A006095 -id:A006095 - cp's OEIS frontend

A171476 a(n) = 6a(n-1) - 8a(n-2) for n > 1, a(0)=1, a(1)=6.

1, 6, 28, 120, 496, 2016, 8128, 32640, 130816, 523776, 2096128, 8386560, 33550336, 134209536, 536854528, 2147450880, 8589869056, 34359607296, 137438691328, 549755289600, 2199022206976, 8796090925056, 35184367894528

Offset: 0

Klaus Brockhaus, Dec 09 2009

Binomial transform of A048473; second binomial transform of A151821; third binomial transform of A010684; fourth binomial transform of A084633 without second term 0; fifth binomial transform of A168589.
Inverse binomial transform of A081625; second inverse binomial transform of A081626; third inverse binomial transform of A081627.
Partial sums of A010036.
Essentially first differences of A006095.
a(n) = A109241(n) converted from binary to decimal. - Robert Price, Jan 19 2016
a(n) is the area enclosed by a Hilbert curve with unit length segments after n iterations, when the start and end points are joined. - Jennifer Buckley, Apr 23 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jennifer Buckley, The Area Enclosed by a Hilbert Curve with Unit Length Segments
Gordon Hamilton, Cookie Monster Counts Cookies, Youtube video, 2010.
Index entries for linear recurrences with constant coefficients, signature (6,-8).

Cf. A006516 (2^(n-1)*(2^n-1)), A020522 (4^n-2^n), A048473 (2*3^n-1), A151821 (powers of 2, omitting 2 itself), A010684 (repeat 1, 3), A084633 (inverse binomial transform of repeated odd numbers), A168589 ((2-3^n)*(-1)^n), A081625 (2*5^n-3^n), A081626 (2*6^n-4^n), A081627 (2*7^n-5^n), A010036 (sum of 2^n, ..., 2^(n+1)-1), A006095 (Gaussian binomial coefficient [n, 2] for q=2), A171472, A171473.

[2*4^n-2^n: n in [0..30]]; // Vincenzo Librandi, Jul 17 2011

LinearRecurrence[{6,-8},{1,6},30] (* Harvey P. Dale, Aug 02 2020 *)

m=23; v=concat([1, 6], vector(m-2)); for(n=3, m, v[n]=6*v[n-1]-8*v[n-2]); v

a(n) = Sum_{k=1..2^n-1} k.
a(n) = 2*4^n - 2^n.
G.f.: 1/((1-2*x)*(1-4*x)).
a(n) = A006516(n+1).
a(n) = 4*a(n-1) + 2^n for n > 0, a(0)=1. - Vincenzo Librandi, Jul 17 2011
a(n) = Sum_{k=0..n} 2^(n+k). - Bruno Berselli, Aug 07 2013
a(n) = A020522(n+1)/2. - Hussam al-Homsi, Jun 06 2021
E.g.f.: exp(2*x)*(2*exp(2*x) - 1). - Stefano Spezia, Dec 10 2021

A060486 Tricoverings of an n-set.

Original entry on oeis.org

1, 0, 0, 5, 205, 11301, 904580, 101173251, 15207243828, 2975725761202, 738628553556470, 227636079973503479, 85554823285296622543, 38621481302086460057613, 20669385794052533823555309, 12966707189875262685801947906, 9441485712482676603570079314728

Offset: 0

Vladeta Jovovic, Mar 20 2001

nonn

A covering of a set is a tricovering if every element of the set is covered by exactly three blocks of the covering.

			There are 1 4-block tricovering, 3 5-block tricoverings and 1 6-block tricovering of a 3-set (cf. A060487), so a(3)=5.

Andrew Howroyd, Table of n, a(n) for n = 0..100

Row 3 of A188445.

Cf. A006095, A060483-A060485, (row sums of) A060487, A060090-A060095, A060069, A060070, A060051-A060053, A002718, A059443, A003462, A059945-A059951.

E.g.f. for k-block tricoverings of an n-set is exp(-x+x^2/2+(exp(y)-1)*x^3/3)*Sum_{k=0..inf}x^k/k!*exp(-1/2*x^2*exp(k*y))*exp(binomial(k, 3)*y).

Terms a(11) and beyond from Andrew Howroyd, Dec 15 2018

A134057 a(n) = binomial(2^n-1,2).

