A002001 -id:A002001 - cp's OEIS frontend

A200139 Triangle T(n,k), read by rows, given by (1,1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

1, 1, 1, 2, 3, 1, 4, 8, 5, 1, 8, 20, 18, 7, 1, 16, 48, 56, 32, 9, 1, 32, 112, 160, 120, 50, 11, 1, 64, 256, 432, 400, 220, 72, 13, 1, 128, 576, 1120, 1232, 840, 364, 98, 15, 1, 256, 1280, 2816, 3584, 2912, 1568, 560, 128, 17, 1, 512, 2816, 6912, 9984, 9408, 6048, 2688, 816, 162, 19, 1

Offset: 0

Philippe Deléham, Nov 13 2011

Riordan array ((1-x)/(1-2x),x/(1-2x)).
Product A097805*A007318 as infinite lower triangular arrays.
Product A193723*A130595 as infinite lower triangular arrays.
T(n,k) is the number of ways to place n unlabeled objects into any number of labeled bins (with at least one object in each bin) and then designate k of the bins. - Geoffrey Critzer, Nov 18 2012
Apparently, rows of this array are unsigned diagonals of A028297. - Tom Copeland, Oct 11 2014
Unsigned A118800, so my conjecture above is true. - Tom Copeland, Nov 14 2016

			Triangle begins:
   1
   1,   1
   2,   3,   1
   4,   8,   5,   1
   8,  20,  18,   7,   1
  16,  48,  56,  32,   9,   1
  32, 112, 160, 120,  50,  11,   1

Cf. A118800 (signed version), A081277, A039991, A001333 (antidiagonal sums), A025192 (row sums); diagonals: A000012, A005408, A001105, A002492, A072819l; columns: A011782, A001792, A001793, A001794, A006974, A006975, A006976.

Cf. A007318, A028297, A059260, A097805, A118801, A130595.

nn=15;f[list_]:=Select[list,#>0&];Map[f,CoefficientList[Series[(1-x)/(1-2x-y x) ,{x,0,nn}],{x,y}]]//Grid  (* Geoffrey Critzer, Nov 18 2012 *)

T(n,k) = 2*T(n-1,k)+T(n-1,k-1) with T(0,0)=T(1,0)=T(1,1)=1 and T(n,k)=0 for k<0 or for n

T(n,k) = A011782(n-k)*A135226(n,k) = 2^(n-k)*(binomial(n,k)+binomial(n-1,k-1))/2.

Sum_{k, 0<=k<=n} T(n,k)*x^k = A000007(n), A011782(n), A025192(n), A002001(n), A005054(n), A052934(n), A055272(n), A055274(n), A055275(n), A052268(n), A055276(n), A196731(n) for n=-1,0,1,2,3,4,5,6,7,8,9,10 respectively.

G.f.: (1-x)/(1-(2+y)*x).

T(n,k) = Sum_j>=0 T(n-1-j,k-1)*2^j.

T = A007318*A059260, so the row polynomials of this entry are given umbrally by p_n(x) = (1 + q.(x))^n, where q_n(x) are the row polynomials of A059260 and (q.(x))^k = q_k(x). Consequently, the e.g.f. is exp[tp.(x)] = exp[t(1+q.(x))] = e^t exp(tq.(x)) = [1 + (x+1)e^((x+2)t)]/(x+2), and p_n(x) = (x+1)(x+2)^(n-1) for n > 0. - Tom Copeland, Nov 15 2016

T^(-1) = A130595*(padded A130595), differently signed A118801. Cf. A097805. - Tom Copeland, Nov 17 2016

The n-th row polynomial in descending powers of x is the n-th Taylor polynomial of the rational function (1 + x)/(1 + 2*x) * (1 + 2*x)^n about 0. For example, for n = 4, (1 + x)/(1 + 2*x) * (1 + 2*x)^4 = (8*x^4 + 20*x*3 + 18*x^2 + 7*x + 1) + O(x^5). - Peter Bala, Feb 24 2018

A110594 a(1) = 4, a(2) = 12, for n>1: a(n) = 3*4^(n-1).

