A074323 -id:A074323 - cp's OEIS frontend

A137206 First differences of A074323.

0, 2, -1, 4, -2, 8, -4, 16, -8, 32, -16, 64, -32, 128, -64, 256, -128, 512, -256, 1024, -512, 2048, -1024, 4096, -2048, 8192, -4096, 16384, -8192, 32768, -16384, 65536, -32768, 131072, -65536, 262144, -131072, 524288, -262144, 1048576, -524288

Offset: 0

Paul Curtz, Mar 05 2008

sign

Matthew House, Table of n, a(n) for n = 0..6605
Index entries for linear recurrences with constant coefficients, signature (0,2).

Cf. A000079, A131577, A135530.

From R. J. Mathar, Apr 22 2009: (Start)
G.f.: x*(2-x)/(1-2*x^2).
a(n) = (-1)^(n+1)*A135530(n-1). (End)
a(n) = 2*a(n-2). - Matthew House, Jan 15 2017

More terms from R. J. Mathar, Apr 22 2009

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

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

A162255 a(n) = 2*a(n-2) for n > 2; a(1) = 3, a(2) = 2.

Original entry on oeis.org

3, 2, 6, 4, 12, 8, 24, 16, 48, 32, 96, 64, 192, 128, 384, 256, 768, 512, 1536, 1024, 3072, 2048, 6144, 4096, 12288, 8192, 24576, 16384, 49152, 32768, 98304, 65536, 196608, 131072, 393216, 262144, 786432, 524288, 1572864, 1048576, 3145728, 2097152

Offset: 1

Klaus Brockhaus, Jun 29 2009

Apparently a(n) = A074323(n+1). a(n) = A072946(n-1) for n > 1.
Partial sums are in A164053.
Binomial transform is A135532 without initial term -1. Second binomial transform is A161938.

Index entries for linear recurrences with constant coefficients, signature (0, 2).

Cf. A074323, A072946, A164053, A135532, A161938.

LinearRecurrence[{0,2},{3,2},50] (* Harvey P. Dale, Aug 28 2012 *)

m=42; u=concat([3, 2], vector(m-2)); for(n=3, m, u[n]=2*u[n-2]); u

a(n) = (2^(1/4))^(3+2*n+(-1)^n) * (2-(-1)^n)/2.
G.f.: x*(3+2*x)/(1-2*x^2).
E.g.f.: cosh(sqrt(2)*x) + 3*sinh(sqrt(2)*x)/sqrt(2) - 1. - Stefano Spezia, May 26 2024

G.f. corrected, comments and cross-references added by Klaus Brockhaus, Aug 08 2009

Corrected by Harvey P. Dale, Aug 28 2012

A098790 a(n) = 2*a(n-1) + a(n-2) + 1, a(0) = 1, a(1) = 2.

Original entry on oeis.org

1, 2, 6, 15, 37, 90, 218, 527, 1273, 3074, 7422, 17919, 43261, 104442, 252146, 608735, 1469617, 3547970, 8565558, 20679087, 49923733, 120526554, 290976842, 702480239, 1695937321, 4094354882, 9884647086, 23863649055, 57611945197

Offset: 0

Creighton Dement, Oct 30 2004

nonn

Previous name was: a(n) = A048739(n) - A000129(n).

Partial sums of Pell numbers A000129 except omit next-to-last Pell number. E.g., 37 = 0+1+2+5+12+29 - 12.

M. Bicknell-Johnson and G. E. Bergum, The Generalized Fibonacci Numbers {C(n)}, C(n)=C(n-1)+C(n-2)+K, Applications of Fibonacci Numbers, 1986, pp. 193-205.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. Bicknell, A Primer on the Pell Sequence and related sequences, Fibonacci Quarterly, Vol. 13, No. 4, 1975, pp. 345-349.
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434.
Hermann Stamm-Wilbrandt, 4 interlaced bisections
Index entries for linear recurrences with constant coefficients, signature (3, -1, -1).

