A033484 -id:A033484 - cp's OEIS frontend

A093381 Expansion of (1 - 2x - 3x^2 - 4x^3)/((1-x)(1-2x)(1-3x)(1-4*x)).

1, 8, 42, 186, 766, 3058, 12062, 47426, 186606, 735858, 2909182, 11528866, 45781646, 182104658, 725311902, 2891845506, 11539011886, 46070609458, 184025468222, 735329653346, 2938999333326, 11749034250258, 46975237266142

Offset: 0

Paul Barry, Apr 28 2004

Second binomial transform of A093380.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Cf. A033484.

[4/3-5*2^n+2*3^n+8*4^n/3: n in [0..30]]; // Vincenzo Librandi, May 31 2011

CoefficientList[Series[(1-2x-3x^2-4x^3)/((1-x)(1-2x)(1-3x)(1-4x)),{x,0,30}],x] (* or *) LinearRecurrence[{10,-35,50,-24},{1,8,42,186},30] (* Harvey P. Dale, May 29 2013 *)

a(n) = 4/3 - 5*2^n + 2*3^n + 8*4^n/3;
a(n) = 2*A000244(n) - 5*A000079(n) + 4*A001045(2n+1).
a(n) = 10*a(n-1) - 35*a(n-2) + 50*a(n-3) - 24*a(n-4), n > 3. - Harvey P. Dale, May 29 2013

A130452 Triangle read by rows: A097806 * A130321 as infinite lower triangular matrices.

Original entry on oeis.org

1, 3, 1, 6, 3, 1, 12, 6, 3, 1, 24, 12, 6, 3, 1, 48, 24, 12, 6, 3, 1, 96, 48, 24, 12, 6, 3, 1, 192, 96, 48, 24, 12, 6, 3, 1, 384, 192, 96, 48, 24, 12, 6, 3, 1, 768, 384, 192, 96, 48, 24, 12, 6, 3, 1, 1536, 768, 384, 192, 96, 48, 24, 12, 6, 3, 1, 3072, 1536, 768, 384, 192, 96, 48, 24, 12, 6, 3, 1

Offset: 1

Gary W. Adamson, May 26 2007

Row sums = A033484: (1, 4, 10, 22, 46, 94, 190, ...).

			First few rows of the triangle:
   1;
   3,  1;
   6,  3,  1;
  12,  6,  3,  1;
  24, 12,  6,  3,  1;
  48, 24, 12,  6,  3,  1;
  ...

Cf. A003945, A033484, A097806, A130321.

A097806 * A130321 as infinite lower triangular matrices. A097806 = the pairwise operator, A130321 = [1; 2,1; 4,2,1; ...]. Triangle, A003945 (1, 3, 6, 12, 24, 48, ...) in every column.

a(28) = 1 inserted and more terms from Georg Fischer, May 29 2023

A133093 A007318 * A097806 * A133080.

Original entry on oeis.org

1, 3, 1, 6, 3, 1, 10, 6, 5, 1, 15, 10, 15, 5, 1, 21, 15, 35, 15, 7, 1, 28, 21, 70, 35, 28, 7, 1, 36, 28, 126, 70, 84, 28, 9, 1, 45, 36, 210, 126, 210, 84, 45, 9, 1, 55, 45, 330, 210, 462, 210, 165, 45, 11, 1

Offset: 1

Gary W. Adamson, Sep 09 2007

Row sums = A033484: (1, 4, 10, 22, 46, 94, ...).

			First few rows of the triangle:
   1;
   3,  1;
   6,  3,  1;
  10,  6,  5,  1;
  15, 10, 15,  5,  1;
  21, 15, 35, 15,  7,  1;
  28, 21, 70, 35, 28,  7,  1;
  ...

Cf. A133080, A097806, A033484. Duplicate of A131110.

A007318 * A097806 * A133080 as infinite lower triangular matrices.

Binomial transform of an infinite lower triangular matrix with (1,1,1,...) in the main diagonal, (2,1,2,1,...) in the subdiagonal and (1,0,1,0,...) in the subsubdiagonal.

A358125 Triangle read by rows: T(n, k) = 2^n - 2^(n-k-1) - 2^k, 0 <= k <= n-1.

Original entry on oeis.org

0, 1, 1, 3, 4, 3, 7, 10, 10, 7, 15, 22, 24, 22, 15, 31, 46, 52, 52, 46, 31, 63, 94, 108, 112, 108, 94, 63, 127, 190, 220, 232, 232, 220, 190, 127, 255, 382, 444, 472, 480, 472, 444, 382, 255, 511, 766, 892, 952, 976, 976, 952, 892, 766, 511, 1023, 1534, 1788, 1912, 1968, 1984, 1968, 1912, 1788, 1534, 1023

Offset: 1

Ambrosio Valencia-Romero, Dec 20 2022

T(n, k) is the expanded number of player-reduced static games within an n-player two-strategy game scenario in which one player (the "standpoint") faces a specific combination of other players' individual strategies with the possibility of anti-coordination between them -- the total number of such combinations is 2^(n-1). The value of k represents the number of other players who (are expected to) agree on one of the two strategies in S, while the other n-k-1 choose the other strategy; the standpoint player is not included.

