A084633 -id:A084633 - cp's OEIS frontend

A155559 a(n) = 2*A131577(n).

0, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592

Offset: 0

Paul Curtz, Jan 24 2009

Essentially the same as A131577, A046055, A011782, A000079 and A034008.

Index entries for linear recurrences with constant coefficients, signature (2).

Cf. A000079, A007582, A011782, A034008, A046055, A063376, A084633, A090129, A131577, A137173.

CoefficientList[ Series[ 2x/(1 - 2x), {x, 0, 32}], x] (* Robert G. Wilson v, Aug 08 2018 *)

a(n)=if(n,2^n,0) \\ Charles R Greathouse IV, Aug 01 2016

def A155559(n): return 1<Chai Wah Wu, Sep 05 2024

a(n) = A000079(n), n>0.
a(n) = (-1)^(n+1)*A084633(n+1).
a(n) + A155543(n) = 2^n+4^n = A063376(n) = 2*A007582(n) =2*A137173(2n+1).
Conjecture: a(n) = A090129(n+3)-A090129(n+2).
G.f.: 2*x/(1-2*x). - R. J. Mathar, Jul 23 2009
E.g.f.: exp(2*x) - 1. - Stefano Spezia, Aug 26 2025

Edited by R. J. Mathar, Jul 23 2009

Extended by Omar E. Pol, Nov 19 2012

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

Original entry on oeis.org

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

A010703 Period 2: repeat (3,5).

Original entry on oeis.org

3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5, 3, 5

Offset: 0

N. J. A. Sloane

From Klaus Brockhaus, Dec 10 2009: (Start)
Interleaving of A010701 and A010716.
Also continued fraction expansion of (15+sqrt(285))/10.
Also decimal expansion of 35/99.
Binomial transform of 3 followed by A084633 without initial terms 1,0.
Inverse binomial transform of A171497. (End)

Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A010701 (all 3's sequence), A010716 (all 5's sequence), A084633 (inverse binomial transform of repeated odd numbers), A171497.

&cat[ [3, 5]: n in [1..53] ]; // Klaus Brockhaus, Dec 10 2009

Table[If[OddQ[n], 3, 5], {n, 1, 50}] (* Stefan Steinerberger, Apr 10 2006 *)
PadRight[{},120,{3,5}] (* Harvey P. Dale, Sep 03 2012 *)

a(n)=3+n%2*2 \\ Charles R Greathouse IV, Nov 20 2011

From Klaus Brockhaus, Dec 10 2009: (Start)
a(n) = a(n-2) for n > 1; a(0) = 3, a(1) = 5.
G.f.: (3+5*x)/((1-x)*(1+x)). (End)
a(n) = 4 - (-1)^n. - Aaron J Grech, Aug 02 2024
E.g.f.: 3*cosh(x) + 5*sinh(x). - Stefano Spezia, Aug 04 2024

A118400 Triangle T, read by rows, where all columns of T are different and yet all columns of the matrix square T^2 (A118401) are equal; a signed version of triangle A087698.

Original entry on oeis.org

1, 1, -1, 1, 0, 1, -1, -1, -1, -1, 1, 2, 2, 2, 1, -1, -3, -4, -4, -3, -1, 1, 4, 7, 8, 7, 4, 1, -1, -5, -11, -15, -15, -11, -5, -1, 1, 6, 16, 26, 30, 26, 16, 6, 1, -1, -7, -22, -42, -56, -56, -42, -22, -7, -1, 1, 8, 29, 64, 98, 112, 98, 64, 29, 8, 1, -1, -9, -37, -93, -162, -210, -210, -162, -93, -37, -9, -1

Offset: 0

Paul D. Hanna, Apr 27 2006

Matrix inverse equals A118404. Row sums equal A084633. Signed version of: A087698 = maximum number of Boolean inputs at Hamming distance 2 for symmetric Boolean functions. This is an example of the fact that special matrices (cf. A118401) can have more than 2 signed matrix square-roots if the main diagonal is allowed to be signed.

