A007583 -id:A007583 - cp's OEIS frontend

A239130 Smallest positive integer solution x = a(n) of (3^4)x - 2^ny = 1 for n >= 0.

1, 1, 1, 1, 1, 17, 49, 49, 177, 177, 177, 177, 2225, 2225, 2225, 18609, 18609, 84145, 84145, 84145, 608433, 1657009, 1657009, 1657009, 1657009, 1657009, 1657009, 1657009, 135874737, 404310193, 941181105, 2014922929, 2014922929

Offset: 0

Wolfdieter Lang, Mar 22 2014

This is instance m=4 of the m-family of smallest positive solutions [x0(m,n), y0(m,n)] of 3^m*x - 2^n*y = 1, n >= 0, m >= 0, described in a comment on A239125.
The companion sequence is y(n) = y0(4, n) = A239131(n), which is periodic with period length phi(3^4) = 54, where phi(n) = A000010(n) (Euler's totient).
The G.f. can be found from that of the periodic sequence y(n).

			n=0: 81*1 - 1*80 = 1;
n=1: 81*1 - 2*40 = 1;
n=2: 81*1 - 4*20 = 1;
n=3: 81*1 - 8*10 = 1;
n=4: 81*1 - 16*5 = 1;
n=5: 81*17 - 32*5 =1; ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Wolfdieter Lang, On Collatz' Words, Sequences and Trees, arXiv preprint arXiv:1404.2710 [math.NT], 2014 and J. Int. Seq. 17 (2014) # 14.11.7

Cf. A000010, A007583 (m=1), A234038 (m=2), A239125 (m=3), A239131.

[Floor(2^n*((41^(n+27) mod 81)/81))+1: n in [0..40]]; // Vincenzo Librandi, Mar 23 2014

Floor[Table[(2^n Mod[(41^(n + 27)), 81])/81 + 1, {n, 0, 40}]] (* Vincenzo Librandi, Mar 23 2014 *)

a(n) = (1 + 2^n*y0(4, n))/3^4, with y0(4, n) == ((3^4+1)/2)^(n + 3^3) (mod 3^4) = A239131(n), n >= 0.

a(n + 54) = 2^(54)*a(n) - (2^(54)-1)/3^4, n >= 0, from the y0(4, n) periodicity.

A321358 a(n) = (2*4^n + 7)/3.

Original entry on oeis.org

3, 5, 13, 45, 173, 685, 2733, 10925, 43693, 174765, 699053, 2796205, 11184813, 44739245, 178956973, 715827885, 2863311533, 11453246125, 45812984493, 183251937965, 733007751853, 2932031007405, 11728124029613, 46912496118445, 187649984473773, 750599937895085, 3002399751580333

Offset: 0

Paul Curtz, Nov 07 2018

Difference table:
3,  5, 13,  45,  173,  685,  2733, ...   (this sequence)
2,  8, 32, 128,  512, 2048,  8192, ...    A004171
6, 24, 96, 384, 1536, 6144, 24576, ...    A002023

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A010701, A010727, A020988, A083594, A002023, A004171, A155980, A171382, A193579.

a[n_]:= (2*4^n + 7)/3; Array[a, 20, 0] (* or *)
CoefficientList[Series[1/3 (7 E^x + 2 E^(4 x)), {x, 0, 20}], x]*Table[n!, {n, 0, 20}] (* Stefano Spezia, Nov 10 2018 *)

a(n) = (2*4^n + 7)/3; \\ Michel Marcus, Nov 08 2018

Vec((3 - 10*x) / ((1 - x)*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Nov 10 2018

O.g.f.: (3 - 10*x) / ((1 - x)*(1 - 4*x)). - Colin Barker, Nov 10 2018
E.g.f.: (1/3)*(7*exp(x) + 2*exp(4*x)). - Stefano Spezia, Nov 10 2018
a(n) = 5*a(n-1) - 4*a(n-2), a(0) = 3, a(1) = 5.
a(n) = 4*a(n-1) - 7, a(0) = 3.
a(n) = (2/3)*(4^n-1)/3 + 3.
a(n) = A171382(2*n) = A155980(2*n+2).
a(n) = A193579(n)/3.
a(n) = A007583(n) + 2 = A001045(2*n+1) + 2.

