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 1-10 of 16 results. Next

A122757 Process number as a vertex through put triangular product function: m (In)-> {n-states}->m (Out) T(n,m)=m^2*g(n): g(n)=A084221[n].

Original entry on oeis.org

0, 1, 3, 4, 12, 16, 9, 27, 36, 108, 16, 48, 64, 192, 256, 25, 75, 100, 300, 400, 1200, 36, 108, 144, 432, 576, 1728, 2304, 49, 147, 196, 588, 784, 2352, 3136, 9408, 64, 192, 256, 768, 1024, 3072, 4096, 12288, 16384, 81, 243, 324, 972, 1296, 3888, 5184, 15552
Offset: 1

Views

Author

Roger L. Bagula, Sep 21 2006

Keywords

Comments

0 1, 3 4, 12, 16 9, 27, 36, 108 16, 48, 64, 192, 256 25, 75, 100, 300, 400, 1200

Crossrefs

Cf. A084221.

Programs

  • Mathematica
    g[n_] := If[Mod[n, 2] == 0, 2^(n), 2^n + 2^(n - 1)]; t[n_, m_] := m^2*g[n]; a = Table[Table[t[n, m], {n, 0, m}], {m, 0, 10}]; Flatten[a]

Formula

T(n,m)=m^2*g(n): g(n)=A084221[n]

A122758 Triangle read by rows: T(n,m) = 2*n^2*A084221(n) (n>=0, 0 <= m <= n).

Original entry on oeis.org

0, 2, 6, 8, 24, 32, 18, 54, 72, 216, 32, 96, 128, 384, 512, 50, 150, 200, 600, 800, 2400, 72, 216, 288, 864, 1152, 3456, 4608, 98, 294, 392, 1176, 1568, 4704, 6272, 18816, 128, 384, 512, 1536, 2048, 6144, 8192, 24576, 32768, 162, 486, 648, 1944, 2592, 7776
Offset: 1

Views

Author

Roger L. Bagula, Sep 21 2006

Keywords

Examples

			Triangle begins:
0
2, 6
8, 24, 32
18, 54, 72, 216
32, 96, 128, 384, 512
50, 150, 200, 600, 800, 2400

Crossrefs

Cf. A084221.

Programs

  • Mathematica
    f[n_] := If[Mod[n, 2] == 0, 2^(n + 1), 2^n + 2^(n + 1)]; T[n_, m_] := m^2*f[n]; a = Table[Table[T[n, m], {n, 0, m}], {m, 0, 10}]; Flatten[a]

Extensions

Edited by N. J. A. Sloane, Jun 23 2009

A137344 a(n)=4a(n-2). Also 3*A084221.

Original entry on oeis.org

3, 9, 12, 36, 48, 144, 192, 576, 768, 2304, 3072, 9216, 12288, 36864, 49152, 147456, 196608, 589824, 786432, 2359296, 3145728, 9437184, 12582912, 37748736, 50331648, 150994944, 201326592, 603979776, 805306368, 2415919104
Offset: 0

Views

Author

Paul Curtz, Apr 21 2008

Keywords

Links

Programs

  • Mathematica
    RecurrenceTable[{a[1]==3,a[2]==9,a[n]==4*a[n-2]},a,{n,30}] (* or *) LinearRecurrence[ {0,4},{3,9},30] (* Harvey P. Dale, Sep 20 2018 *)

Extensions

More terms from Harvey P. Dale, Sep 20 2018

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

A084222 a(n) = -2*a(n-1) + 3*a(n-2), with a(0)=1, a(1)=2.

Original entry on oeis.org

1, 2, -1, 8, -19, 62, -181, 548, -1639, 4922, -14761, 44288, -132859, 398582, -1195741, 3587228, -10761679, 32285042, -96855121, 290565368, -871696099, 2615088302, -7845264901, 23535794708, -70607384119, 211822152362, -635466457081, 1906399371248
Offset: 0

Views

Author

Paul Barry, May 21 2003

Keywords

Links

Crossrefs

Cf. A211866.

Programs

Formula

Binomial transform is A084221.
a(n) = (5-(-3)^n)/4.
G.f.: (1+4*x)/((1-x)*(1+3*x)).
E.g.f.: (5*exp(x)-exp(-3*x))/4.
For n > 1, abs(a(n) - a(n+1)) = 3^n. - Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jul 15 2003; corrected by Philippe Deléham, Dec 16 2007
a(n) = 9*a(n-2) - 10 with a(0) = 1 and a(1) = 2. - Philippe Deléham, Feb 24 2014
a(2n) = -A211866(n), n>0. - Philippe Deléham, Feb 24 2014

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.

A133632 a(1)=1, a(n) = (p-1)*a(n-1), if n is even, otherwise a(n) = p*a(n-2), where p = 5.