Original entry on oeis.org

0, 0, 3, 21, 105, 465, 1953, 8001, 32385, 130305, 522753, 2094081, 8382465, 33542145, 134193153, 536821761, 2147385345, 8589737985, 34359345153, 137438167041, 549754241025, 2199020109825, 8796086730753, 35184359505921

Offset: 0

Ross La Haye, Jan 11 2008, Jun 01 2008

Let P(A) be the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which x is not a subset of y and y is not a subset of x, or 1) x and y are intersecting but for which x is not a subset of y and y is not a subset of x, or 2) x and y are intersecting and for which either x is a proper subset of y or y is a proper subset of x.
Or: Number of connections between the nodes of the perfect depth n binary tree and the nodes of a perfect depth (n-1) binary tree. - Alex Ratushnyak, Jun 02 2013
a(n) is the number of positive entries in the positive rows and columns of a Walsh matrix of order 2^n. It is also the size of the smallest nontrivial conjugacy class in the general linear group GL(n,2). See the link "3-bit Walsh permutation...". - Tilman Piesk, Sep 15 2022

			a(2) = 3 because for P(A) = {{},{1},{2},{1,2}} we have for case 0 {{1},{2}} and we have for case 2 {{1},{1,2}}, {{2},{1,2}}. There are 0 {x,y} of P(A) in this example that fall under case 1.

Robert Israel, Table of n, a(n) for n = 0..1600
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6. - _Ross La Haye_, Feb 22 2009
Tilman Piesk, 3-bit Walsh permutation; conjugacy class 2+2 (Wikiversity)
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A000217, A000225, A000392, A032263, A028243, A171477.

[Binomial(2^n-1, 2): n in [0..30]]; // Vincenzo Librandi, Nov 30 2015

seq((2^n-1)*(2^(n-1)-1), n=0..100); # Robert Israel, Nov 30 2015

Table[Binomial[2^n - 1, 2], {n, 0, 30}] (* Vincenzo Librandi, Nov 30 2015 *)

a(n) = binomial(2^n-1, 2); \\ Michel Marcus, Nov 30 2015

print([(2**n-1)*(2**(n-1)-1) for n in range(23)])
# Alex Ratushnyak, Jun 02 2013

a(n) = (1/2)*(4^n - 3*2^n + 2) = 3*(Stirling2(n+1,4) + Stirling2(n+1,3)).
a(n) = 3 *A006095(n).
a(n) = (2^n-1)*(2^(n-1)-1). - Alex Ratushnyak, Jun 02 2013
a(n) = Stirling2(2^n - 1,2^n - 2).
G.f.: 3*x^2/(1-x)/(1-2*x)/(1-4*x). - Colin Barker, Feb 22 2012
a(n) = A000225(n)*A000225(n-1). - Michel Marcus, Nov 30 2015
a(n) = A000217(2^n-2). - Michel Marcus, Nov 30 2015
a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3). - Wesley Ivan Hurt, May 17 2021
E.g.f.: exp(x)*(exp(x) - 1)^2*(exp(x) + 2)/2. - Stefano Spezia, Apr 06 2022

A340030 Triangle read by rows: T(n,k) is the number of hypergraphs on n labeled vertices with k edges and all vertices having even degree, 0 <= k < 2^n.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 7, 7, 0, 0, 1, 1, 0, 0, 35, 105, 168, 280, 435, 435, 280, 168, 105, 35, 0, 0, 1, 1, 0, 0, 155, 1085, 5208, 22568, 82615, 247845, 628680, 1383096, 2648919, 4414865, 6440560, 8280720, 9398115, 9398115, 8280720, 6440560, 4414865, 2648919, 1383096, 628680, 247845, 82615, 22568, 5208, 1085, 155, 0, 0, 1

Offset: 0

Andrew Howroyd, Jan 09 2021

Hypergraphs are graphs in which an edge is connected to a nonempty subset of vertices rather than exactly two of them. An edge is a nonempty subset of vertices.
Equivalently, T(n,k) is the number of subsets of {1..2^n-1} with k elements such that the bitwise-xor of the elements is zero.
Also the coefficients of polynomials p_{n}(x) which have the representation
  p_{n}(x) = (x + 1)^(2*(n - 1) - 1)*q_{n - 1}(x), where q_{n}(x) are the polynomials defined in A340263, and n >= 2. - Peter Luschny, Jan 10 2021

			Triangle begins:
[0]  1;
[1]  1, 0;
[2]  1, 0, 0,  1;
[3]  1, 0, 0,  7,   7,   0,   0,   1;
[4]  1, 0, 0, 35, 105, 168, 280, 435, 435, 280, 168, 105, 35, 0, 0, 1;

Andrew Howroyd, Table of n, a(n) for n = 0..2046 (rows 0..10)
Wikipedia, Hypergraph.

