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.

A134417 A007318 * Triangle with A133632 as the diagonal and the rest zeros.

Original entry on oeis.org

1, 1, 4, 1, 8, 5, 1, 12, 15, 20, 1, 16, 30, 80, 25, 1, 20, 50, 200, 125, 100, 1, 24, 75, 400, 375, 600, 125, 1, 28, 105, 700, 875, 2100, 875, 500, 1, 32, 140, 1120, 1750, 5600, 3500, 4000, 625, 1, 36, 180, 1680, 3150, 12600, 10500, 18000, 5625, 2500
Offset: 1

Views

Author

Gary W. Adamson, Oct 24 2007

Keywords

Comments

Row sums = A134418 (1, 5, 14, 48, 152, 496, 2600, 5184, ...).

Examples

			First few rows of the triangle:
  1;
  1,  4;
  1,  8,  5;
  1, 12, 15,  20;
  1, 16, 30,  80,  25;
  1, 20, 50,  20, 125, 100;
  1, 24, 75, 400, 375, 600, 125;
  ...

Crossrefs

Cf. A133632, A134418.

Formula

Binomial transform of an infinite lower triangular matrix with A133632 (1, 4, 5, 20, 25, 100, ...) as the main diagonal and the rest zeros.

A084221 a(n+2) = 4*a(n), with a(0)=1, a(1)=3.

Original entry on oeis.org

1, 3, 4, 12, 16, 48, 64, 192, 256, 768, 1024, 3072, 4096, 12288, 16384, 49152, 65536, 196608, 262144, 786432, 1048576, 3145728, 4194304, 12582912, 16777216, 50331648, 67108864, 201326592, 268435456, 805306368, 1073741824, 3221225472, 4294967296, 12884901888
Offset: 0

Views

Author

Paul Barry, May 21 2003

Keywords

Comments

Binomial transform is A060925. Binomial transform of A084222.
Sequences with similar recurrence rules: A016116 (multiplier 2), A038754 (multiplier 3), A133632 (multiplier 5). See A133632 for general formulas. - Hieronymus Fischer, Sep 19 2007
Equals A133080 * A000079. A122756 is a companion sequence. - Gary W. Adamson, Sep 19 2007

Examples

			Binary...............Decimal
1..........................1
11.........................3
100........................4
1100......................12
10000.....................16
110000....................48
1000000...................64
11000000.................192
100000000................256
1100000000...............768
10000000000.............1024
110000000000............3072, etc. - _Philippe Deléham_, Mar 21 2014

Links

Crossrefs

For partial sums see A133628. Partial sums for other multipliers p: A027383(p=2), A087503(p=3), A133629(p=5).
Other related sequences: A132666, A132667, A132668, A132669.
Cf. A000302, A133080, A133087, A164346.

Programs

Formula

a(n) = (5*2^n-(-2)^n)/4.
G.f.: (1+3*x)/((1-2*x)(1+2*x)).
E.g.f.: (5*exp(2*x) - exp(-2*x))/4.
a(n) = A133628(n) - A133628(n-1) for n>1. - Hieronymus Fischer, Sep 19 2007
Equals A133080 * [1, 2, 4, 8, ...]. Row sums of triangle A133087. - Gary W. Adamson, Sep 08 2007
a(n+1)-2a(n) = A000079 signed. a(n)+a(n+2)=5*a(n). First differences give A135520. - Paul Curtz, Apr 22 2008
a(n) = A074323(n+1)*A016116(n). - R. J. Mathar, Jul 08 2009
a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011
a(n) = Sum_{k=0..n+1} A181650(n+1,k)*2^k. - Philippe Deléham, Nov 19 2011
a(2*n) = A000302(n); a(2*n+1) = A164346(n). - Philippe Deléham, Mar 21 2014

Extensions

Edited by N. J. A. Sloane, Dec 14 2007

A087503 a(n) = 3*(a(n-2) + 1), with a(0) = 1, a(1) = 3.

Original entry on oeis.org

1, 3, 6, 12, 21, 39, 66, 120, 201, 363, 606, 1092, 1821, 3279, 5466, 9840, 16401, 29523, 49206, 88572, 147621, 265719, 442866, 797160, 1328601, 2391483, 3985806, 7174452, 11957421, 21523359, 35872266, 64570080, 107616801, 193710243, 322850406, 581130732
Offset: 0

Views

Author

Reinhard Zumkeller, Sep 11 2003

Keywords

Links

Crossrefs

Sequences with similar recurrence rules: A027383 (p=2), A133628 (p=4), A133629 (p=5).
Other related sequences for different p: A016116 (p=2), A038754 (p=3), A084221 (p=4), A133632 (p=5).
See A133629 for general formulas with respect to the recurrence rule parameter p.
Related sequences: A132666, A132667, A132668, A132669.

