A008592 -id:A008592 - cp's OEIS frontend

A127544 Partition numbers (A000041) which are multiples of 10 (A008592).

30, 490, 3010, 12310, 715220, 831820, 1741630, 2323520, 7089500, 13848650, 26543660, 92669720, 133230930, 271248950, 541946240, 1844349560, 2841940500, 4351078600, 4835271870, 5371315400, 10015581680, 18440293320, 37027355200

Offset: 1

Zak Seidov, Apr 01 2007

nonn

Intersection of A000041 and A008592.

Partition numbers of the form 10k. - Omar E. Pol, May 08 2013

Cf. A000041, A052001, A008592, A087183.

Select[Table[PartitionsP[n],{n,0,200}],Mod[ #,10]==0&]

a(n) = 10*A225317(n). - Omar E. Pol, May 08 2013

Corrected by Omar E. Pol, May 05 2013

A008593 Multiples of 11.

Original entry on oeis.org

0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132, 143, 154, 165, 176, 187, 198, 209, 220, 231, 242, 253, 264, 275, 286, 297, 308, 319, 330, 341, 352, 363, 374, 385, 396, 407, 418, 429, 440, 451, 462, 473, 484, 495, 506, 517, 528, 539, 550, 561, 572, 583

Offset: 0

N. J. A. Sloane

Numbers for which the sum of "digits" in base 100 is divisible by 11. For instance, 193517302 gives 1 + 93 + 51 + 73 + 02 = 220, and 2 + 20 = 22 = 2 * 11. - Daniel Forgues, Feb 22 2016

Numbers in which the sum of the digits in the even positions equals the sum of the digits in the odd positions. - Stefano Spezia, Jan 05 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 323.
Tanya Khovanova, Recursive Sequences.
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. A008591, A008592, A008604, A059632; union of A135499 and A060979.

a008593 = (* 11)
a008593_list = [0, 11 ..]  -- Reinhard Zumkeller, Jul 05 2014

[11*n: n in [0..60]]; // Vincenzo Librandi, Sep 18 2011

g:=(1+10*z)/((1-z)): gser:=series(g, z=0, 88): seq((coeff(gser, z, n))*n, n=0..77); # Zerinvary Lajos, Feb 25 2009

Range[0, 1000, 11] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)

makelist(11*n,n,0,20); /* Martin Ettl, Dec 17 2012 */

a(n)=11*n \\ Charles R Greathouse IV, Nov 06 2014

a(n) = 11*n.
G.f.: 11*x/(1-x)^2. - David Wilding, Jun 21 2014
E.g.f.: 11*x*exp(x). - Stefano Spezia, Oct 08 2022
From Elmo R. Oliveira, Apr 10 2025: (Start)
a(n) = 2*a(n-1) - a(n-2).
a(n) = A008604(n)/2. (End)

A008594 Multiples of 12.

Original entry on oeis.org

0, 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, 156, 168, 180, 192, 204, 216, 228, 240, 252, 264, 276, 288, 300, 312, 324, 336, 348, 360, 372, 384, 396, 408, 420, 432, 444, 456, 468, 480, 492, 504, 516, 528, 540, 552, 564, 576, 588, 600, 612, 624, 636

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 36 ).
The positive terms are the differences of consecutive star numbers (A003154). - Mihir Mathur, Jun 07 2013
A089911(a(n)) = 0. - Reinhard Zumkeller, Jul 05 2013
a(1) = 12 is a primitive abundant number, thus all a(n), n >= 2, are nonprimitive abundant numbers. - Daniel Forgues, Sep 24 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Tanya Khovanova, Recursive Sequences.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 324.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N)).
William A. Stein, The modular forms database.
Wikipedia, Star Number.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Subsequence of A072065 and A121032.

Cf. A003154, A008588, A008592, A008593, A008606, A089911.

a008594 = (* 12)
a008594_list = [0, 12 ..]  -- Reinhard Zumkeller, Dec 12 2012

[12*n: n in [0..50]]; // Vincenzo Librandi, Jun 11 2011

A008594:=n->12*n: seq(A008594(n), n=0..100); # Wesley Ivan Hurt, Sep 24 2016

12*Range[0,200] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2011 *)
NestList[12+#&,0,60] (* Harvey P. Dale, Feb 02 2022 *)

a(n)=12*n \\ Charles R Greathouse IV, Apr 21 2015

From Vincenzo Librandi, Jun 11 2011: (Start)
a(n) = 12*n.
a(n) = 2*a(n-1) - a(n-2) for n > 1.
G.f.: 12*x/(1-x)^2. (End)
a(n) = A003154(n) - A003154(n-1). - Mihir Mathur, Jun 07 2013
From Elmo R. Oliveira, Apr 10 2025: (Start)
E.g.f.: 12*x*exp(x).
a(n) = 2*A008588(n) = A008606(n)/2. (End)

A017377 a(n) = 10*n + 9.