Rows 3..8 are A002394, A010085, A010086, A010087, A010088, A010089.
Row sums are A016031(n+1).
Column k=3 gives A006095.
Cf. A058878, A281123, A340312, A340263.

T(n,k) = {(binomial(2^n-1, k) + (-1)^((k+1)\2)*(2^n-1)*binomial(2^(n-1)-1, k\2))/2^n}
{ for(n=0, 5, print(vector(2^n, k, T(n,k-1)))) }

T(n,k) = (binomial(2^n-1, k) + (-1)^ceiling(k/2)*(2^n-1)*binomial(2^(n-1)-1, floor(k/2)))/2^n.
T(n,2*k) + T(n,2*k+1) = binomial(2^n-1, k)/2^n = A281123(n,k).
T(n, k) = T(n, 2^n-1-k) for n >= 2.

A006118 Sum of Gaussian binomial coefficients [ n,k ] for q=4.

Original entry on oeis.org

1, 2, 7, 44, 529, 12278, 565723, 51409856, 9371059621, 3387887032202, 2463333456292207, 3557380311703796564, 10339081666350180289849, 59703612489554311631068958

Offset: 0

N. J. A. Sloane

nonn

J. Goldman and G.-C. Rota, The number of subspaces of a vector space, pp. 75-83 of W. T. Tutte, editor, Recent Progress in Combinatorics. Academic Press, NY, 1969.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.

Vincenzo Librandi, Table of n, a(n) for n = 0..80
S. Hitzemann, W. Hochstattler, On the combinatorics of Galois numbers, Discr. Math. 310 (2010) 3551-3557.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)

Row sums of triangle A022168.

[n le 2 select n else 2*Self(n-1)+(4^(n-2)-1)*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Aug 13 2016

Flatten[{1,RecurrenceTable[{a[n]==2*a[n-1]+(4^(n-1)-1)*a[n-2],a[0]==1,a[1]==2},a,{n,1,15}]}] (* Vaclav Kotesovec, Aug 21 2013 *)
Table[Sum[QBinomial[n, k, 4], {k, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Aug 13 2016 *)

a(n) = 2*a(n-1)+(4^(n-1)-1)*a(n-2), n>1. [Hitzemann and Hochstattler]. - R. J. Mathar, Aug 21 2013

a(n) ~ c * 4^(n^2/4), where c = EllipticTheta[3,0,1/4]/QPochhammer[1/4,1/4] = 2.189888057761... if n is even and c = EllipticTheta[2,0,1/4]/QPochhammer[1/4,1/4] = 2.182810929357... if n is odd. - Vaclav Kotesovec, Aug 21 2013

A006119 Sum of Gaussian binomial coefficients [ n,k ] for q=5.

Original entry on oeis.org

1, 2, 8, 64, 1120, 42176, 3583232, 666124288, 281268665344, 260766671206400, 549874114073747456, 2547649010961476288512, 26854416724405008878829568

Offset: 0

N. J. A. Sloane

nonn

J. Goldman and G.-C. Rota, The number of subspaces of a vector space, pp. 75-83 of W. T. Tutte, editor, Recent Progress in Combinatorics. Academic Press, NY, 1969.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.

Vincenzo Librandi, Table of n, a(n) for n = 0..75
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)

Row sums of triangle A022169.

[n le 2 select n else 2*Self(n-1)+(5^(n-2)-1)*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 13 2016

Flatten[{1,RecurrenceTable[{a[n]==2*a[n-1]+(5^(n-1)-1)*a[n-2],a[0]==1,a[1]==2},a,{n,1,15}]}] (* Vaclav Kotesovec, Aug 21 2013 *)
Table[Sum[QBinomial[n, k, 5], {k, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Aug 13 2016 *)

a(n) = 2*a(n-1)+(5^(n-1)-1)*a(n-2), (Goldman + Rota, 1969). - Vaclav Kotesovec, Aug 21 2013

a(n) ~ c * 5^(n^2/4), where c = EllipticTheta[3,0,1/5]/QPochhammer[1/5,1/5] = 1.845509008203... if n is even and c = EllipticTheta[2,0,1/5]/QPochhammer[1/5,1/5] = 1.829548121746... if n is odd. - Vaclav Kotesovec, Aug 21 2013

A169720 a(n) = (32^(n-1)-1)(3*2^n-1).

