cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A153237 a(n) = A000079(n) - A153130(n).

Original entry on oeis.org

0, 0, 0, 0, 9, 27, 63, 126, 252, 504, 1017, 2043, 4095, 8190, 16380, 32760, 65529, 131067, 262143, 524286, 1048572, 2097144, 4194297, 8388603, 16777215, 33554430, 67108860, 134217720, 268435449, 536870907, 1073741823, 2147483646
Offset: 0

Views

Author

Paul Curtz, Dec 21 2008

Keywords

Links

Programs

  • Mathematica
    LinearRecurrence[{3,-2,-1,3,-2},{0,0,0,0,9},40] (* Harvey P. Dale, Dec 26 2021 *)

Formula

a(n) = 9 *A153234(n). G.f. 9*x^4 / ( (x-1)*(2*x-1)*(1+x)*(x^2-x+1) ). - R. J. Mathar, Dec 17 2012

Extensions

Definition corrected by Omar E. Pol, Dec 24 2008
Edited by N. J. A. Sloane, Dec 31 2008

A172241 a(n) = (1/18)*(8^n - (-1)^n - 9).

Original entry on oeis.org

0, 3, 28, 227, 1820, 14563, 116508, 932067, 7456540, 59652323, 477218588, 3817748707, 30541989660, 244335917283, 1954687338268, 15637498706147, 125099989649180, 1000799917193443, 8006399337547548, 64051194700380387
Offset: 1

Views

Author

Ralf Stephan, Nov 20 2010

Keywords

Comments

It appears that a(n) = A153234(3*n-1). - Bruno Berselli, May 03 2011

Links

Programs

Formula

G.f.: x^2*(3 + 4*x)/((1 - x)*(1 + x)*(1 - 8*x)). - Adapted to the offset by Bruno Berselli, May 03 2011
a(2*k) = 8*a(2*k-1) + 3 and a(2*k+1) = 8*a(2*k) + 4 for k>0, a(1)=0. - Yosu Yurramendi, Dec 30 2016

A178742 Partial sums of floor(2^n/9).

Original entry on oeis.org

0, 0, 0, 0, 1, 4, 11, 25, 53, 109, 222, 449, 904, 1814, 3634, 7274, 14555, 29118, 58245, 116499, 233007, 466023, 932056, 1864123, 3728258, 7456528, 14913068, 29826148, 59652309, 119304632, 238609279, 477218573, 954437161
Offset: 0

Views

Author

Mircea Merca, Dec 26 2010

Keywords

Comments

Partial sums of A153234.

Examples

			a(6) = 0 + 0 + 0 + 0 + 1 + 3 + 7 = 11.

Links

Crossrefs

Cf. A153234.

Programs

  • Magma
    [&+[Floor(2^k/9): k in [0..n]]: n in [0..25]];  // Bruno Berselli, Apr 26 2011
  • Magma
    I:=[0,0,0,0,1,4]; [n le 6 select I[n] else 4*Self(n-1)-5*Self(n-2)+Self(n-3)+4*Self(n-4)-5*Self(n-5)+2*Self(n-6): n in [1..40]]; // Vincenzo Librandi, Mar 26 2014
  • Maple
    A178742 := proc(n) add( floor(2^i/9),i=0..n) ; end proc:
  • Mathematica
    CoefficientList[Series[x^4/((1-2x)(1+x)(1-x+x^2)(1-x)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 26 2014 *)
LinearRecurrence[{4,-5,1,4,-5,2},{0,0,0,0,1,4},40] (* Harvey P. Dale, Jan 25 2015 *)
  • PARI
    vector(30, n, n--; ((4*2^n-9*n+2)/18)\1) \\ G. C. Greubel, Jan 24 2019
  • Sage
    [floor((4*2^n-9*n+2)/18) for n in (0..30)] # G. C. Greubel, Jan 24 2019

Formula

a(n) = round((8*2^n - 18*n - 9)/36).
a(n) = floor((4*2^n - 9*n + 2)/18).
a(n) = ceiling((4*2^n - 9*n - 11)/18).
a(n) = round((4*2^n - 9*n - 4)/18).
a(n) = a(n-6) + 7*2^(n-5) - 3, n > 5.
a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 4*a(n-4) - 5*a(n-5) + 2*a(n-6).
G.f.: x^4 / ( (1-2*x)*(1+x)*(1-x+x^2)*(1-x)^2 ).

A224520 Numbers a(n) with property a(n) + a(n+4) = 2^(n+4) - 1 = A000225(n+4).

Original entry on oeis.org

0, 1, 3, 7, 15, 30, 60, 120, 240, 481, 963, 1927, 3855, 7710, 15420, 30840, 61680, 123361, 246723, 493447, 986895, 1973790, 3947580, 7895160, 15790320, 31580641, 63161283, 126322567, 252645135, 505290270, 1010580540
Offset: 0

Views

Author

Arie Bos, Apr 09 2013

Keywords

Comments

This is the case k=4 of a(n) + a(n+k) = 2^(n+k) - 1 = A000225(n+k). The sequences A000975, A077854 and A153234 correspond to cases k=1,2 and 3, respectively.