Original entry on oeis.org

9, 19, 29, 39, 49, 59, 69, 79, 89, 99, 109, 119, 129, 139, 149, 159, 169, 179, 189, 199, 209, 219, 229, 239, 249, 259, 269, 279, 289, 299, 309, 319, 329, 339, 349, 359, 369, 379, 389, 399, 409, 419, 429, 439, 449, 459, 469, 479, 489, 499, 509, 519, 529, 539

Offset: 0

N. J. A. Sloane

Numbers k such that k^k ends with 9. - Bruno Berselli, Dec 11 2018

Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pp. 126-127.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 979.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008592, A016897, A017365, A190876.

[10*n+9: n in [0..60]]; // Vincenzo Librandi, May 29 2011

Range[9, 1000, 10] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)

a(n)=10*n+9 \\ Charles R Greathouse IV, Feb 12 2017

def A017377(n): return 10*n+9
print([A017377(n) for n in range(0,54)]) # Stefano Spezia, May 26 2025

a(n) = 10*n + 9; a(n) = 2*a(n-1) - a(n-2). - Vincenzo Librandi, May 29 2011
G.f.: (9+x)/(x-1)^2. - R. J. Mathar, Oct 16 2015
From Elmo R. Oliveira, Apr 05 2025: (Start)
E.g.f.: exp(x)*(9 + 10*x).
a(n) = A016897(2*n+1). (End)

A067251 Numbers with no trailing zeros in decimal representation.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 102, 103, 104

Offset: 1

Reinhard Zumkeller, Mar 10 2002

Or, decimated numbers: every 10th number has been omitted from the natural numbers. - Cino Hilliard, Feb 21 2005. For example, The 10th number starting with 1 is 10 and is missing from the table because it was decimated.
The word "decimated" can be interpreted in several ways and should be used with caution. - N. J. A. Sloane, Feb 21 2005
Not the same as A052382, as 101 is included.
Numbers in here but not in A043095 are 81, 91, 92, 93, 94,... for example. - R. J. Mathar, Sep 30 2008
The integers 100*a(n) are precisely the numbers whose square ends with exactly 4 identical digits while the integers 10*a(n) form just a subsequence of the numbers whose square ends with exactly 2 identical digits (A346678). - Bernard Schott, Oct 04 2021

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,1,-1).

Complement of A008592.
Cf. A076641 (reversed).
Cf. A004086, A043095, A052382, A168184.
Cf. A039685 (a subsequence), A346678, A346940, A346942.

a067251 n = a067251_list !! (n-1)
a067251_list = filter ((> 0) . flip mod 10) [0..]
-- Reinhard Zumkeller, Jul 11 2015, Dec 29 2011

S := seq(n + floor((n-1)/9), n=1..100); # Bernard Schott, Oct 04 2021

DeleteCases[Range[110],?(Divisible[#,10]&)] (* _Harvey P. Dale, May 16 2016 *)

f(n) = for(x=1,n,if(x%10,print1(x","))) \\ Cino Hilliard, Feb 21 2005

Vec(x*(x+1)*(x^4-x^3+x^2-x+1)*(x^4+x^3+x^2+x+1)/((x-1)^2*(x^2+x+1)*(x^6+x^3+1)) + O(x^100)) \\ Colin Barker, Sep 28 2015

def a(n): return n + (n-1)//9
print([a(n) for n in range(1, 95)]) # Michael S. Branicky, Oct 04 2021

a(n) = n + floor((n-1)/9).
a(n) mod 10 > 0 for all n.
A004086(A004086(a(n))) = a(n).
A168184(a(n)) = 1. - Reinhard Zumkeller, Nov 30 2009
From Colin Barker, Sep 28 2015: (Start)
a(n) = a(n-1) + a(n-9) - a(n-10) for n>10.
G.f.: x*(x+1)*(x^4-x^3+x^2-x+1)*(x^4+x^3+x^2+x+1) / ((x-1)^2*(x^2+x+1)*(x^6+x^3+1)). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (1/20 + 1/sqrt(5) - sqrt(1+2/sqrt(5))/5) * Pi. - Amiram Eldar, May 11 2025

Edited by N. J. A. Sloane, Sep 06 2008 at the suggestion of R. J. Mathar

Typos corrected in a comment line by Reinhard Zumkeller, Apr 04 2010

A121030 Multiples of 10 containing a 10 in their decimal representation.