Original entry on oeis.org

1, 10, 55, 253, 1081, 4465, 18145, 73153, 293761, 1177345, 4713985, 18865153, 75479041, 301953025, 1207885825, 4831690753, 19327057921, 77308821505, 309236465665, 1236948221953, 4947797606401, 19791199862785, 79164818325505, 316659311050753, 1266637319700481

Offset: 0

Alice V. Kleeva (alice27353(AT)gmail.com), Jan 19 2010

A subsequence of the triangular numbers A000217.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Alice V. Kleeva, Grid for this sequence
Alice V. Kleeva, Illustration of initial terms
Robert Munafo, Sequence A169720, and two others by Alice V. Kleeva
Robert Munafo, Sequence A169720, and two others by Alice V. Kleeva [Cached copy, in pdf format, included with permission]
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A169721-A169727, A033484, A000217.

I:=[1, 10, 55]; [n le 3 select I[n] else 7*Self(n-1)-14*Self(n-2)+8*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Dec 03 2012

CoefficientList[Series[(1 + 3*x - x^2)/((1-x)*(1-2*x)*(1-4*x)), {x, 0, 30}], x] (* or *) LinearRecurrence[{7, -14, 8}, {1, 10, 55}, 30] (* Vincenzo Librandi, Dec 03 2012 *)

a(n)=polcoeff((1+3*x-x^2)/((1-x)*(1-2*x)*(1-4*x)+x*O(x^n)),n) \\ Paul D. Hanna, Apr 29 2010

G.f.: (1 + 3*x - x^2)/((1-x)*(1-2*x)*(1-4*x)). - Paul D. Hanna, Apr 29 2010
a(n) = A000217(A033484(n)). - Mitch Harris, Dec 02 2012
a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3). - Vincenzo Librandi, Dec 03 2012
a(n) = (3*A169726(n)-1)/2. - L. Edson Jeffery, Dec 03 2012
a(n) = A006095(n+2) +3*A006095(n+1) - A006905(n). - R. J. Mathar, Dec 04 2016

A006121 Sum of Gaussian binomial coefficients [ n,k ] for q=7.

Original entry on oeis.org

1, 2, 10, 116, 3652, 285704, 61946920, 33736398032, 51083363186704, 194585754101247008, 2061787082699360148640, 54969782721182164414355264

Offset: 0

N. J. A. Sloane

nonn

J. Goldman and G.-C. Rota, The number of subspaces of a vector space, pp. 75-83 of W. T. Tutte, editor, Recent Progress in Combinatorics. Academic Press, NY, 1969.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.

Vincenzo Librandi, Table of n, a(n) for n = 0..65
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)

[n le 2 select n else 2*Self(n-1)+(7^(n-2)-1)*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 13 2016

Flatten[{1,RecurrenceTable[{a[n]==2*a[n-1]+(7^(n-1)-1)*a[n-2],a[0]==1,a[1]==2},a,{n,1,15}]}] (* Vaclav Kotesovec, Aug 21 2013 *)
Table[Sum[QBinomial[n, k, 7], {k, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Aug 13 2016 *)

a(n) = 2*a(n-1)+(7^(n-1)-1)*a(n-2), (Goldman + Rota, 1969). - Vaclav Kotesovec, Aug 21 2013

a(n) ~ c * 7^(n^2/4), where c = EllipticTheta[3,0,1/7]/QPochhammer[1/7,1/7] = 1.537469386940... if n is even and c = EllipticTheta[2,0,1/7]/QPochhammer[1/7,1/7] = 1.499386995418... if n is odd. - Vaclav Kotesovec, Aug 21 2013

A006122 Sum of Gaussian binomial coefficients [ n,k ] for q=8.