More terms from Michel Marcus, Nov 08 2018

A362783 Square array A(n,k) = (n^(2*k + 1) + 1)/(n + 1), n >= 0, k >= 0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 11, 7, 1, 1, 1, 43, 61, 13, 1, 1, 1, 171, 547, 205, 21, 1, 1, 1, 683, 4921, 3277, 521, 31, 1, 1, 1, 2731, 44287, 52429, 13021, 1111, 43, 1, 1, 1, 10923, 398581, 838861, 325521, 39991, 2101, 57, 1, 1, 1, 43691, 3587227, 13421773, 8138021, 1439671

Offset: 0

Juri-Stepan Gerasimov, May 03 2023

			Array begins:
=====================================================================
n/k |  0    1      2       3         4          5            6   ...
----+----------------------------------------------------------------
0   |  1    1      1       1         1          1            1   ...
1   |  1    1      1       1         1          1            1   ...
2   |  1    3     11      43       171        683         2731   ...
3   |  1    7     61     547      4921      44287       398581   ...
4   |  1   13    205    3277     52429     838861     13421773   ...
5   |  1   21    521   13021    325521    8138021    203450521   ...
6   |  1   31   1111   39991   1439671   51828151   1865813431   ...
   ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Columns k=0..3 are A000012, A002061, A060884, A060888.
Rows n=2..4 are A007583, A066443, A299960.
Main diagonal is A179897.

/* as array */ [[&+[(-n)^j: j in [0..2*k]]: k in [0..6]]: n in [0..6]]; // Juri-Stepan Gerasimov, May 06 2023

A(n,k) = (n^(2*k + 1) + 1)/(n + 1) \\ Andrew Howroyd, May 03 2023

A(n,k) = Sum_{j=0..2*k} (-n)^j.

a(49) corrected by Andrew Howroyd, Jan 20 2024

A377626 Cogrowth sequence of the 12-element group A4 = .

Original entry on oeis.org

1, 0, 1, 1, 1, 5, 4, 14, 21, 43, 91, 165, 354, 676, 1373, 2741, 5445, 10965, 21808, 43738, 87381, 174735, 349647, 698901, 1398318, 2796080, 5592445, 11185081, 22369145, 44740069, 89477724, 178957606, 357914197, 715826739, 1431658435, 2863308229, 5726626746

Offset: 0

Sean A. Irvine, Nov 02 2024

			a(6)=4 corresponds to the words SSSSSS = TTTTTT = STSTST = TSTSTS = 1.

Cf. A007583 (D6), A377627 (C6 x C2), A377656 (Dic2).

A087449 a(n) = n * 4^(n-1) + (2*4^n + 1) / 3.

Original entry on oeis.org

1, 4, 19, 91, 427, 1963, 8875, 39595, 174763, 764587, 3320491, 14330539, 61516459, 262843051, 1118481067, 4742359723, 20043180715, 84467690155, 355050629803, 1488921995947, 6230565890731, 26021775190699, 108485147273899

Offset: 0

Paul Barry, Sep 05 2003

Binomial transform of A064017.

Index entries for linear recurrences with constant coefficients, signature (9,-24,16).

Cf. A002697, A007583, A064017.

LinearRecurrence[{9,-24,16},{1,4,19},30] (* Harvey P. Dale, Apr 15 2018 *)

a(n) = my(p4 = 1<<(2*n)); n * p4 / 4 + (2*p4 + 1) / 3 \\ David A. Corneth, Apr 15 2018

G.f.: (1-5x+7x^2)/((1-x)(1-4x)^2).

a(n) = A002697(n) + A007583(n).

Name clarified by David A. Corneth, Apr 15 2018

A103279 Array read by antidiagonals, generated by the matrix M = [1,1,1;1,N,1;1,1,1].