Original entry on oeis.org

4, 12, 48, 192, 768, 3072, 12288, 49152, 196608, 786432, 3145728, 12582912, 50331648, 201326592, 805306368, 3221225472, 12884901888, 51539607552, 206158430208, 824633720832, 3298534883328, 13194139533312, 52776558133248

Offset: 1

Jonathan Vos Post, Jul 29 2005

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4).

Cf. A000302, A007090, A081604, A110591, A110593.

Concatenation([4],List([2..25],n->3*4^(n-1))); # Muniru A Asiru, Oct 21 2018

[4] cat [3*4^(n-1): n in [2..30]]; // Vincenzo Librandi, May 29 2014

seq(coeff(series(4*x*(1-x)/(1-4*x),x,n+1), x, n), n = 1 .. 25); # Muniru A Asiru, Oct 21 2018

CoefficientList[Series[4 (1 - x)/(1 - 4 x), {x, 0, 40}], x] (* Vincenzo Librandi, May 29 2014 *)

x='x+O('x^50); Vec(4*x*(1 - x)/(1 - 4*x)) \\ G. C. Greubel, Sep 01 2017

a(n) = A002001(n), n>1. - R. J. Mathar, Aug 18 2008

G.f.: 4*x*(1 - x)/(1 - 4*x). - Vincenzo Librandi, May 29 2014

Definition corrected by R. J. Mathar, Aug 18 2008

A178789 a(n) = 4^(n-1) + 2: Number of acute angles after n iterations of the Koch snowflake construction.

Original entry on oeis.org

3, 6, 18, 66, 258, 1026, 4098, 16386, 65538, 262146, 1048578, 4194306, 16777218, 67108866, 268435458, 1073741826, 4294967298, 17179869186, 68719476738, 274877906946, 1099511627778, 4398046511106, 17592186044418, 70368744177666

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 14 2010

Starting from an equilateral triangle, at each step each straight segment is replaced by a "/\" shape of four segments of equal length, with the acute angle in the middle pointing to the exterior. The sequence counts the angles which are (i.e., already were) at both extremities, plus the one newly created acute angle in the middle of each former segment. At step n, there are 3*4^(n-1) straight segments, therefore a(n+1) = a(n) + 3*4^(n-1). - M. F. Hasler, Dec 17 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Mario Raso, Integer Sequences in Cryptography: A New Generalized Family and its Application, Ph. D. Thesis, Sapienza University of Rome (Italy 2025). See p. 113.
Larry Riddle, Koch Curve.
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A002001, A000302, A010502, A047849, A164346.

[2^(2*(n-1)) + 2: n in [1..30]]; // Vincenzo Librandi, Feb 02 2013

A178789:=n->2+4^(n-1); seq(A178789(n), n=1..30); # Wesley Ivan Hurt, Dec 17 2013

a=b=3;lst={a};Do[a=a+b;b*=4;AppendTo[lst,a],{n,40}];lst
Flatten[Table[2^(2*(n-1)) + 2, {n, 1, 50}]](* or *)   CoefficientList[Series[(3 - 9*x)/(1 - 5*x + 4*x^2),{x, 0, 100}], x] (* Vincenzo Librandi, Feb 02 2013 *)

A178789=n->4^(n-1)+2  \\ - M. F. Hasler, Dec 17 2013

G.f.: 3*x*(1 - 3*x)/(1 - 5*x + 4*x^2).
a(n) = 3 * A047849(n-1).
a(n) = 2^(2*(n-1)) + 2. - Vincenzo Librandi, Feb 02 2013
a(n+1) = a(n) + 3*4^(n-1) = a(n) + A002001(n) for n > 0. - M. F. Hasler, Dec 17 2013
a(n) = 2 + A000302(n-1). - Omar E. Pol, Dec 18 2013

A196661 Expansion of g.f. (1-2x)/(1-7x).