Original entry on oeis.org

1, 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

Hieronymus Fischer, Sep 19 2007

Keywords

Comments

Binomial transform = A134418: (1, 5, 14, 48, 152, 496, 1600, ...). Double binomial transform = A048875: (1, 6, 25, 106, 449, 1902, ...) - Gary W. Adamson, Oct 24 2007

Links

Crossrefs

For the partial sums see A133629.
Sequences with similar recurrence rules: A016116(p=2), A038754(p=3), A084221(p=4).
Partial sums for other p: A027383(p=2), A087503(p=3), A133628(p=4).
Other related sequences: A132666, A132667, A132668, A132669.
Cf. A134418, A048875.

Programs

  • Mathematica
    RecurrenceTable[{a[1]==1,a[2]==4,a[n]==If[EvenQ[n],4a[n-1],5a[n-2]]},a,{n,30}] (* Harvey P. Dale, Jan 14 2013 *)

Formula

The following formulas are given for a general natural parameter p > 1 (p = 5 for this sequence).
G.f.: g(x) = x(1+(p-1)x)/(1-px^2).
a(n) = p^floor((n-1)/2)*(p+(p-2)*(-1)^n)/2.
a(n) = A133629(n) - A133629(n-1) for n > 1.
a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011

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)

A097164 Expansion of (1+3x)/((1-x)(1-4x^2)).

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: 0

Views

Author

Paul Barry, Jul 30 2004

Keywords

Comments

Partial sums of A084221. a(n) = A097163(n+1)/4. Third binomial transform is A097165.
a(n+1) = 4*A097163(n). - Zerinvary Lajos, Mar 17 2008
See A133628 for an essentially identical sequence. - R. J. Mathar, Jun 08 2008

Links

Programs

  • 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
    CoefficientList[Series[(1+3x)/((1-x)(1-4x^2)),{x,0,50}],x] (* or *) LinearRecurrence[{1,4,-4},{1,4,8},50] (* Harvey P. Dale, Jul 11 2023 *)

Formula

a(n) = 5*2^n/2 - (-2)^n/6 - 4/3;
a(n) = a(n-1) + 4a(n-2) - 4a(n-3).
G.f. ( 1+3*x ) / ( (x-1)*(2*x+1)*(2*x-1) ). - R. J. Mathar, Jul 06 2011

A122756 Odd-indexed terms, a(n) = 2^n. Even-indexed terms, a(n) = floor(2^n+2^(n-1)).

Original entry on oeis.org

1, 2, 6, 8, 24, 32, 96, 128, 384, 512, 1536, 2048, 6144, 8192, 24576, 32768, 98304, 131072, 393216, 524288, 1572864, 2097152, 6291456, 8388608, 25165824, 33554432, 100663296, 134217728, 402653184, 536870912, 1610612736, 2147483648
Offset: 0

Views

Author

Roger L. Bagula, Sep 21 2006

Keywords

Comments

Row sums of triangle A133569. - Gary W. Adamson, Sep 16 2007

Examples

			Binary.................Decimal
1............................1
10...........................2
110..........................6
1000.........................8
11000.......................24
100000......................32
1100000.....................96
10000000...................128
110000000..................384
1000000000.................512
11000000000...............1536
100000000000..............2048
1100000000000.............6144
10000000000000............8192, etc. - _Philippe Deléham_, Mar 20 2014

Links

Crossrefs

Cf. A000079, A004171, A084221, A133569, A164908.

Programs

  • Magma
    [1] cat [(5*2^n-(-2)^n)/8: n in [2..40]]; // Vincenzo Librandi, Feb 10 2018
  • Mathematica
    a[n_] := If[Mod[n, 2] == 0, 2^(n + 1), 2^n + 2^(n + 1)] Table[a[n], {n, 0, 30}]
Join[{1, 2}, LinearRecurrence[{0, 4}, {6, 8}, 40]] (* Vincenzo Librandi, Feb 10 2018 *)
  • PARI
    A122756(n)=(3-bittest(n,0))<<(n-1) \\ M. F. Hasler, Feb 09 2018

Formula

a(n) = 2*A084221(n-1) for all n >= 1. [Corrected by M. F. Hasler, Feb 09 2018]
a(0)=1, a(1)=2, a(2)=6, a(n)=4*a(n-2) for n>=3. G.f.: (1+2*x+2*x^2)/(1-4*x^2). - Philippe Deléham, Dec 14 2007
a(n-1) = (5*2^n - (-2)^n)/8 for n>1. - Ralf Stephan, Jul 18 2013
a(2*n) = A164908(n), a(2*n+1) = A004171(n). - Philippe Deléham, Mar 20 2014

Extensions

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