A164073 -id:A164073 - cp's OEIS frontend

A164031 a(n) = ((2+3sqrt(2))(5+sqrt(2))^n+(2-3sqrt(2))(5-sqrt(2))^n)/4.

1, 8, 57, 386, 2549, 16612, 107493, 692854, 4456201, 28626368, 183771057, 1179304106, 7566306749, 48539073052, 311365675293, 1997258072734, 12811170195601, 82174766283128, 527090748332457, 3380887858812626

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 08 2009

nonn

Binomial transform of A164072. Fifth binomial transform of A164073.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-23).

Cf. A164072, A164073 (1, 3, 2, 6, 4, 12).

a:=[1,8];; for n in [3..25] do a[n]:=10*a[n-1]-23*a[n-2]; od; a; # Muniru A Asiru, Apr 04 2018

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((2+3*r)*(5+r)^n+(2-3*r)*(5-r)^n)/4: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 09 2009

CoefficientList[Series[(1 - 2*x)/(1 - 10*x + 23*x^2), {x,0,50}], x] (* or *) LinearRecurrence[{10,-23}, {1,8}, 50] (* G. C. Greubel, Sep 07 2017 *)

x='x+O('x^50); Vec((1-2*x)/(1-10*x+23*x^2)) \\ G. C. Greubel, Sep 07 2017

a(n) = 10*a(n-1) - 23*a(n-2) for n > 1; a(0) = 1, a(1) = 8.
G.f.: (1-2*x)/(1-10*x+23*x^2).
E.g.f.: exp(5*x)*(2*cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x))/2. - G. C. Greubel, Apr 03 2018

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 09 2009

A239628 Factored over the Gaussian integers, the least positive number having n prime factors counted multiply, including units -1, i, and -i.

Original entry on oeis.org

1, 9, 2, 6, 4, 12, 8, 16, 48, 144, 32, 96, 64, 192, 128, 256, 768, 2304, 512, 1536, 1024, 3072, 2048, 4096, 12288, 36864, 8192, 24576, 16384, 49152, 32768, 65536, 196608, 589824, 131072, 393216, 262144, 786432, 524288, 1048576, 3145728, 9437184, 2097152

Offset: 1

T. D. Noe, Mar 31 2014

nonn

Here -1, i, and -i are counted as factors. The factor 1 is counted only in a(1). All these numbers of products of 2^k, 3, and 9.

Similar to A164073, which gives the least integer having n prime factors (over the Gaussian integers) shifted by 1.

			a(2) = 9 because 9 = 3 * 3.
a(3) = 2 because 2 = -i * (1 + i)^2.
a(4) = 6 because 6 = -i * (1 + i)^2 * 3.

Cf. A001221, A001222 (integer factorizations).
Cf. A078458, A086275 (Gaussian factorizations).
Cf. A164073 (least number having n Gaussian factors, excluding units);
Cf. A239627 (number of Gaussian factors of n, including units).
Cf. A239629, A239630 (similar, but count distinct prime factors).

nn = 30; t = Table[0, {nn}]; n = 0; found = 0; While[found < nn, n++; cnt = Total[Transpose[FactorInteger[n, GaussianIntegers -> True]][[2]]]; If[cnt <= nn && t[[cnt]] == 0, t[[cnt]] = n; found++]]; t

A072946 Coefficient of the highest power of q in the expansion of nu(0)=1, nu(1)=b and for n>=2, nu(n)=bnu(n-1)+lambda(n-1)_q*nu(n-2) with (b,lambda)=(2,2), where (n)_q=(1+q+...+q^(n-1)) and q is a root of unity.

Original entry on oeis.org

1, 2, 6, 4, 12, 8, 24, 16, 48, 32, 96, 64, 192, 128, 384, 256, 768, 512, 1536, 1024, 3072, 2048, 6144, 4096, 12288, 8192, 24576, 16384, 49152, 32768, 98304, 65536, 196608, 131072, 393216, 262144, 786432, 524288, 1572864, 1048576, 3145728, 2097152

Offset: 0

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002

Instead of listing the coefficients of the highest power of q in each nu(n), if we listed the coefficients of the smallest power of q (i.e., constant terms), we get a weighted Fibonacci numbers described by f(0)=1, f(1)=1, for n>=2, f(n)=2f(n-1)+2f(n-2).

			nu(0)=1,
nu(1)=2,
nu(2)=6,
nu(3)=16+4q,
nu(4)=44+20q+12q^2,
nu(5)=120+80q+64q^2+40q^3+8q^4,
nu(6)=328+288q+280q^2+232q^3+168q^4+64q^5+24q^6.
By listing the coefficients of the highest power in each nu(n) we get 1,2,6,4,12,8,24,...

M. Beattie, S. Dăscălescu and S. Raianu, Lifting of Nichols Algebras of Type B_2, arXiv:math/0204075 [math.QA], 2002.
Index entries for linear recurrences with constant coefficients, signature (0, 2).

Essentially the same as A162255 and A164073.

Cf. A002605.

LinearRecurrence[{0,2},{1,2,6},50] (* Harvey P. Dale, Dec 31 2015 *)

For given b and lambda, the recurrence relation is given by; t(0)=1, t(1)=b, t(2)=b^2+lambda and for n>=3, t(n)=lambda*T(n-2).

O.g.f.: (1+2*x+4*x^2)/(1-2*x^2). - R. J. Mathar, Dec 05 2007

More terms from R. J. Mathar, Dec 05 2007

A135247 a(n) = 3a(n-1) + 2a(n-2) - 6*a(n-3).