Links

Crossrefs

Cf. A000975, A077854, A153234.

Programs

  • Mathematica
    CoefficientList[Series[x/((1 - x)*(1 - 2*x)*(1 + x^4)), {x, 0, 50}], x] (* G. C. Greubel, Oct 11 2017 *)
LinearRecurrence[{3,-2,0,-1,3,-2},{0,1,3,7,15,30},40] (* Harvey P. Dale, Aug 23 2021 *)
  • PARI
    x='x+O('x^50); concat([0], Vec(x/((1-x)*(1-2*x)*(1+x^4)))) \\ G. C. Greubel, Oct 11 2017
  • Python
    print([2**(n+4)//17 for n in range(31)]) # Karl V. Keller, Jr., Jun 30 2021

Formula

a(n) + a(n+4) = 2^(n+4) - 1.
From Joerg Arndt, Apr 09 2013: (Start)
G.f.: x/((1-x)*(1-2*x)*(1+x^4)).
a(n) = +3*a(n-1) -2*a(n-2) -1*a(n-4) +3*a(n-5) -2*a(n-6). (End)
a(n) = floor(2^(n+4)/17). - Karl V. Keller, Jr., Jun 30 2021

A281166 a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) for n>2, a(0)=a(1)=1, a(2)=3.

Original entry on oeis.org

1, 1, 3, 8, 17, 33, 64, 127, 255, 512, 1025, 2049, 4096, 8191, 16383, 32768, 65537, 131073, 262144, 524287, 1048575, 2097152, 4194305, 8388609, 16777216, 33554431, 67108863, 134217728, 268435457, 536870913, 1073741824, 2147483647, 4294967295, 8589934592
Offset: 0

Views

Author

Paul Curtz, Jan 16 2017

Keywords

Comments

a(n) is the first sequence on three (with its first and second differences):
1, 1, 3, 8, 17, 33, 64, 127, ...;
0, 2, 5, 9, 16, 31, 63, 128, ..., that is 0 followed by A130752;
2, 3, 4, 7, 15, 32, 65, 129, ..., that is 2 followed by A130755;
1, 1, 3, 8, 17, 33, 64, 127, ..., this sequence.
The main diagonal is 2^n.
The sum of the first three lines is 3*2^n.
Alternated sum and subtraction of a(n) and its inverse binomial transform (period 3: repeat [1, 0, 2]) gives the autosequence of the first kind b(n):
0, 1, 1, 9, 17, 35, 63, 127, ...
1, 0, 8, 8, 18, 28, 64, 126, ...
-1, 8, 0, 10, 10, 36, 62, 134, ...
9, -8, 10, 0, 26, 26, 72, 118, ... .
The main diagonal is 0's. The first two upper diagonals are A259713.
The sum of the first three lines gives 9*A001045.
a(n) mod 9 gives a periodic sequence of length 6: repeat [1, 1, 3, 8, 8, 6].
a(n) = A130750(n-1) for n > 2. - Georg Fischer, Oct 23 2018

Links

Crossrefs

Cf. A000079, A001045, A005010, A007283, A014551 (a diagonal), A057079, A062510 (a diagonal), A128834, A130750, A130752, A130755, A153234, A153237, A259713.

Programs

  • Magma
    I:=[1,1,3]; [n le 3 select I[n] else 3*Self(n-1) - 3*Self(n-2) + 2*Self(n-3): n in [1..30]]; // G. C. Greubel, Jan 15 2018
  • Mathematica
    LinearRecurrence[{3, -3, 2}, {1, 1, 3}, 30] (* Jean-François Alcover, Jan 16 2017 *)
  • PARI
    Vec((1 - 2*x + 3*x^2) / ((1 - 2*x)*(1 - x + x^2)) + O(x^40)) \\ Colin Barker, Jan 16 2017

Formula

Binomial transform of the sequence of length 3: repeat [1, 0, 2].
a(n+3) = -a(n) + 9*2^n.
a(n) = 2^n - periodic 6: repeat [0, 1, 1, 0, -1, -1, 0].
a(n+6) = a(n) + 63*2^n.
a(n+1) = 2*a(n) - period 6: repeat [1, -1, -2, -1, 1, 2].
a(n) = 2^n - 2*sin(Pi*n/3)/sqrt(3). - Jean-François Alcover and Colin Barker, Jan 16 2017
G.f.: (1 - 2*x + 3*x^2)/((1 - 2*x)*(1 - x + x^2)). - Colin Barker, Jan 16 2017

A224521 Numbers a(n) with property a(n) + a(n+5) = 2^(n+5) - 1 = A000225(n+5).