Original entry on oeis.org

1, 5, 35, 245, 1715, 12005, 84035, 588245, 4117715, 28824005, 201768035, 1412376245, 9886633715, 69206436005, 484445052035, 3391115364245, 23737807549715, 166164652848005, 1163152569936035, 8142067989552245, 56994475926865715, 398961331488060005

Offset: 0

Philippe Deléham, Oct 05 2011

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

Cf. A002001, A193577 (which is the same except for the initial 1), A193722.

a(n)=if(n,5*7^(n-1),1) \\ Charles R Greathouse IV, Nov 21 2011

a(0) = 1, a(n) = 5*7^(n-1) for n>0.
a(n) = Sum_{k=0..n} A193722(n,k)*2^k.
From Elmo R. Oliveira, Mar 18 2025: (Start)
E.g.f.: (5*exp(7*x) + 2)/7.
a(n) = 7*a(n-1). (End)

A196662 Expansion of g.f. (1-3x)/(1-10x).

Original entry on oeis.org

1, 7, 70, 700, 7000, 70000, 700000, 7000000, 70000000, 700000000, 7000000000, 70000000000, 700000000000, 7000000000000, 70000000000000, 700000000000000, 7000000000000000, 70000000000000000, 700000000000000000, 7000000000000000000, 70000000000000000000

Offset: 0

Philippe Deléham, Oct 05 2011

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

Cf. A002001, A193722, A196661.

CoefficientList[Series[(1-3x)/(1-10x),{x,0,20}],x] (* or *) LinearRecurrence[ {10},{1,7},30] (* or *) Join[{1},NestList[10#&,7,20]] (* Harvey P. Dale, Dec 18 2021 *)

a(0) = 1, a(n) = 7*10^(n-1) for n>0.
a(n) = Sum_{k=0..n} A193722(n,k)*3^k.
From Elmo R. Oliveira, Mar 18 2025: (Start)
E.g.f.: (7*exp(10*x) + 3)/10.
a(n) = 10*a(n-1). (End)

A006342 Coloring a circuit with 4 colors.

Original entry on oeis.org

1, 1, 4, 10, 31, 91, 274, 820, 2461, 7381, 22144, 66430, 199291, 597871, 1793614, 5380840, 16142521, 48427561, 145282684, 435848050, 1307544151, 3922632451, 11767897354, 35303692060, 105911076181, 317733228541, 953199685624, 2859599056870, 8578797170611

Offset: 0

N. J. A. Sloane, Simon Plouffe

Also equal to the number of set partitions of {1,2,...,n+2} with at most 4 parts such that each part does not contain both i,i+1 for 1<=iMike Zabrocki, Sep 08 2020

Also a(n) equals the number of color-complete multipoles with n terminals (that is, having all the states allowed by the Parity Lemma). - Miquel A. Fiol, May 27 2024

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
F. R. Bernhart, Topics in Graph Theory Related to the Five Color Conjecture, Ph.D. Dissertation, Kansas State Univ., 1974.
F. R. Bernhart, Fundamental chromatic numbers, Unpublished. (Annotated scanned copy)
F. R. Bernhart & N. J. A. Sloane, Correspondence, 1977
G. D. Birkhoff, D. C. Lewis, Chromatic polynomials, Trans. Amer. Math. Soc. 60, (1946). 355-451.
Gesualdo Delfino and Jacopo Viti, Potts q-color field theory and scaling random cluster model, arXiv preprint arXiv:1104.4323 [hep-th], 2011.
M. A. Fiol and J. Vilaltella, Some results on the structure of multipoles in the study of snarks, Electron. J. Combin. 22(1) (2015), #P1.45.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (3,1,-3).

A214142, A214167

[3*3^n/8+1/4+3*(-1)^n/8: n in [0..30]]; // Vincenzo Librandi, Aug 20 2011

A006342:=-(-1+2*z)/(z-1)/(3*z-1)/(z+1); # conjectured by Simon Plouffe in his 1992 dissertation
a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]+3*a[n-2]-1 od: seq(a[n], n=1..26); # Zerinvary Lajos, Apr 28 2008

CoefficientList[Series[(1-2 x)/((1-x^2) (1-3 x)),{x,0,30}],x] (* or *) LinearRecurrence[{3,1,-3},{1,1,4},30] (* Harvey P. Dale, Aug 16 2016 *)

Vec((1 - 2*x) / ((1 - x)*(1 + x)*(1 - 3*x)) + O(x^30)) \\ Colin Barker, Nov 07 2017

