,J_2,J_3) in the Jordan-Schwinger form (cf. Harter, Klee, Schwinger). The triangle terms T(n,k) = T(2j,j+m) satisfy: (1/2)T(2j,j+m)^(1/2) =  =  = i  = -i . Matrices for J_1 and J_2 are sparse. These equalities determine the only nonzero entries. - Bradley Klee, Jan 29 2016
T(n+1,k+1) is the number of degrees of freedom of a k-dimensional affine subspace within an n-dimensional vector space. This is most readily interpreted geometrically: e.g. in 3 dimensions a line (1-dimensional subspace) has T(4,2) = 4 degrees of freedom and a plane has T(4,3) = 3. T(n+1,1) = n indicates that points in n dimensions have n degrees of freedom. T(n,n) = 0 for any n as all n-dimensional spaces in an n-dimensional space are equivalent. - Daniel Leary, Apr 29 2020

			From _M. F. Hasler_, Feb 02 2013: (Start)
Triangle begins:
  0;
  1, 0;
  2, 2, 0;
  3, 4, 3, 0;
  4, 6, 6, 4, 0;
  5, 8, 9, 8, 5, 0;
  (...)
If an additional 0 was added at the beginning, this would become:
  0;
  0, 1;
  0, 2, 2;
  0, 3, 4; 3;
  0, 4, 6, 6, 4;
  0, 5, 8, 9, 8, 5;
  ... (End)

W. Harter, Principles of Symmetry, Dynamics, Spectroscopy, Wiley, 1993, Ch. 5, page 345-346.
B. Klee, Quantum Angular Momentum Matrices, Wolfram Demonstrations Project, 2016.
J. Schwinger, On Angular Momentum , Cambridge: Harvard University, Nuclear Development Associates, Inc., 1952.

J_3: A114327; J_1^2, J_2^2: A141387, A268759.
Cf. A000005, A002378, A003991, A005408.
Cf. A000292 (row sums), A000332 (triangle sums).
T(n,k) for values of k:
  A005843 (k=2), A008585 (k=3), A008586 (k=4), A008587 (k=5), A008588 (k=6), A008589 (k=7), A008590 (k=8), A008591 (k=9), A008592 (k=10), A008593 (k=11), A008594 (k=12), A008595 (k=13), A008596 (k=14), A008597 (k=15), A008598 (k=16), A008599 (k=17), A008600 (k=18), A008601 (k=19), A008602 (k=20).

/* As triangle */ [[k*(n-k): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Jan 30 2016

Flatten[Table[(j - m) (j + m + 1), {j, 0, 10, 1/2}, {m, -j, j}]] (* Bradley Klee, Jan 29 2016 *)

{for(n=1, 13, for(k=1, n, print1(k*(n - k)," ");); print(););} \\ Indranil Ghosh, Mar 12 2017

A010690 Period 2: repeat (1,9).

Original entry on oeis.org

1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1

Offset: 0

N. J. A. Sloane

Digital roots of the nonzero square triangular numbers. - Ant King, Jan 21 2012
Continued fraction expansion of A176019. - R. J. Mathar, Mar 08 2012
Exp( Sum_{n >= 1} a(n-1)*x^n/n ) = 1 + x + 5*x^2 + 5*x^3 + 15*x^4 + 15*x^5 + ... is the o.g.f. for A189976 (taken with an offset of 0). - Peter Bala, Mar 13 2015
Final digit of 9^n. - Martin Renner, Jun 11 2020
Decimal expansion of 19/99. - Stefano Spezia, Feb 09 2025

			0.191919191919191919191919191919191919191...

Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A008592, A001019 (9^n), A014393, A189976.

5+4*(-1)^# &/@Range[81] (* Ant King, Jan 21 2012 *)

a(n)=1; if(n%2==1, 9, 1) \\ Felix Fröhlich, Aug 11 2014

G.f.: (1+9x)/((1-x)(1+x)). - R. J. Mathar, Nov 21 2011
a(n) = 9^n mod 10. - Martin Renner, Jun 11 2020
E.g.f.: cosh(x) + 9*sinh(x). - Stefano Spezia, Feb 09 2025
From Amiram Eldar, Jun 09 2025: (Start)
With offset 1:
Multiplicative with a(2^e) = 9, a(p^e) = 1 for an odd prime p.
Dirichlet g.f.: zeta(s) * (1 + 1/2^(s-3)). (End)

cp's OEIS Frontend

A008602 Multiples of 20.

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Mathematica

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A017353 a(n) = 10*n + 7.

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Maple

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A034048 Numbers with multiplicative digital root value 0.

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A195817 Multiples of 10 and odd numbers interleaved.

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Magma

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A168184 Characteristic function of numbers that are not multiples of 10.

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Haskell

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A017317 a(n) = 10*n + 4.

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Magma

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A179051 Number of partitions of n into powers of 10 (cf. A011557).

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