Original entry on oeis.org

0, 1, 3, 7, 15, 31, 62, 124, 248, 496, 992, 1985, 3971, 7943, 15887, 31775, 63550, 127100, 254200, 508400, 1016800, 2033601, 4067203, 8134407, 16268815, 32537631, 65075262, 130150524, 260301048, 520602096, 1041204192, 2082408385, 4164816771, 8329633543
Offset: 0

Views

Author

Arie Bos, Apr 09 2013

Keywords

Comments

This is the case k=5 of a(n) + a(n+k) = 2^(n+k) - 1 = A000225(n+k). The sequences A000975, A077854, A153234 and A224520 correspond to cases k=1,2,3 and 4, respectively.

Links

Crossrefs

Cf. A000975, A077854, A153234, A224520.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x/((1-x)*(1-2*x)*(1+x^5)) )); // G. C. Greubel, Jun 06 2019
  • Mathematica
    CoefficientList[Series[x/((1-x)*(1-2*x)*(1+x^5)), {x,0,40}], x] (* G. C. Greubel, Oct 11 2017 *)
LinearRecurrence[{3,-2,0,0,-1,3,-2},{0,1,3,7,15,31,62},40] (* Harvey P. Dale, Apr 29 2020 *)
  • PARI
    my(x='x+O('x^40)); concat([0], Vec(x/((1-x)*(1-2*x)*(1+x^5)))) \\ G. C. Greubel, Oct 11 2017
  • Python
    print([2**(n+5)//33 for n in range(31)]) # Karl V. Keller, Jr., Jul 03 2021
  • Sage
    (x/((1-x)*(1-2*x)*(1+x^5))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jun 06 2019

Formula

a(n) + a(n+5) = 2^(n+5) - 1.
From Joerg Arndt, Apr 09 2013: (Start)
G.f.: x/((1-x)*(1+x)*(1-2*x)*(1-x+x^2-x^3+x^4)).
a(n) = +3*a(n-1) -2*a(n-2) -1*a(n-5) +3*a(n-6) -2*a(n-7). (End)
a(n) = floor(2^(n+5)/33). - Karl V. Keller, Jr., Jul 03 2021

A282153 Expansion of x*(1 - 2*x + 3*x^2)/((1 - x)*(1 - 2*x)*(1 - x + x^2)).

Original entry on oeis.org

0, 1, 2, 5, 13, 30, 63, 127, 254, 509, 1021, 2046, 4095, 8191, 16382, 32765, 65533, 131070, 262143, 524287, 1048574, 2097149, 4194301, 8388606, 16777215, 33554431, 67108862, 134217725, 268435453, 536870910, 1073741823, 2147483647, 4294967294, 8589934589
Offset: 0

Views

Author

Paul Curtz, Feb 07 2017

Keywords

Comments

After 0, partial sums of A281166.
Table of the first differences:
0, 1, 2, 5, 13, 30, 63, 127, 254, 509, 1021, 2046, ...
1, 1, 3, 8, 17, 33, 64, 127, 255, 512, 1025, 2049, ... A281166
0, 2, 5, 9, 16, 31, 63, 128, 257, 513, 1024, 2047, ...
2, 3, 4, 7, 15, 32, 65, 129, 256, 511, 1023, 2048, ...
repeat A281166.

Links

Crossrefs

Cf. A000079, A153237, A281166.

Programs

  • Mathematica
    LinearRecurrence[{4, -6, 5, -2}, {0, 1, 2, 5}, 34] (* Robert P. P. McKone, Feb 07 2021 *)
  • PARI
    concat(0, Vec(x*(1 - 2*x + 3*x^2) / ((1 - x)*(1 - 2*x)*(1 - x + x^2)) + O(x^50))) \\ Colin Barker, Feb 10 2017

Formula

From Colin Barker, Feb 10 2017: (Start)
G.f.: x*(1 - 2*x + 3*x^2)/((1 - x)*(1 - 2*x)*(1 - x + x^2)).
a(n) = 4*a(n-1) - 6*a(n-2) + 5*a(n-3) - 2*a(n-4) for n>3. (End)
From Bruno Berselli, Feb 10 2017: (Start)
a(n) = 2^n + ((-1)^floor(n/3) + (-1)^floor((n+1)/3))/2 - 2. Therefore:
a(3*k) = 8^k + (-1)^k - 2,
a(3*k+1) = 2*8^k + (-1)^k - 2,
a(3*k+2) = 4*8^k - 2. (End)
a(n+6*h) = a(n) + 2^n*(64^h - 1) with h>=0. For h=1, a(n+6) = a(n) + 63*2^n.
a(n) - (a(n) mod 9) = A153237(n) = 9*A153234(n).

Extensions

More terms from Colin Barker, Feb 10 2017
Showing 1-7 of 7 results.

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