Programs

  • Magma
    [(3/2)*(3^Floor((n+1)/2)+3^Floor(n/2)-3^Floor((n-1)/2)-1): n in [0..40]]; // Vincenzo Librandi, Aug 16 2011
  • Maple
    A087503 := proc(n)
    option remember;
    if n <=1 then
        op(n+1,[1,3]) ;
    else
        3*procname(n-2)+3 ;
    end if;
end proc:
seq(A087503(n),n=0..20) ; # R. J. Mathar, Sep 10 2021
  • Mathematica
    RecurrenceTable[{a[0]==1,a[1]==3,a[n]==3(a[n-2]+1)},a,{n,40}] (* or *) LinearRecurrence[{1,3,-3},{1,3,6},40] (* Harvey P. Dale, Jan 01 2015 *)
  • Python
    def A087503(n): return (3+((n+1&1)<<1))*3**(n+1>>1)-3>>1 # Chai Wah Wu, Sep 02 2025

Formula

a(n) = a(n-1) + A038754(n). (i.e., partial sums of A038754).
From Hieronymus Fischer, Sep 19 2007, formulas adjusted to offset, Dec 29 2012: (Start)
G.f.: (1+2*x)/((1-3*x^2)*(1-x)).
a(n) = (3/2)*(3^((n+1)/2)-1) if n is odd, else a(n) = (3/2)*(5*3^((n-2)/2)-1).
a(n) = (3/2)*(3^floor((n+1)/2) + 3^floor(n/2) - 3^floor((n-1)/2)-1).
a(n) = 3^floor((n+1)/2) + 3^floor((n+2)/2)/2 - 3/2.
a(n) = A132667(a(n+1)) - 1.
a(n) = A132667(a(n-1) + 1) for n > 0.
A132667(a(n)) = a(n-1) + 1 for n > 0.
Also numbers such that: a(0)=1, a(n) = a(n-1) + (p-1)*p^((n+1)/2 - 1) if n is odd, else a(n) = a(n-1) + p^(n/2), where p=3. (End)
a(n) = A052993(n)+2*A052993(n-1). - R. J. Mathar, Sep 10 2021

Extensions

Additional comments from Hieronymus Fischer, Sep 19 2007
Edited by N. J. A. Sloane, May 04 2010. I merged two essentially identical entries with different offsets, so some of the formulas may need to be adjusted.
Formulas and MAGMA prog adjusted to offset 0 by Hieronymus Fischer, Dec 29 2012

A133628 a(1)=1, a(n) = a(n-1) + (p-1)*p^(n/2-1) if n is even, else a(n) = a(n-1) + p^((n-1)/2), where p=4.

Original entry on oeis.org

1, 4, 8, 20, 36, 84, 148, 340, 596, 1364, 2388, 5460, 9556, 21844, 38228, 87380, 152916, 349524, 611668, 1398100, 2446676, 5592404, 9786708, 22369620, 39146836, 89478484, 156587348, 357913940, 626349396, 1431655764, 2505397588
Offset: 1

Views

Author

Hieronymus Fischer, Sep 19 2007

Keywords

Comments

This is essentially a duplicate of A097164. - R. J. Mathar, Jun 08 2008
Partial sums of A084221.

Links

Crossrefs

Sequences with similar recurrence rules: A027383(p=2), A087503(p=3), A133629(p=5).
See A133629 for general formulas with respect to the recurrence rule parameter p.
Related sequences: A132666, A132667, A132668, A132669.
Other related sequences for different p: A016116(p=2), A038754(p=3), A084221(p=4), A133632(p=5).

Programs

  • Magma
    [4^Floor(n/2)+4^Floor((n+1)/2)/3-4/3: n in [1..40]]; // Vincenzo Librandi, Aug 17 2011
  • Maple
    a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=4*a[n-2]+4 od: seq(a[n], n=1..31); # Zerinvary Lajos, Mar 17 2008
  • Mathematica
    nxt[{n_,a_}]:={n+1,If[OddQ[n],a+3*4^((n+1)/2-1),a+4^(n/2)]}; Transpose[ NestList[ nxt,{1,1},30]][[2]] (* Harvey P. Dale, Mar 31 2013 *)
  • PARI
    vector(40, n, (3*4^floor(n/2) + 4^floor((n+1)/2) - 4)/3) \\ G. C. Greubel, Nov 08 2018

Formula

a(n) = Sum_{k=1..n} A084221(k).
G.f.: x*(1+3*x)/((1-4*x^2)*(1-x)).
a(n) = (4/3)*(4^(n/2)-1) if n is even, otherwise a(n) = (4/3)*(7*4^((n-3)/2)-1).
a(n) = (4/3)*(4^floor(n/2) + 4^floor((n-1)/2) - 4^floor((n-2)/2) - 1).
a(n) = 4^floor(n/2) + 4^floor((n+1)/2)/3 - 4/3.
a(n) = A132668(a(n+1)) - 1.
a(n) = A132668(a(n-1) + 1) for n > 0.
A132668(a(n)) = a(n-1) + 1 for n > 0.

A133629 a(1)=1, a(n) = a(n-1) + (p-1)*p^(n/2-1) if n is even, otherwise a(n) = a(n-1) + p^((n-1)/2), where p=5.

Original entry on oeis.org

1, 5, 10, 30, 55, 155, 280, 780, 1405, 3905, 7030, 19530, 35155, 97655, 175780, 488280, 878905, 2441405, 4394530, 12207030, 21972655, 61035155, 109863280, 305175780, 549316405, 1525878905, 2746582030, 7629394530, 13732910155, 38146972655, 68664550780
Offset: 1

Views

Author

Hieronymus Fischer, Sep 19 2007

Keywords

Comments

Partial sums of A133632.

Links

Crossrefs

Sequences with similar recurrence rules: A027383 (p=2), A087503 (p=3), A133628 (p=4).
Related sequences: A132666, A132667, A132668, A132669.
Other related sequences for different p: A016116 (p=2), A038754 (p=3), A084221 (p=4), A133632 (p=5).

Programs

  • Maple
    a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=5*a[n-2]+5 od: seq(a[n], n=1..29); # Zerinvary Lajos, Mar 17 2008
  • PARI
    Vec(x*(1 + 4*x) / ((1 - x) * (1 - 5*x^2)) + O(x^40)) \\ Colin Barker, Nov 25 2016
  • Python
    def A133629(n): return (5+((n&1)<<2))*5**(n>>1)-5>>2 # Chai Wah Wu, Sep 02 2025

Formula

a(n) = Sum_{k=1..n} A133632(k).
The following formulas are given for a general natural parameter p > 1 (p=5 for this sequence).
G.f.: x(1+(p-1)x)/((1-px^2)(1-x)).
a(n) = (p/(p-1))*(p^(n/2)-1) if n is even, otherwise a(n)=(p/(p-1))*((2p-1)*p^((n-3)/2)-1).
a(n) = (p/(p-1))*(p^floor(n/2) + p^floor((n-1)/2) - p^floor((n-2)/2)-1).
a(n) = p^floor(n/2) + (p^floor((n+1)/2)-p)/(p-1).
a(n) = A132669(a(n+1)) - 1.
a(n) = A132669(a(n-1)+1) for n > 0.
A132669(a(n)) = a(n-1)+1 for n > 0.
From Colin Barker, Nov 25 2016: (Start)
a(n) = 5*(5^(n/2) - 1)/4 for n even.
a(n) = (9*5^(n/2-1/2) - 5)/4 for n odd.
a(n) = a(n-1) + 5*a(n-2) - 5*a(n-3) for n > 3.
G.f.: x*(1 + 4*x) / ((1 - x) * (1 - 5*x^2)).
(End)

A163141 a(n) = 5*a(n-2) for n > 2; a(1) = 4, a(2) = 5.

Original entry on oeis.org

4, 5, 20, 25, 100, 125, 500, 625, 2500, 3125, 12500, 15625, 62500, 78125, 312500, 390625, 1562500, 1953125, 7812500, 9765625, 39062500, 48828125, 195312500, 244140625, 976562500, 1220703125, 4882812500, 6103515625, 24414062500
Offset: 1

Views

Author

Klaus Brockhaus, Jul 21 2009

Keywords

Comments

Apparently the same as A133632 without initial 1.
Binomial transform is A163069, second binomial transform is A163070, third binomial transform is A163071, fourth binomial transform is A108404 without initial 1, fifth binomial transform is A163072.

Links

Crossrefs

Cf. A133632, A108404, A163069, A163070, A163071, A163072.

Programs

  • Magma
    [ n le 2 select n+3 else 5*Self(n-2): n in [1..29] ];
  • Mathematica
    LinearRecurrence[{0,5},{4,5},30] (* Harvey P. Dale, Dec 20 2021 *)
  • PARI
    Vec(x*(4+5*x)/(1-5*x^2) + O(x^30)) \\ Jinyuan Wang, Mar 23 2020

Formula

a(n) = (5-3*(-1)^n)*5^(1/4*(2*n-1+(-1)^n))/2.
G.f.: x*(4+5*x)/(1-5*x^2).

A134418 Row sums of triangle A134417.

Original entry on oeis.org

1, 5, 14, 48, 152, 496, 1600, 5184, 16768, 54272, 175616, 568320, 1839104, 5951488
Offset: 1

Views

Author

Gary W. Adamson, Oct 24 2007

Keywords

Examples

			a(4) = 48 = sum of row 4 terms of triangle A134417: (1 + 12 + 15 + 20).
a(4) = 48 = (1, 3, 3, 1) dot (1, 4, 5, 20), given A133632 = (1, 4, 5, 20, 25, 100, ...).
a(3) = 14 = (1, -2, 1) dot (1, 6, 25), = (1, -12, 25), where A048875 = (1, 6, 25, 106, 449, 1902, ...).

Links

Crossrefs

Cf. A133632, A134417, A048875.
Cf. A048875.

Formula

Row sums of triangle A134417. Binomial transform of A133632. Inverse binomial transform of A048875.
Inverse binomial transform of A048875: (l, 6, 25, 106, 449, ...). - Gary W. Adamson, Nov 23 2007
G.f.: (1+3*x)/(1-2*x-4*x^2). - Vladimir Kruchinin, May 06 2011
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.