			Triangle T begins:
1;
1,-1;
1, 0, 1;
-1,-1,-1,-1;
1, 2, 2, 2, 1;
-1,-3,-4,-4,-3,-1;
1, 4, 7, 8, 7, 4, 1;
-1,-5,-11,-15,-15,-11,-5,-1;
1, 6, 16, 26, 30, 26, 16, 6, 1;
-1,-7,-22,-42,-56,-56,-42,-22,-7,-1;
1, 8, 29, 64, 98, 112, 98, 64, 29, 8, 1;
-1,-9,-37,-93,-162,-210,-210,-162,-93,-37,-9,-1; ...
The matrix square is A118401:
1;
0, 1;
2, 0, 1;
-2, 2, 0, 1;
4,-2, 2, 0, 1;
-6, 4,-2, 2, 0, 1;
8,-6, 4,-2, 2, 0, 1;
-10, 8,-6, 4,-2, 2, 0, 1;
12,-10, 8,-6, 4,-2, 2, 0, 1; ...
in which all columns are equal.

Cf. A118401 (matrix square), A084633 (row sums), A087698 (unsigned version); A118404 (matrix inverse).

T(n,k)=polcoeff(polcoeff((1+2*x+2*x^2)/(1+x+x*y+x*O(x^n)),n,x)+y*O(y^k),k,y)

T(n,k)=if(n==1&k==0,1,(-1)^n*(binomial(n,k)-2*binomial(n-2,k-1)))

G.f.: A(x,y) = (1+2*x+2*x^2)/(1+x+x*y). G.f. of column k = (-1)^k*(1+2*x+2*x^2)/(1+x)^(k+1) for k>=0. T(n,k) = (-1)^n*[C(n,k) - 2*C(n-2,k-1)] for n>=k>=0 except that T(1,0)=1.

A130265 Triangle read by rows: matrix product A007318 * A051340.

Original entry on oeis.org

1, 2, 2, 4, 5, 3, 8, 10, 10, 4, 16, 19, 23, 17, 5, 32, 36, 46, 46, 26, 6, 64, 69, 87, 102, 82, 37, 7, 128, 134, 162, 204, 204, 134, 50, 8, 256, 263, 303, 387, 443, 373, 205, 65, 9, 512, 520, 574, 718, 886, 886, 634, 298, 82, 10

Offset: 0

Gary W. Adamson, May 18 2007

			First few rows of the triangle are:
   1;
   2,  2;
   4,  5,  3;
   8, 10, 10,   4;
  16, 19, 23,  17,  5;
  32, 36, 46,  46, 26,  6;
  64, 69, 87, 102, 82, 37,  7;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A001787 (row sums), A007318, A051340, A058877, A084633, A258431.

A130265:= func< n,k | k eq n select n+1 else (k+1)*Binomial(n,k) + (&+[Binomial(n, j+k): j in [1..n-k]]) >;
[A130265(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 18 2023

A051340 := proc(n,k)
    if k = n then
        n+1 ;
    elif k <= n then
        1;
    else
        0;
    end if;
end proc:
A130265 := proc(n,k)
    add( binomial(n,j)*A051340(j,k),j=k..n) ;
end proc:
seq(seq(A130265(n,k),k=0..n),n=0..15) ; # R. J. Mathar, Aug 06 2016

T[n_, k_]:= (k+1)*Binomial[n,k] + Binomial[n,k+1]*Hypergeometric2F1[1, k-n+1, k+2, -1];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 18 2023 *)

def A130265(n,k): return (k+1)*binomial(n,k) + sum(binomial(n, j+k) for j in range(1,n-k+1))
flatten([[A130265(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Mar 18 2023

Binomial transform of A051340.
From G. C. Greubel, Mar 18 2023: (Start)
T(n, k) = (k+1)*binomial(n,k) + Sum_{j=1..n-k} binomial(n, j+k).
T(n, k) = (k+1)*binomial(n,k) + binomial(n,k+1)*Hypergeometric2F1([1, k-n+1], [k+2], -1).
T(2*n, n) = (1/2)*T(2*n+1, n) = A258431(n+1).
Sum_{k=0..n} T(n, k) = A001787(n+1).
Sum_{k=0..n-1} T(n, k) = A058877(n+1), for n >= 1.
Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n*A084633(n). (End)

Missing term inserted by R. J. Mathar, Aug 06 2016

cp's OEIS Frontend

A155559 a(n) = 2*A131577(n).

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

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A010703 Period 2: repeat (3,5).

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A118400 Triangle T, read by rows, where all columns of T are different and yet all columns of the matrix square T^2 (A118401) are equal; a signed version of triangle A087698.

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A130265 Triangle read by rows: matrix product A007318 * A051340.

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