Cf. A006451, A000129, A048739, A124124, A024537, A074323, A052542.

a[0] = 1; a[1] = 2; a[n_] := a[n] = 2a[n - 1] + a[n - 2] + 1; Table[ a[n], {n, 0, 28}] (* Robert G. Wilson v, Nov 04 2004 *)
LinearRecurrence[{3,-1,-1},{1,2,6},31] (* Harvey P. Dale, Oct 15 2011 *)
CoefficientList[Series[(x^2 - x + 1)/((1 - x) (1 - 2 x - x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 14 2014 *)

a(n) = 2*a(n-1) + a(n-2) + 1, a(0) = 1, a(1) = 2.
G.f.: (x^2-x+1)/((1-x)(1-2x-x^2)).
a(n+1) = - A024537(n+1) + 2*A048739(n+1) - 2*A048739(n).
a(n) = - A024537(n) + A052542(n+1).
Partial sums of A074323. - Paul Barry, Mar 11 2007
a(n) = (sqrt(2)+1)^n*(3/4+sqrt(2)/4)+(sqrt(2)-1)^n*(3/4-sqrt(2)/4)*(-1)^n-1/2; - Paul Barry, Mar 11 2007
a(0)=1, a(1)=2, a(2)=6, a(n)=3*a(n-1)-a(n-2)-a(n-3). [Harvey P. Dale, Oct 15 2011]
a(2*n) = A124124(2*n+1). - Hermann Stamm-Wilbrandt, Aug 03 2014
a(2*n+1) = A006451(2*n+1). - Hermann Stamm-Wilbrandt, Aug 26 2014
a(n) = 7*a(n-2) - 7*a(n-4) + a(n-6), for n>5. - Hermann Stamm-Wilbrandt, Aug 26 2014
2*a(n) = A135532(n+1)-1. - R. J. Mathar, Jan 13 2023

More terms from Robert G. Wilson v, Nov 04 2004
Definition edited by N. J. A. Sloane, Aug 03 2014
New name from existing formula by Joerg Arndt, Aug 13 2014

A164073 a(n) = 2*a(n-2) for n > 2; a(1) = 1, a(2) = 3.

Original entry on oeis.org

1, 3, 2, 6, 4, 12, 8, 24, 16, 48, 32, 96, 64, 192, 128, 384, 256, 768, 512, 1536, 1024, 3072, 2048, 6144, 4096, 12288, 8192, 24576, 16384, 49152, 32768, 98304, 65536, 196608, 131072, 393216, 262144, 786432, 524288, 1572864, 1048576, 3145728, 2097152, 6291456

Offset: 1

Klaus Brockhaus, Aug 09 2009

Interleaving of A000079 and A007283.
Binomial transform is A048654. Second binomial transform is A111567. Third binomial transform is A081179 without initial 0. Fourth binomial transform is A164072. Fifth binomial transform is A164031.
Absolute second differences are the sequence itself. - Eric Angelini, Jul 30 2013
Least number having n - 1 Gaussian prime factors, counted with multiplicity, excluding units. See A239628 for a similar sequence. - T. D. Noe, Mar 31 2014
Writing the prime factorizations of the terms of this sequence, the exponents of prime factor 2 give the integers repeated (A004526), while the exponents of prime factor 3 give the sequence of alternating 0's and 1's (A000035). - Alonso del Arte, Nov 30 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Index entries for linear recurrences with constant coefficients, signature (0, 2).

Cf. A000079 (powers of 2), A007283 (3*2^n), A074323, A162255, A048654, A111567, A081179, A164072, A164031, A239628.

[ n le 2 select 2*n-1 else 2*Self(n-2): n in [1..42] ];

terms = 50; CoefficientList[Series[x * (1 + 3 * x)/(1 - 2 * x^2), {x, 0, terms}], x] (* T. D. Noe, Mar 31 2014 *)
Flatten[Table[{2^n, 3 * 2^n}, {n, 0, 31}]] (* Alonso del Arte, Nov 30 2016 *)
CoefficientList[Series[x (1 + 3 x)/(1 - 2 x^2), {x, 0, 44}], x] (* Michael De Vlieger, Dec 13 2016 *)