			Triangle begins:
  0;
  1,     1;
  3,     4,    3;
  7,    10,   10,    7;
  15,   22,   24,   22,   15;
  31,   46,   52,   52,   46,   31;
  63,   94,  108,  112,  108,   94,   63;
 127,  190,  220,  232,  232,  220,  190,  127;
 255,  382,  444,  472,  480,  472,  444,  382,  255;
 511,  766,  892,  952,  976,  976,  952,  892,  766,  511;
1023, 1534, 1788, 1912, 1968, 1984, 1968, 1912, 1788, 1534, 1023;
2047, 3070, 3580, 3832, 3952, 4000, 4000, 3952, 3832, 3580, 3070, 2047;
  ...

Column k=0 gives A000225(n-1).
Column k=1 gives A033484(n-2).
Column k=2 gives A053208(n-3).
Cf. A005061, A359200.

T := n -> seq(2^n - 2^(n - k - 1) - 2^k, k = 0 .. n - 1);
seq(T(n), n=1..12);

T[n_, k_] := 2^n - 2^(n - k - 1) - 2^k; Table[T[n, k], {n, 1, 11}, {k, 0, n - 1}] // Flatten (* Amiram Eldar, Dec 20 2022 *)

T(n, k) = 2^n - 2^(n-k-1) - 2^k.

Sum_{k=0..n-1} T(n,k)*binomial(n-1,k) = 2*A005061(n-1)

A369491 a(n) = n! * [x^n] (2x - 4exp(x) + 3exp(2x) + 3) / 2.

Original entry on oeis.org

1, 2, 4, 10, 22, 46, 94, 190, 382, 766, 1534, 3070, 6142, 12286, 24574, 49150, 98302, 196606, 393214, 786430, 1572862, 3145726, 6291454, 12582910, 25165822, 50331646, 100663294, 201326590, 402653182, 805306366, 1610612734, 3221225470, 6442450942, 12884901886, 25769803774

Offset: 0

Peter Luschny, Jan 24 2024

Paolo Xausa, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A033484 (similar, 'missing' 2).

gf := ((x + 1)*(2*x^2 - 2*x + 1))/((2*x - 1)*(x - 1)):
ser := series(gf, x, 40): seq(coeff(ser, x, n), n = 0..34);
a := proc(n) option remember; ifelse(n < 4, [1, 2, 4, 10][n+1],
3*a(n - 1) - 2*a(n - 2)) end: seq(a(n), n = 0..34);

LinearRecurrence[{3, -2}, {1, 2, 4, 10}, 50] (* Paolo Xausa, Feb 27 2024 *)

a(n) = [x^n] ((x + 1)*(2*x^2 - 2*x + 1))/((2*x - 1)*(x - 1)).

a(n) = 3*a(n - 1) - 2*a(n - 2) for n >= 4.

A090842 Square array of numbers read by antidiagonals where T(n,k) = ((k+3)*(k+2)^n-2)/(k+1).

Original entry on oeis.org

1, 1, 4, 1, 5, 10, 1, 6, 17, 22, 1, 7, 26, 53, 46, 1, 8, 37, 106, 161, 94, 1, 9, 50, 187, 426, 485, 190, 1, 10, 65, 302, 937, 1706, 1457, 382, 1, 11, 82, 457, 1814, 4687, 6826, 4373, 766, 1, 12, 101, 658, 3201, 10886, 23437, 27306, 13121, 1534, 1, 13, 122, 911, 5266

Offset: 0

Paul Barry, Dec 09 2003

Nodes on a tree with degree k interior nodes and degree 1 boundary nodes.

			Rows begin:
  1 4 10 22 ...
  1 5 17 53 ...
  1 6 26 106 ...
  1 7 37 187 ...

L. He, X. Liu and G. Strang, Trees with Cantor Eigenvalue Distribution, Studies in Applied Mathematics 110 (2), 123-138, 2003.

Rows include A033484, A048473, A020989, A057651, A061801, A090843.

The total number of nodes on a tree with degree k interior nodes and degree 1 boundary nodes is given by N(k, r) = (k*(k-1)^r-2)/(k-2).

G.f.: Sum_{k>=0} (1+x*y)/(1-x*y)/(1-(k+2)*x*y)*y^k. - Vladeta Jovovic, Dec 12 2003

A099942 Start with 1, then alternately double or add 2.