Original entry on oeis.org

1, 2, 11, 148, 5917, 617894, 195118127, 162366823096, 409516908802369, 2724882133766162378, 54969878431787791720019, 2925929849527072623051175132, 472193512063977840212540697627493, 201069312609841845828101079279279809006

Offset: 0

N. J. A. Sloane

Generally, a(n) ~ c * q^(n^2/4), where c = EllipticTheta[3,0,1/q]/QPochhammer[1/q,1/q] if n is even and c = EllipticTheta[2,0,1/q]/QPochhammer[1/q,1/q] if n is odd. - Vaclav Kotesovec, Aug 21 2013

J. Goldman and G.-C. Rota, The number of subspaces of a vector space, pp. 75-83 of W. T. Tutte, editor, Recent Progress in Combinatorics. Academic Press, NY, 1969.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.

Vincenzo Librandi, Table of n, a(n) for n = 0..65
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)

[n le 2 select n else 2*Self(n-1)+(8^(n-2)-1)*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Aug 13 2016

Flatten[{1,RecurrenceTable[{a[n]==2*a[n-1]+(8^(n-1)-1)*a[n-2],a[0]==1,a[1]==2},a,{n,1,15}]}] (* Vaclav Kotesovec, Aug 21 2013 *)
Table[Sum[QBinomial[n, k, 8], {k, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Aug 13 2016 *)

a(n) = 2*a(n-1)+(8^(n-1)-1)*a(n-2), (Goldman + Rota, 1969). - Vaclav Kotesovec, Aug 21 2013

a(n) ~ c * 8^(n^2/4), where c = EllipticTheta[3,0,1/8]/QPochhammer[1/8,1/8] = 1.455061175158... if n is even and c = EllipticTheta[2,0,1/8]/QPochhammer[1/8,1/8] = 1.405381182498... if n is odd. - Vaclav Kotesovec, Aug 21 2013

A060483 Number of 5-block tricoverings of an n-set.

Original entry on oeis.org

3, 57, 717, 7845, 81333, 825237, 8300757, 83202645, 832809813, 8331237717, 83324947797, 833299785045, 8333199127893, 83332796486997, 833331185898837, 8333324743497045, 83333298973791573, 833333195894773077, 8333332783578305877, 83333331134311650645

Offset: 3

Vladeta Jovovic, Mar 20 2001

A covering of a set is a tricovering if every element of the set is covered by exactly three blocks of the covering.

Andrew Howroyd, Table of n, a(n) for n = 3..500
Index entries for linear recurrences with constant coefficients, signature (17,-84,148,-80).

Column k=5 of A060487.

Cf. A006095, A060484, A060485, A060486, A060090-A060095, A060069, A060070, A060051-A060053, A002718, A059443, A003462, A059945-A059951.

Vec(3*(1+2*x)/(x-1)/(2*x-1)/(4*x-1)/(10*x-1)+O(x^99)) \\ Charles R Greathouse IV, Jan 11 2013

a(n) = (1/5!)*(10^n - 15*4^n + 45*2^n - 40).
Generally, e.g.f. for k-block tricoverings of an n-set is exp(-x+x^2/2+(exp(y)-1)*x^3/3)*Sum_{k=0..inf}x^k/k!*exp(-1/2*x^2*exp(k*y))*exp(binomial(k, 3)*y).
G.f.: 3*x^3*(2*x+1) / ((x-1)*(2*x-1)*(4*x-1)*(10*x-1)). - Colin Barker, Jan 11 2013

More terms from Colin Barker, Jan 11 2013

cp's OEIS Frontend

A171476 a(n) = 6*a(n-1) - 8*a(n-2) for n > 1, a(0)=1, a(1)=6.

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A060486 Tricoverings of an n-set.

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A134057 a(n) = binomial(2^n-1,2).

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A340030 Triangle read by rows: T(n,k) is the number of hypergraphs on n labeled vertices with k edges and all vertices having even degree, 0 <= k < 2^n.

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A006118 Sum of Gaussian binomial coefficients [ n,k ] for q=4.

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Magma

Mathematica

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A006119 Sum of Gaussian binomial coefficients [ n,k ] for q=5.

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A169720 a(n) = (3*2^(n-1)-1)*(3*2^n-1).

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A171476 a(n) = 6a(n-1) - 8a(n-2) for n > 1, a(0)=1, a(1)=6.

A169720 a(n) = (32^(n-1)-1)(3*2^n-1).