Original entry on oeis.org

10, 100, 110, 210, 310, 410, 510, 610, 710, 810, 910, 1000, 1010, 1020, 1030, 1040, 1050, 1060, 1070, 1080, 1090, 1100, 1110, 1210, 1310, 1410, 1510, 1610, 1710, 1810, 1910, 2010, 2100, 2110, 2210, 2310, 2410, 2510, 2610, 2710, 2810, 2910, 3010, 3100

Offset: 1

Reinhard Zumkeller, Jul 21 2006

R. Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.

Cf. A121041, A008592, A011531, A121022, A121023, A121024, A121025, A121026, A121027, A121028, A121029, A121031, A121032, A121033, A121034, A121035, A121036, A121037, A121038, A121039, A121040.

Select[10*Range[1000], StringContainsQ[IntegerString[#], "10"] &] (* Paolo Xausa, Feb 25 2024 *)

a(n) ~ 10n. - Charles R Greathouse IV, Feb 12 2017

A017305 a(n) = 10*n + 3.

Original entry on oeis.org

3, 13, 23, 33, 43, 53, 63, 73, 83, 93, 103, 113, 123, 133, 143, 153, 163, 173, 183, 193, 203, 213, 223, 233, 243, 253, 263, 273, 283, 293, 303, 313, 323, 333, 343, 353, 363, 373, 383, 393, 403, 413, 423, 433, 443, 453, 463, 473, 483, 493, 503, 513, 523, 533

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0(38).

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, A016885, A017198, A017281, A017293, A156677.

[10*n+3: n in [0..60]]; // Vincenzo Librandi, May 28 2011

Range[3, 1000, 10] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)

a(n) = A017198(n) - A156677(n+2). - Reinhard Zumkeller, Jul 13 2010
a(n) = 2*a(n-1) - a(n-2). - Vincenzo Librandi, May 28 2011
G.f.: (3+7*x)/(x-1)^2. - R. J. Mathar, Apr 11 2016
E.g.f.: exp(x)*(3 + 10*x). - Stefano Spezia, Aug 22 2023
a(n) = A016885(2*n). - Elmo R. Oliveira, Apr 10 2025

A017341 a(n) = 10*n + 6.

Original entry on oeis.org

6, 16, 26, 36, 46, 56, 66, 76, 86, 96, 106, 116, 126, 136, 146, 156, 166, 176, 186, 196, 206, 216, 226, 236, 246, 256, 266, 276, 286, 296, 306, 316, 326, 336, 346, 356, 366, 376, 386, 396, 406, 416, 426, 436, 446, 456, 466, 476, 486, 496, 506, 516, 526, 536

Offset: 0

N. J. A. Sloane

Number of 4 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (10;0) and (11;0). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1Sergey Kitaev, Nov 11 2004

Numbers k such that k and (4^h)^k end with the same digit, where h > 0. - Bruno Berselli, Dec 13 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences.
S. Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.
S. Kitaev, On multi-avoidance of right angled numbered polyomino patterns, University of Kentucky Research Reports (2004).
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000400 (powers of 6), A008592, A016861, A016885, A017329.

[10*n+6: n in [0..60]]; // Vincenzo Librandi, May 29 2011

Range[6, 1000, 10] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)

a(n)=10*n+6 \\ Charles R Greathouse IV, Jul 10 2016

a(n) = 2*a(n-1) - a(n-2) with n>1, a(0)=6, a(1)=16. - Vincenzo Librandi, May 29 2011
a(n) = (n+1)*A016861(n+1) - n*A016861(n). - Bruno Berselli, Jan 18 2013
From Stefano Spezia, May 31 2021: (Start)
G.f.: 2*(3 + 2*x)/(1 - x)^2.
E.g.f.: 2*(3 + 5*x)*exp(x). (End)
a(n) = 2*A016885(n) = A016861(2*n+1). - Elmo R. Oliveira, Apr 10 2025

A017293 a(n) = 10*n + 2.