Original entry on oeis.org

1, 1, 3, 1, 3, 8, 1, 3, 9, 22, 1, 3, 10, 27, 60, 1, 3, 11, 34, 81, 164, 1, 3, 12, 43, 116, 243, 448, 1, 3, 13, 54, 171, 396, 729, 1224, 1, 3, 14, 67, 252, 683, 1352, 2187, 3344, 1, 3, 15, 82, 365, 1188, 2731, 4616, 6561, 9136, 1, 3, 16, 99, 516, 2019, 5616, 10923, 15760

Offset: 0

Lambert Klasen (lambert.klasen(AT)gmx.net), Jan 27 2005

Consider the matrix M = [1,1,1;1,N,1;1,1,1]; Characteristic polynomial of M is x^3 + (-N - 2)*x^2 + (2*N - 2)*x.
Now (M^n)[1,1] is equivalent to the recursion a(1) = 1, a(2) = 3, a(n) = (N+2)a(n-1)+(2N-2)a(n-2). (This also holds for negative N and fractional N.)
a(n+1)/a(n) converges to the upper root of the characteristic polynomial ((N + 2) + sqrt((N - 2)^2 + 8))/2 for n to infinity.
Columns of array follow the polynomials:
1,
3,
N + 8,
N^2 + 4*N + 22,
N^3 + 4*N^2 + 16*N + 60,
N^4 + 4*N^3 + 18*N^2 + 56*N + 164,
N^5 + 4*N^4 + 20*N^3 + 68*N^2 + 188*N + 448,
N^6 + 4*N^5 + 22*N^4 + 80*N^3 + 248*N^2 + 608*N + 1224,
N^7 + 4*N^6 + 24*N^5 + 92*N^4 + 312*N^3 + 864*N^2 + 1920*N + 3344,
N^8 + 4*N^7 + 26*N^6 + 104*N^5 + 380*N^4 + 1152*N^3 + 2928*N^2 + 5952*N + 9136,
etc.

			Array begins:
1,3,8,22,60,164,448,1224,3344,9136,...
1,3,9,27,81,243,729,2187,6561,19683,...
1,3,10,34,116,396,1352,4616,15760,53808,...
1,3,11,43,171,683,2731,10923,43691,174763,...
1,3,12,54,252,1188,5616,26568,125712,594864,...
...

Cf. A103280 (for (M^n)[1, 2]), A028859 (for N=0), A000244 (for N=1), A007052 (for N=2), A007583 (for N=3), A083881 (for N=4), A026581 (for N=-1), A026532 (for N=-2), A026568.

T11(N, n) = if(n==1,1,if(n==2,3,(N+2)*r1(N,n-1)-(2*N-2)*r1(N,n-2))) for(k=0,10,print1(k,": ");for(i=1,10,print1(T11(k,i),","));print())

T(N, 1)=1, T(N, 2)=3, T(N, n)=(N+2)*T(N, n-1)-(2*N-2)*T(N, n-2).

A107733 Column 2 of the array in A107735.

Original entry on oeis.org

1, 3, 13, 11, 141, 43, 1485, 171, 15565, 683, 163021, 2731, 1707213, 10923, 17878221, 43691, 187223245, 174763, 1960627405, 699051, 20531956941, 2796203, 215013444813, 11184811, 2251650026701, 44739243, 23579585203405, 178956971, 246928622013645, 715827883, 2585870100909261, 2863311531

Offset: 3

N. J. A. Sloane, Jun 10 2005

nonn

The second bisection [3, 11, 43, 171, 683, ...] is A007583. - Jean-François Alcover, Oct 22 2019 [noticed by Paul Curtz in a private e-mail].

S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 483.

Index entries for linear recurrences with constant coefficients, signature (0,17,0,-80,0,128,0,-64).

Cf. A007583, A107735.

LinearRecurrence[{0, 17, 0, -80, 0, 128, 0, -64}, {1, 3, 13, 11, 141, 43, 1485, 171}, 32] (* