A055841 -id:A055841 - cp's OEIS frontend

A002063 a(n) = 9*4^n.

9, 36, 144, 576, 2304, 9216, 36864, 147456, 589824, 2359296, 9437184, 37748736, 150994944, 603979776, 2415919104, 9663676416, 38654705664, 154618822656, 618475290624, 2473901162496, 9895604649984, 39582418599936, 158329674399744, 633318697598976

Offset: 0

N. J. A. Sloane

a(n) is twice the area of the trapezoid created by the four points (2^n,2^(n+1)), (2^(n+1), 2^n), (2^(n+1), 2^(n+2)), (2^(n+2), 2^(n+1)). - J. M. Bergot, May 23 2014
These are squares that can be expressed as sum of exactly two distinct powers of two. For instance, a(4) = 9*4^4 = 2304 = 2^11 + 2^8 . It is conjectured that these are the only squares with this characteristic (tested on squares up to (10^7)^2). - Andres Cicuttin, Apr 23 2016
Conjecture is true. It is equivalent to prove that the Diophantine equation m^2 = 2^k*(1+2^h), where h>0, has solutions only when h=3. Dividing by 2^k we must obtain an odd square on the left, since 1+2^h is odd, so we can write (2*r+1)^2 = 1+2^h. Expanding, we have 4*r*(r+1) = 2^h, from which it follows that r must be equal to 1 and thus h=3, since r and r+1 must be powers of 2. - Giovanni Resta, Jul 27 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (4).

Essentially the same as A055841. First differences of A002001.

Cf. A000302.

[9*4^n: n in [0..30]]; // Vincenzo Librandi, May 19 2011

9*4^Range[0, 100] (* Vladimir Joseph Stephan Orlovsky, Jun 09 2011 *)
NestList[4#&,9,30] (* Harvey P. Dale, Jan 15 2019 *)

a(n)=9<Charles R Greathouse IV, Apr 17 2012

def A002063(n): return 9<<(n<<1) # Chai Wah Wu, Apr 09 2025

From Philippe Deléham, Nov 23 2008: (Start)
a(n) = 4*a(n-1), n > 0; a(0)=9.
G.f.: 9/(1-4*x). (End)
a(n) = 9*A000302(n). - Michel Marcus, Apr 23 2016
E.g.f.: 9*exp(4*x). - Ilya Gutkovskiy, Apr 23 2016
a(n) = 2^(2*n+3) + 2^(2*n). - Andres Cicuttin, Apr 26 2016
a(n) = A004171(n+1) + A000302(n). - Zhandos Mambetaliyev, Nov 19 2016

A201780 Riordan array ((1-x)^2/(1-2x), x/(1-2x)).

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 2, 5, 4, 1, 4, 12, 13, 6, 1, 8, 28, 38, 25, 8, 1, 16, 64, 104, 88, 41, 10, 1, 32, 144, 272, 280, 170, 61, 12, 1, 64, 320, 688, 832, 620, 292, 85, 14, 1, 128, 704, 1696, 2352, 2072, 1204, 462, 113, 16, 1

Offset: 0

Philippe Deléham, Dec 05 2011

Diagonals ascending: 1, 0, 1, 1, 2, 2, 4, 5, 1, 8, 12, 4, ... (see A201509).

			Triangle begins:
  1;
  0,  1;
  1,  2,  1;
  2,  5,  4,  1;
  4, 12, 13,  6,  1;
  8, 28, 38, 25,  8,  1;

Benjamin Braun, W. K. Hough, Matching and Independence Complexes Related to Small Grids, arXiv preprint arXiv:1606.01204 [math.CO], 2016.
Wesley K. Hough, On Independence, Matching, and Homomorphism Complexes, (2017), Theses and Dissertations--Mathematics, 42.

Diagonals: A000012, A005843, A001844, A035597,
Columns: A034008, A045623, A084851, A055585,
Row sums: A052156

CoefficientList[#, y]& /@ CoefficientList[(1-x)^2/(1-(y+2)*x) + O[x]^10, x] // Flatten (* Jean-François Alcover, Nov 03 2018 *)

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) with T(0,0) = 0, T(1,0) = 0, T(2,0) = 0 and T(n,k)= 0 if k < 0 or if n < k.
Sum_{k=0..n} T(n,k)*x^k = A154955(n+1), A034008(n), A052156(n), A055841(n), A055842(n), A055846(n), A055270(n), A055847(n), A055995(n), A055996(n), A056002(n), A056116(n) for x = -1,0,1,2,3,4,5,6,7,8,9,10 respectively.
G.f.: (1-x)^2/(1-(y+2)*x).

A056120 a(n) = (3^3)*4^(n-3) with a(0)=1, a(1)=1 and a(2)=7.

Original entry on oeis.org

1, 1, 7, 27, 108, 432, 1728, 6912, 27648, 110592, 442368, 1769472, 7077888, 28311552, 113246208, 452984832, 1811939328, 7247757312, 28991029248, 115964116992, 463856467968, 1855425871872

Offset: 0

Barry E. Williams, Jul 05 2000

For n>=3, a(n) is equal to the number of functions f:{1,2,...,n}->{1,2,3,4} such that for fixed, different x_1, x_2, x_3 in {1,2,...,n} and fixed y_1, y_2, y_3 in {1,2,3,4} we have f(x_i)<>y_i, (i=1,2,...,n). - Milan Janjic, May 13 2007

G. C. Greubel, Table of n, a(n) for n = 0..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (4).

Cf. A055841.

First differences of A002063.

Concatenation([1,1,7], List([3..25], n-> 27*4^(n-3) )); # G. C. Greubel, Jan 18 2020

[1,1,7] cat [27*4^(n-3): n in [3..25]]; // G. C. Greubel, Jan 18 2020

1,1,7, seq( 27*4^(n-3), n=3..25); # G. C. Greubel, Jan 18 2020

Table[If[n<2, 1, If[n==2, 7, 27*4^(n-3)]], {n,0,25}] (* G. C. Greubel, Jan 18 2020 *)

vector(26, n, if(n<2, 1, if(n==2, 7, 27*4^(n-3))) ) \\ G. C. Greubel, Jan 18 2020

[1,1,7]+[27*4^(n-3) for n in (3..25)] # G. C. Greubel, Jan 18 2020

a(n) = 4*a(n-1) + (-1)^n*binomial(3, 3-n).
G.f.: (1-x)^3/(1-4*x).
E.g.f.: (37 - 44*x + 8*x^2 + 27*exp(4*x))/64. - G. C. Greubel, Jan 18 2020

a(21) corrected by R. J. Mathar, Dec 03 2014

A143787 Number of compositions of n into floor((3*j)/2) kinds of j's for all j>=1.

Original entry on oeis.org

1, 1, 4, 11, 33, 95, 278, 808, 2355, 6856, 19969, 58151, 169353, 493190, 1436288, 4182793, 12181260, 35474611, 103310209, 300862991, 876181998, 2551642760, 7430968523, 21640683328, 63022629465, 183536340391, 534499885849, 1556586163406, 4533135643968, 13201529892305, 38445880553108, 111963215139163, 326062542045345

Offset: 0

Joerg Arndt, Jul 06 2011

nonn

The g.f. for compositions of k_1 kinds of 1's, k_2 kinds of 2's, ..., k_j kinds of j's, ... is 1/(1-sum(j>=1, k_j * x^j )).

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,3,-1).

Cf. A121907 (floor((3*j-1)/2)), A055841 (3*j-1), A052156 (2*j-1), A006053 (floor(j/2)), A176848 (floor(j/3)).

LinearRecurrence[{2,3,-1},{1,1,4,11},50] (* Paolo Xausa, Nov 14 2023 *)

a(n) = +2*a(n-1) +3*a(n-2) -1*a(n-3).
G.f.: ((1-x)^2*(1+x))/(1-2*x-3*x^2+x^3).
G.f.: 1/(1-sum(j>=1, floor((3*j)/2)*x^j )).