Original entry on oeis.org

1, 2, 4, 8, 10, 20, 22, 44, 46, 92, 94, 188, 190, 380, 382, 764, 766, 1532, 1534, 3068, 3070, 6140, 6142, 12284, 12286, 24572, 24574, 49148, 49150, 98300, 98302, 196604, 196606, 393212, 393214, 786428, 786430, 1572860, 1572862, 3145724, 3145726

Offset: 0

N. J. A. Sloane, Nov 12 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-2).

Cf. A033484, A061776, A075427, A083416.

[3*2^Ceiling(n/2) + (-1)^n - 3: n in [0..50]]; // Vincenzo Librandi, Aug 17 2011

LinearRecurrence[{0,3,0,-2},{1,2,4,8},50] (* Harvey P. Dale, May 03 2016 *)

print1(a=1,",");for(n=1,20,print1(a=2*a,",",a=a+2,","))

a(0)=1; for n > 0, a(n) = a(n-1)*(1 + n mod 2) + 2*((n+1) mod 2).
G.f.: (2*x^3 + x^2 + 2*x + 1)/(2*x^4 - 3*x^2 + 1).
3*2^ceiling(n/2) + (-1)^n - 3. - Ralf Stephan, Dec 04 2004
a(2*n) = A033484(n).
a(n-1) + a(n) = A061776(n) for n > 0.
E.g.f.: -2*cosh(x) + 3*cosh(sqrt(2)*x) - 4*sinh(x) + 3*sqrt(2)*sinh(sqrt(2)*x). - Franck Maminirina Ramaharo, Nov 08 2018

Edited and extended by Klaus Brockhaus, Nov 13 2004

A111526 Number triangle T(n,k)=C((n+k)/2,k)(n+1)(1+(-1)^(n-k))/(2(k+1)); T(n,k)=(-1)^((n-k)/2)*A053120(n+1,k+1)/2^k; Riordan array ((1+x^2)/(1-x^2)^2,x/(1-x^2)).

Original entry on oeis.org

1, 0, 1, 3, 0, 1, 0, 4, 0, 1, 5, 0, 5, 0, 1, 0, 9, 0, 6, 0, 1, 7, 0, 14, 0, 7, 0, 1, 0, 16, 0, 20, 0, 8, 0, 1, 9, 0, 30, 0, 27, 0, 9, 0, 1, 0, 25, 0, 50, 0, 35, 0, 10, 0, 1, 11, 0, 55, 0, 77, 0, 44, 0, 11, 0, 1, 0, 36, 0, 105, 0, 112, 0, 54, 0, 12, 0, 1, 13, 0, 91, 0, 182, 0, 156, 0, 65, 0, 13, 0, 1, 0

Offset: 0

Paul Barry, Aug 05 2005

A scaled Chebyshev triangle.

Row sums are A001350(n+1). Diagonal sums are A033484, with interpolated zeros. Inverse is A111527.

			Triangle starts
1;
0,1;
3,0,1;
0,4,0,1;
5,0,5,0,1;
0,9,0,6,0,1;
7,0,14,0,7,0,1;

Cf. A110813.

A126269 Numbers n such that hcl(n,n) < hcl(n,n-1) where hcl(n,i) is the Huffman code length; see comments.

Original entry on oeis.org

3, 4, 9, 10, 21, 22, 45, 46, 93, 94, 189, 190, 381, 382, 765, 766, 1533, 1534, 3069, 3070, 6141, 6142, 12285, 12286, 24573, 24574, 49149, 49150

Offset: 3

Serhat Sevki Dincer (mesti_mudam(AT)yahoo.com), Dec 22 2006

Consider a string which consists of n distinct symbols such that symbol(i) has frequency i (i=1,2,...,n). Then hcl(n,i) is the Huffman code length of symbol(i).

			Possible Huffman codes for n = 3,4,5 are:
1 : 00
2 : 01
3 : 1
--------
1 : 100
2 : 101
3 : 11
4 : 0
--------
1 : 000
2 : 001
3 : 01
4 : 10
5 : 11
hcl(3,3)=1 < 2=hcl(3,2) and hcl(4,4)=1 < 2=hcl(4,3); so 3,4 are in the sequence.
hcl(5,5)=2=hcl(5,4) so 5 is not in the sequence.

Sean A. Irvine, Java program (github)

Cf. A126268 A126014 A068156 A033484.

Conjecture: a(2k) = A033484(k-1) and a(2k-1) = A068156(k-1), k >= 2.
Conjectures from Colin Barker, Aug 06 2019: (Start)
G.f.: x^3*(3 + 4*x - 2*x^3) / ((1 - x)*(1 + x)*(1 - 2*x^2)).
a(n) = 3*a(n-2) - 2*a(n-4) for n>6.
a(n) = -5/2 + (-1)^n/2 + 3*2^((1/2)*(n-5))*(2-2*(-1)^n + sqrt(2) + (-1)^n*sqrt(2)) for n>2.
(End)

More terms from Sean A. Irvine, Aug 05 2019

A137215 a(n) = 3(10^n) + (n^2 + 1)(10^n - 1)/9.