Original entry on oeis.org

1, 3, 11, 33, 103, 309, 935, 2805, 8431, 25293, 75911, 227733, 683263, 2049789, 6149495, 18448485, 55345711, 166037133, 498111911, 1494335733, 4483008223, 13449024669, 40347076055, 121041228165, 363123688591, 1089371065773, 3268113205511, 9804339616533

Offset: 0

Paul Curtz, Feb 15 2008

This sequence interleaves A016133 and 3*A016133, see formulas. - Mathew Englander, Jan 08 2024

a(n) is the number of partitions of n into parts 1 (in three colors) and 2 (in two colors) where the order of colors matters. For example, the a(2)=11 such partitions (using parts 1, 1', 1'', 2, and 2') are 2, 2', 1+1, 1+1', 1+1'', 1'+1, 1'+1', 1'+1'', 1''+1, 1''+1', 1''+1''. For such partitions where the order of colors does not matter see A002624. - Joerg Arndt, Jan 18 2024

Sean A. Irvine, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,2,-6).

Cf. A016133.

a:=[1,3,11];; for n in [4..30] do a[n]:=3*a[n-1]+2*a[n-2]-6*a[n-3]; od; a; # G. C. Greubel, Nov 20 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/(1-3*x-2*x^2+6*x^3) )); // G. C. Greubel, Nov 20 2019

seq(coeff(series(1/(1-3*x-2*x^2+6*x^3), x, n+1), x, n), n = 0..30); # G. C. Greubel, Nov 20 2019

LinearRecurrence[{3,2,-6},{1,3,11},30] (* Harvey P. Dale, Jun 27 2015 *)

my(x='x+O('x^30)); Vec(1/(1-3*x-2*x^2+6*x^3)) \\ G. C. Greubel, Nov 20 2019

def A135247_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-3*x-2*x^2+6*x^3) ).list()
A135247_list(30) # G. C. Greubel, Nov 20 2019

G.f.: 1/((1-3*x)*(1-2*x^2)). - G. C. Greubel, Oct 04 2016
From Mathew Englander, Jan 08 2024: (Start)
a(n) = A010684(n) * A016133(floor(n/2)).
a(n) = 3*a(n-1) + A077957(n) for n >= 1.
a(n) = (A000244(n+2) - A164073(n+3))/7.
(End)

More terms from Harvey P. Dale, Jun 27 2015

Dropped two leading terms = 0. - Joerg Arndt, Jan 18 2024

A260211 Irregular triangle read by rows, T(n,k) is the decimal number conversion from an n-bit symmetric binary table arranged in ascending order for n > 1.

Original entry on oeis.org

0, 1, 0, 3, 0, 2, 5, 7, 0, 6, 9, 15, 0, 4, 10, 14, 17, 21, 27, 31, 0, 12, 18, 30, 33, 45, 51, 63, 0, 8, 20, 28, 34, 42, 54, 62, 65, 73, 85, 93, 99, 107, 119, 127, 0, 24, 36, 60, 66, 90, 102, 126, 129, 153, 165, 189, 195, 219, 231, 255

Offset: 1

Kival Ngaokrajang, Jul 19 2015

The sequence of row lengths is A060546(n).

Column 2 is A164073. See illustration.

			The irregular triangle begins:
n\k 0  1  2  3  4  5  6  7 ...
1   0  1
2   0  3
3   0  2  5  7
4   0  6  9 15
5   0  4 10 14 17 21 27 31
6   0 12 18 30 33 45 51 63
...

Kival Ngaokrajang, Illustration of initial terms

Cf. A060546, A164073.

Flatten[Table[Select[Range[0,2^k -1], #==IntegerReverse[#,2,k]&],{k,1,8}]] (* Ed Pegg Jr, May 03 2021 *)

cp's OEIS Frontend

A164031 a(n) = ((2+3*sqrt(2))*(5+sqrt(2))^n+(2-3*sqrt(2))*(5-sqrt(2))^n)/4.

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A239628 Factored over the Gaussian integers, the least positive number having n prime factors counted multiply, including units -1, i, and -i.

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A072946 Coefficient of the highest power of q in the expansion of nu(0)=1, nu(1)=b and for n>=2, nu(n)=b*nu(n-1)+lambda*(n-1)_q*nu(n-2) with (b,lambda)=(2,2), where (n)_q=(1+q+...+q^(n-1)) and q is a root of unity.

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A135247 a(n) = 3*a(n-1) + 2*a(n-2) - 6*a(n-3).

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GAP

Magma

Maple

Mathematica

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Sage

Formula

Extensions

A260211 Irregular triangle read by rows, T(n,k) is the decimal number conversion from an n-bit symmetric binary table arranged in ascending order for n > 1.

Views

Author

Keywords

Comments

Examples

Links

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Programs

Mathematica

A164031 a(n) = ((2+3sqrt(2))(5+sqrt(2))^n+(2-3sqrt(2))(5-sqrt(2))^n)/4.

A072946 Coefficient of the highest power of q in the expansion of nu(0)=1, nu(1)=b and for n>=2, nu(n)=bnu(n-1)+lambda(n-1)_q*nu(n-2) with (b,lambda)=(2,2), where (n)_q=(1+q+...+q^(n-1)) and q is a root of unity.

A135247 a(n) = 3a(n-1) + 2a(n-2) - 6*a(n-3).