A132668 -id:A132668 - cp's OEIS frontend

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

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

Paul Barry, May 21 2003

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

			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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Hester Graves, The Minimal Euclidean Function on the Gaussian Integers, arXiv:1802.08281 [math.NT], 2018. See Definition 2.3 p.3.
Index entries for linear recurrences with constant coefficients, signature (0,4)

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.

[(5*2^n-(-2)^n)/4: n in [0..40]]; // Vincenzo Librandi, Aug 13 2011

CoefficientList[Series[(-3*x - 1)/(4*x^2 - 1), {x, 0, 200}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)

a(n)=([0,1; 4,0]^n*[1;3])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

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

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

Reinhard Zumkeller, Sep 11 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (1,3,-3).

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.

[(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

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

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 *)

def A087503(n): return (3+((n+1&1)<<1))*3**(n+1>>1)-3>>1 # Chai Wah Wu, Sep 02 2025

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

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

Hieronymus Fischer, Sep 19 2007

nonn

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

Partial sums of A084221.

Vincenzo Librandi, Table of n, a(n) for n = 1..3000
Index entries for linear recurrences with constant coefficients, signature (1, 4, -4).

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).

[4^Floor(n/2)+4^Floor((n+1)/2)/3-4/3: n in [1..40]]; // Vincenzo Librandi, Aug 17 2011

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

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 *)

vector(40, n, (3*4^floor(n/2) + 4^floor((n+1)/2) - 4)/3) \\ G. C. Greubel, Nov 08 2018

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) = pa(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

Hieronymus Fischer, Sep 19 2007

nonn

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

Index entries for linear recurrences with constant coefficients, signature (0, 5).

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.

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 *)

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

Hieronymus Fischer, Sep 19 2007

Partial sums of A133632.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,5,-5).

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).

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

Vec(x*(1 + 4*x) / ((1 - x) * (1 - 5*x^2)) + O(x^40)) \\ Colin Barker, Nov 25 2016

def A133629(n): return (5+((n&1)<<2))*5**(n>>1)-5>>2 # Chai Wah Wu, Sep 02 2025

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)

cp's OEIS Frontend

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

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

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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.

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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.

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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.

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Maple

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Formula

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