Original entry on oeis.org

3, 32, 355, 4110, 48887, 588886, 7111107, 85555550, 1022222215, 12111111102, 142222222211, 1655555555542, 19111111111095, 218888888888870, 2488888888888867, 28111111111111086, 315555555555555527, 3522222222222222190, 39111111111111111075, 432222222222222222182

Offset: 0

Ctibor O. Zizka, Mar 06 2008

Sequence generalized: a(n) = a(0)*(B^n) + F(n)* ((B^n)-1)/(B-1); a(0), B integers, F(n) arithmetic function.
Examples:
a(0) = 1, B = 10, F(n) = 1 gives A002275, F(n) = 2 gives A090843, F(n) = 3 gives A097166, F(n) = 4 gives A099914, F(n) = 5 gives A099915.
a(0) = 1, B = 2, F(n) = 1 gives A000225, F(n) = 2 gives A033484, F(n) = 3 gives A036563, F(n) = 4 gives A048487, F(n) = 5 gives A048488, F(n) = 6 gives A048489.
a(0) = 1, B = 3, F(n) = 1 gives A003462, F(n) = 2 gives A048473, F(n) = 3 gives A134931, F(n) = 4 gives A058481, F(n) = 5 gives A116952.
a(0) = 1, B = 4, F(n) = 1 gives A002450, F(n) = 2 gives A020989, F(n) = 3 gives A083420, F(n) = 4 gives A083597, F(n) = 5 gives A083584.
a(0) = 1, B = 5, F(n) = 1 gives A003463, F(n) = 2 gives A057651, F(n) = 3 gives A117617, F(n) = 4 gives A081655.
a(0) = 2, B = 10, F(n) = 1 gives A037559, F(n) = 2 gives A002276.

			a(3) = 3*10^3 + (3*3 + 1)*(10^3 - 1)/9 = 4110.

G. C. Greubel, Table of n, a(n) for n = 0..985
Index entries for linear recurrences with constant coefficients, signature (33,-393,1991,-3930,3300,-1000).

Cf. A002275, A011557.

Table[3*10^n +(n^2 +1)*(10^n -1)/9, {n,0,30}] (* G. C. Greubel, Jan 05 2022 *)

a(n) = 3*(10^n) + (n*n+1)*((10^n)-1)/9; \\ Jinyuan Wang, Feb 27 2020

[3*10^n +(1+n^2)*(10^n -1)/9 for n in (0..30)] # G. C. Greubel, Jan 05 2022

a(n) = 3*(10^n) + (n^2 + 1)*(10^n - 1)/9.

O.g.f.: (3 - 67*x + 478*x^2 - 1002*x^3 + 850*x^4 - 100*x^5)/((1-x)^3 * (1-10*x)^3). - R. J. Mathar, Mar 16 2008

More terms from R. J. Mathar, Mar 16 2008

More terms from Jinyuan Wang, Feb 27 2020

cp's OEIS Frontend

A093381 Expansion of (1 - 2*x - 3*x^2 - 4*x^3)/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)).

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A130452 Triangle read by rows: A097806 * A130321 as infinite lower triangular matrices.

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A133093 A007318 * A097806 * A133080.

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A358125 Triangle read by rows: T(n, k) = 2^n - 2^(n-k-1) - 2^k, 0 <= k <= n-1.

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A369491 a(n) = n! * [x^n] (2*x - 4*exp(x) + 3*exp(2*x) + 3) / 2.

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A090842 Square array of numbers read by antidiagonals where T(n,k) = ((k+3)*(k+2)^n-2)/(k+1).

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A099942 Start with 1, then alternately double or add 2.

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A111526 Number triangle T(n,k)=C((n+k)/2,k)(n+1)(1+(-1)^(n-k))/(2(k+1)); T(n,k)=(-1)^((n-k)/2)*A053120(n+1,k+1)/2^k; Riordan array ((1+x^2)/(1-x^2)^2,x/(1-x^2)).

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A126269 Numbers n such that hcl(n,n) < hcl(n,n-1) where hcl(n,i) is the Huffman code length; see comments.

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A093381 Expansion of (1 - 2x - 3x^2 - 4x^3)/((1-x)(1-2x)(1-3x)(1-4*x)).

A369491 a(n) = n! * [x^n] (2x - 4exp(x) + 3exp(2x) + 3) / 2.

A137215 a(n) = 3(10^n) + (n^2 + 1)(10^n - 1)/9.