A176848 Number of compositions of n into floor(j/3) kinds of j's for all j>=1.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 3, 4, 5, 10, 15, 21, 36, 56, 83, 134, 210, 320, 505, 791, 1221, 1911, 2988, 4639, 7240, 11305, 17595, 27436, 42806, 66691, 103968, 162144, 252720, 393965, 614285, 957581, 1492791, 2327396, 3628273, 5656274, 8818275, 13747425, 21431700, 33411976, 52088551, 81204526, 126596778, 197361904, 307682405

Offset: 0

Joerg Arndt, Jul 06 2011

nonn

The g.f. for compositions of k_1 kinds of 1's, k_2 kinds of 2's, ..., k_j kinds of j's, ... is 1/(1-sum(j>=1, k_j * x^j )).

Jarib R. Acosta, Yadira Caicedo, Juan P. Poveda, José L. Ramírez, Mark Shattuck, Some New Restricted n-Color Composition Functions, J. Int. Seq., Vol. 22 (2019), Article 19.6.4.
Index entries for linear recurrences with constant coefficients, signature (1, 0, 2, -1).

Cf. A121907 (floor((3*j-1)/2)), A055841 (3*j-1), A052156 (2*j-1), A006053 (floor(j/2)), A143787 (floor((3*j)/2)).

N=66; x='x+O('x^N) /* that many terms */
gf= 1/(1-sum(j=1,N, floor(j/3)*x^j ))
Vec(gf) /* show terms */

G.f.: 1/(1-sum(j>=1, floor(j/3)*x^j )).
Conjectural g.f.: (x-1)^2*(x^2+x+1) / (x^4-2*x^3-x+1). - Colin Barker, May 15 2013
G.f.: 1 + x^3*Q(0)/2 , where Q(k) = 1 + 1/(1 - x*(4*k+1 + 2*x^2 - x^3)/( x*(4*k+3 + 2*x^2 - x^3 ) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 11 2013

A166976 Array of A002450 in the top row and higher-order differences in subsequent rows, read by antidiagonals.

Original entry on oeis.org

0, 1, 1, 3, 4, 5, 9, 12, 16, 21, 27, 36, 48, 64, 85, 81, 108, 144, 192, 256, 341, 243, 324, 432, 576, 768, 1024, 1365, 729, 972, 1296, 1728, 2304, 3072, 4096, 5461, 2187, 2916, 3888, 5184, 6912, 9216, 12288, 16384, 21845, 6561

Offset: 0

Paul Curtz, Oct 26 2009

			The array starts:
0,   1,   5,  21,  85, 341,1365,5461,21845,87381,349525,    A002450
1,   4,  16,  64, 256,1024,4096,16384,65536,262144,1048576, A000302
3,  12,  48, 192, 768,3072,12288,49152,196608,786432,       A002001, A164346, A110594
9,  36, 144, 576,2304,9216,36864,147456                     A002063, A055841

A002450 := proc(n) (4^n-1)/3 ; end proc:
A166976 := proc(n,k) option remember; if n = 0 then A002450(k) else procname(n-1,k+1)-procname(n-1,k) ; end if; end proc: # R. J. Mathar, Jul 02 2011

T(0,k) = A002450(k). T(n,k) = T(n-1,k+1) - T(n-1,k), n > 0.

cp's OEIS Frontend

A002063 a(n) = 9*4^n.

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A201780 Riordan array ((1-x)^2/(1-2x), x/(1-2x)).

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A056120 a(n) = (3^3)*4^(n-3) with a(0)=1, a(1)=1 and a(2)=7.

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A143787 Number of compositions of n into floor((3*j)/2) kinds of j's for all j>=1.

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A176848 Number of compositions of n into floor(j/3) kinds of j's for all j>=1.

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A166976 Array of A002450 in the top row and higher-order differences in subsequent rows, read by antidiagonals.

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Maple

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