Original entry on oeis.org

2, 12, 22, 32, 42, 52, 62, 72, 82, 92, 102, 112, 122, 132, 142, 152, 162, 172, 182, 192, 202, 212, 222, 232, 242, 252, 262, 272, 282, 292, 302, 312, 322, 332, 342, 352, 362, 372, 382, 392, 402, 412, 422, 432, 442, 452, 462, 472, 482, 492, 502, 512, 522, 532

Offset: 0

N. J. A. Sloane, Dec 11 1996

Number of 5 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (11;0) and (01;1). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1A008574; m=3: A016933; m=4: A022144; m=6: A017569. - Sergey Kitaev, Nov 13 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences.
Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Subsequence of A034709, together with A017281, A139222, A139245, A017329, A139249, A139264, A139279 and A139280.

Cf. A008574, A008592, A016861, A016873, A016933, A017569, A022144.

[10*n+2: n in [0..50]]; // Vincenzo Librandi, May 04 2011

A017293:=n->10*n+2; seq(A017293(n), n=0..100); # Wesley Ivan Hurt, May 03 2014

Range[2, 1000, 10] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)
CoefficientList[Series[(2 + 8 x) / (1 - x)^2, {x, 0, 30}], x] (* Vincenzo Librandi, Jul 23 2016 *)
10 Range[0,60]+2 (* or *) LinearRecurrence[{2,-1},{2,12},60] (* Harvey P. Dale, Jul 04 2019 *)

a(n) = 2*A016861(n) = A008592(n) + 2. - Wesley Ivan Hurt, May 03 2014
G.f.: 2*(1 + 4*x)/(1-x)^2. - Vincenzo Librandi, Jul 23 2016
From Elmo R. Oliveira, Apr 04 2025: (Start)
E.g.f.: 2*exp(x)*(1 + 5*x).
a(n) = 2*a(n-1) - a(n-2) for n >= 2.
a(n) = A016873(2*n). (End)

A249674 a(n) = 30*n.

Original entry on oeis.org

0, 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330, 360, 390, 420, 450, 480, 510, 540, 570, 600, 630, 660, 690, 720, 750, 780, 810, 840, 870, 900, 930, 960, 990, 1020, 1050, 1080, 1110, 1140, 1170, 1200, 1230, 1260, 1290, 1320, 1350, 1380, 1410, 1440

Offset: 0

Kaylan Purisima, Nov 03 2014

Numbers divisible by 2, 3 and 5. - Robert Israel, Nov 19 2014

a(n) is the maximum score of a 10-pin n-frame bowling game and the maximum score of an n-pin 10-frame bowling game, given the rules: a strike is worth the number of pins in each frame plus the number of pins knocked down by the next two balls (except in the last frame), a spare is worth the number of pins in each frame plus the number of pins knocked down by the next ball (except in the last frame), and if a strike or spare is earned in the last frame then the player must continue to throw balls until they have thrown 3 balls in the last frame. - Iain Fox, Mar 02 2018

			a(7) = 7 * 30 = 210.

Iain Fox, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Bowling.
Wikipedia, Ten-pin bowling.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A005843, A008585, A008587, A008588, A008592, A008597, A050519, A169823.

[30*n : n in [0..50]]; // Wesley Ivan Hurt, Nov 18 2014

A249674:=n->30*n: seq(A249674(n), n=0..50); # Wesley Ivan Hurt, Nov 18 2014

30*Range[0, 59] (* Alonso del Arte, Nov 18 2014 *)

vector(100,n,30*(n-1)) \\ Derek Orr, Nov 18 2014

first(n) = Vec(30*x/(x-1)^2 + O(x^n), -n) \\ Iain Fox, Mar 02 2018

G.f.: 30*x/(x-1)^2; a(n) = 2*a(n-1) - a(n-2). - Wesley Ivan Hurt, Nov 18 2014
a(n) = 2*A008597(n) = 3*A008592(n) = 5*A008588(n) = 6*A008587(n) = 10*A008585(n) = 15*A005843(n). - Omar E. Pol, Nov 24 2014
From Elmo R. Oliveira, Apr 08 2025: (Start)
E.g.f.: 30*x*exp(x).
a(n) = A169823(n)/2. (End)

cp's OEIS Frontend

A127544 Partition numbers (A000041) which are multiples of 10 (A008592).

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A008593 Multiples of 11.

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A008594 Multiples of 12.

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A017377 a(n) = 10*n + 9.

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A067251 Numbers with no trailing zeros in decimal representation.

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A121030 Multiples of 10 containing a 10 in their decimal representation.

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A017305 a(n) = 10*n + 3.