G.f.: (1 - 2 x ) / (( 1 - x^2 ) ( 1 - 3 x )).
Binomial transform of A002001 (with interpolated zeros). Partial sums of A054878. E.g.f.: exp(x)(3*cosh(2*x) + 1)/4; a(n) = 3*3^n/8 + 1/4 + 3(-1)^n/8 = Sum_{k=0..n} (3^k + 3(-1)^k)/4. - Paul Barry, Sep 03 2003
a(n) = 2*a(n-1) + 3*a(n-2) - 1, n > 1. - Gary Detlefs, Jun 21 2010
a(n) = a(n-1) + A054878(n-2). - Yuchun Ji, Sep 12 2017
From Colin Barker, Nov 07 2017: (Start)
a(n) = (3^(n+1) + 5) / 8 for n even.
a(n) = (3^(n+1) - 1) / 8 for n odd.
a(n) = 3*a(n-1) + a(n-2) - 3*a(n-3) for n > 2.
(End)
a(n) = 3*a(n-1) + (3*(-1)^n - 1)/2 for n > 0. - Yuchun Ji, Dec 05 2019

A053661 For n > 1: if n is present, 2n is not.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 9, 11, 12, 13, 15, 17, 19, 20, 21, 23, 25, 27, 28, 29, 31, 32, 33, 35, 36, 37, 39, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 57, 59, 60, 61, 63, 65, 67, 68, 69, 71, 73, 75, 76, 77, 79, 80, 81, 83, 84, 85, 87, 89, 91, 92, 93, 95, 97, 99, 100, 101, 103, 105

Offset: 1

Jeevan Chana Rai (Karanjit.Rai(AT)btinternet.com), Feb 16 2000

The Name line gives a property of the sequence, not a definition. The sequence can be defined simultaneously with b(n) := A171945(n) via a(n) = mex{a(i), b(i) : 0 <= i < n} (n >= 0}, b(n)=2a(n). The two sequences are complementary, hence A053661 is identical to A171944 (except for the first terms). Furthmore, A053661 is the same as A003159 except for the replacement of vile by dopey powers of 2. - Aviezri S. Fraenkel, Apr 28 2011
For n >= 2, either n = 2^k where k is odd or n = 2^k*m where m > 1 is odd and k is even (found by Kirk Bresniker and Stan Wagon). [Robert Israel, Oct 10 2010]
Subsequence of A175880; A000040, A001749, A002001, A002042, A002063, A002089, A003947, A004171 and A081294 are subsequences.

Robert Israel, Table of n, a(n) for n = 1..10000
Aviezri S. Fraenkel, The vile, dopey, evil and odious game players, Discrete Math. 312 (2012), no. 1, 42-46.

Essentially identical to A171944 and the complement of A171945.

a053661 n = a053661_list !! (n-1)
a053661_list = filter (> 0) a175880_list -- Reinhard Zumkeller, Feb 09 2011

N:= 1000: # to get all terms <= N
sort([1,seq(2^(2*i+1),i=0..(ilog2(N)-1)/2), seq(seq(2^(2*i)*(2*j+1),j=1..(N/2^(2*i)-1)/2),i=0..ilog2(N)/2)]); # Robert Israel, Jul 24 2015

Clear[T]; nn = 105; T[n_, k_] := T[n, k] = If[n < 1 || k < 1, 0, If[n == 1 || k == 1, 1, If[k > n, T[k, n], If[n > k, T[k, Mod[n, k, 1]], -Product[T[n, i], {i, n - 1}]]]]]; DeleteCases[Table[If[T[n, n] == -1, n, ""], {n, 1, nn}], ""] (* Mats Granvik, Aug 25 2012 *)

More terms from James Sellers, Feb 22 2000

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A067403 Third column of triangle A067402.

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A067410 Triangle with columns built from certain power sequences.

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A140660 a(n) = 3*4^n + 1.

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A200139 Triangle T(n,k), read by rows, given by (1,1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

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A110594 a(1) = 4, a(2) = 12, for n>1: a(n) = 3*4^(n-1).

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A178789 a(n) = 4^(n-1) + 2: Number of acute angles after n iterations of the Koch snowflake construction.

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A196661 Expansion of g.f. (1-2*x)/(1-7*x).

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A196661 Expansion of g.f. (1-2x)/(1-7x).

A196662 Expansion of g.f. (1-3x)/(1-10x).