A179052 -id:A179052 - cp's OEIS frontend

A008592 Multiples of 10: a(n) = 10*n.

0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200, 210, 220, 230, 240, 250, 260, 270, 280, 290, 300, 310, 320, 330, 340, 350, 360, 370, 380, 390, 400, 410, 420, 430, 440, 450, 460, 470, 480, 490, 500, 510, 520, 530

Offset: 0

N. J. A. Sloane

Number of 3 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01,1) 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 i11 and n>1. - Sergey Kitaev, Nov 12 2004
If Y is a 5-subset of an n-set X then, for n>=5, a(n-4) is the number of 3-subsets of X having at least two elements in common with Y. - Milan Janjic, Dec 08 2007
Complement of A067251; A168184(a(n)) = 0. - Reinhard Zumkeller, Nov 30 2009
Where record values occur for the number of partitions of n into powers of 10: A179052(n) = A179051(a(n)). - Reinhard Zumkeller, Jun 27 2010
Numbers ending in 0. - Wesley Ivan Hurt, Apr 10 2016

Ivan Panchenko, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 322.
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.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A005843, A008587, A008590, A008591, A067251, A168184, A179051, A179052.

a008592 = (10 *)  -- Reinhard Zumkeller, Jun 13 2015

[10*n : n in [0..100]]; // Wesley Ivan Hurt, Apr 10 2016

A008592:=n->10*n: seq(A008592(n), n=0..100); # Wesley Ivan Hurt, Apr 10 2016

Range[0, 1000, 10] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)

vector(50, n, n--; 10*n) \\ Michel Marcus, Feb 05 2016

x='x+O('x^999); concat(0, Vec(10*x/(x-1)^2)) \\ Altug Alkan, Apr 11 2016

apply( A008592(n)=10*n, [1..55]) \\ M. F. Hasler, Apr 23 2021

From Vincenzo Librandi, Dec 24 2010: (Start)
G.f.: 10*x/(x-1)^2.
a(n) = 2*a(n-1) - a(n-2) for n > 1. (End)
a(n) = Sum_{i=2n-2..2n+2} i. - Wesley Ivan Hurt, Apr 11 2016
E.g.f.: 10*x*exp(x). - Stefano Spezia, May 31 2021

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

A008728 Molien series for 3-dimensional group [2,n ] = *22n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 156, 162, 168, 174, 180, 186, 192, 198, 204, 210, 217, 224, 231, 238

Offset: 0

N. J. A. Sloane

a(n) = A179052(n) for n < 100. - Reinhard Zumkeller, Jun 27 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 193
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1).

Cf. A001840, A001972, A008724, A008725, A008726, A008727, A008732.

a:=[1,2,3,4,5,6,7,8,9,10,12,14];; for n in [13..70] do a[n]:=2*a[n-1]-a[n-2]+a[n-10]-2*a[n-11]+a[n-12]; od; a; # G. C. Greubel, Jul 30 2019

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/((1-x)^2*(1-x^10)) )); // G. C. Greubel, Jul 30 2019

g:= 1/((1-x)^2*(1-x^10)); gser:= series(g, x=0,72); seq(coeff(gser, x, n), n=0..70); # modified by G. C. Greubel, Jul 30 2019

CoefficientList[Series[1/((1-x)^2(1-x^10)), {x,0,70}], x] (* Vincenzo Librandi, Jun 11 2013 *)

my(x='x+O('x^70)); Vec(1/((1-x)^2*(1-x^10))) \\ G. C. Greubel, Jul 30 2019

(1/((1-x)^2*(1-x^10))).series(x, 70).coefficients(x, sparse=False) # G. C. Greubel, Jul 30 2019

G.f.: 1/((1-x)^2*(1-x^10)).
From Mitch Harris, Sep 08 2008: (Start)
a(n) = Sum_{j=0..n+10} floor(j/10).
a(n-10) = (1/2)*floor(n/10)*(2*n - 8 - 10*floor(n/10)). (End)

More terms from Vladimir Joseph Stephan Orlovsky, Mar 14 2010

A145513 Number of partitions of 10^n into powers of 10.

Original entry on oeis.org

1, 2, 12, 562, 195812, 515009562, 10837901390812, 1899421190329234562, 2851206628197445401265812, 37421114946843687272702534859562, 4362395890943439751990308572939648140812, 4573514084633441973328831327010967245403925484562, 43557001521047571730475817291330175020887917015964570015812

Offset: 0

Alois P. Heinz, Oct 11 2008

nonn

a(n) = A179051(10^n); for n>0: a(n) = A179052(10^(n-1)). - Reinhard Zumkeller, Jun 27 2010

			a(1) = 2, because there are 2 partitions of 10^1 into powers of 10: [1,1,1,1,1,1,1,1,1,1], [10].

Alois P. Heinz, Table of n, a(n) for n = 0..45

Cf. 10th column of A145515, A007318.

Cf. A011557, A002577, A078125.

import Data.MemoCombinators (memo2, list, integral)
a145513 n = a145513_list !! n
a145513_list = f [1] where
   f xs = (p' xs $ last xs) : f (1 : map (* 10) xs)
   p' = memo2 (list integral) integral p
   p  0 = 1; p []  = 0
   p ks'@(k:ks) m = if m < k then 0 else p' ks' (m - k) + p' ks m
-- Reinhard Zumkeller, Nov 27 2015

g:= proc(b,n,k) option remember; local t; if b<0 then 0 elif b=0 or n=0 or k<=1 then 1 elif b>=n then add(g(b-t, n, k) *binomial(n+1, t) *(-1)^(t+1), t=1..n+1); else g(b-1, n, k) +g(b*k, n-1, k) fi end: a:= n-> g(1,n,10): seq(a(n), n=0..13);

g[b_, n_, k_] := g[b, n, k] = Module[{t}, Which[b < 0, 0, b == 0 || n == 0 || k <= 1, 1, b >= n, Sum[g[b - t, n, k]*Binomial[n + 1, t] *(-1)^(t + 1), {t, 1, n + 1}], True, g[b - 1, n, k] + g[b*k, n - 1, k]]]; a[n_] := g[1, n, 10]; Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Feb 01 2017, after Alois P. Heinz *)

a(n) = [x^(10^n)] 1/Product_{j>=0} (1-x^(10^j)).

cp's OEIS Frontend

A008592 Multiples of 10: a(n) = 10*n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

A008728 Molien series for 3-dimensional group [2,n ] = *22n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A145513 Number of partitions of 10^n into powers of 10.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Formula