PARI
```
a(n) = (5 + (-1)^n) * 2^((2*n-9)\/4)
```

Vec(x*(1+3*x)/(1-2*x^2)+O(x^99)) \\ Charles R Greathouse IV, Dec 13 2016

a(n) = (5 + (-1)^n) * 2^(1/4 * (2*n - 1 + (-1)^n))/4.
G.f.: x*(1 + 3 * x)/(1 - 2 * x^2).
a(n) = A074323(n), n>=1.
a(n) = A162255(n-1), n>=2.
a(n) = A072946(n-2), n > 2. - R. J. Mathar, Aug 17 2009
a(n+3) = a(n + 2) * a(n + 1)/a(n). - Reinhard Zumkeller, Mar 04 2011
a(n) = (2/3)a(n - 1) for odd n > 1, a(n) = 3a(n - 1) for even n. - Alonso del Arte, Nov 30 2016

A134626 Sum-fill array starting with (1,2,4,8,16,...), powers of 2.

Original entry on oeis.org

1, 2, 1, 4, 3, 1, 8, 2, 4, 1, 16, 6, 3, 5, 1, 32, 4, 5, 4, 6, 1, 64, 12, 2, 7, 5, 7, 1, 128, 8, 8, 3, 9, 6, 8, 1, 256, 24, 6, 8, 4, 11, 7, 9, 1, 512, 16, 10, 2, 11, 5, 13, 8, 10, 1, 1024, 48, 16, 10, 7, 14, 6

Offset: 1

Clark Kimberling, Nov 04 2007

(Row 2) is possibly A074323 except for an initial 1. The sequence represents the para-sequence in which the "final ordering" << is given by 1 << ... << 4 << 3 << 2. Row n contains 1,2,3,...2n, but not 2n+1. Row n starts like row n of A134625; e.g., row 6 of A123625 and row of A134626 have the same first 16 terms.

			Starting with x = row 3, Step 1 gives
y = (1,5,4,7,3,8,5,7,2,10,8,14,6,...).
Delete the second 5,7,8,... leaving row 4:
(1,5,4,7,3,8,2,10,14,6,...).
Northwest corner:
1 2 4 8 16 32
1 3 2 6 4 12
1 4 3 5 2 8
1 5 4 7 3 8
1 6 5 9 4 11.

C. Kimberling, Proper self-containing sequences, fractal sequences and para-sequences, preprint, 2007.

Clark Kimberling, Self-Containing Sequences, Selection Functions, and Parasequences, J. Int. Seq. Vol. 25 (2022), Article 22.2.1.

Cf. A134625, A134627, A134628.

Row 1 is A000079. Row n>=2 is produced from row n by the sum-fill operation, defined on an arbitrary infinite or finite sequence x = (x(1), x(2), x(3), ...) by the following two steps: Step 1. Form the sequence x(1), x(1)+x(2), x(2), x(2)+x(3), x(3), x(3)+x(4), ...; i.e., fill the space between x(n) and x(n+1) by their sum. Step 2. Delete duplicates; i.e., letting y be the sequence resulting from Step 1, if y(n+h)=y(n) for some h>=1, then delete y(n+h).

A134693 a(n)=A133806(n)+A133806(n+6).

Original entry on oeis.org

3, 9, 6, 18, 12, 36, 24, 72, 48, 144, 96, 288, 192, 576, 384, 1152, 768, 2304, 1536, 4608, 3072, 9216, 6144, 18432, 12288, 36864, 24576, 73728, 49152, 147456, 98304, 294912, 196608, 589824, 393216, 1179648, 786432, 2359296, 1572864, 4718592, 3145728

Offset: 0

Paul Curtz, Jan 27 2008

nonn

a(n)=3*A074323(n+1).
a(2n)=A007283(n). a(2n+1)=3*A007283(n).
O.g.f.: 3(1+3x)/(1-2x^2). - R. J. Mathar, Jul 22 2008

Edited and extended by R. J. Mathar, Jul 22 2008

cp's OEIS Frontend

A137206 First differences of A074323.

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

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A162255 a(n) = 2*a(n-2) for n > 2; a(1) = 3, a(2) = 2.

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A098790 a(n) = 2*a(n-1) + a(n-2) + 1, a(0) = 1, a(1) = 2.

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A164073 a(n) = 2*a(n-2) for n > 2; a(1) = 1, a(2) = 3.

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A134626 Sum-fill array starting with (1,2,4,8,16,...), powers of 2.

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A134693 a(n)=A133806(n)+A133806(n+6).

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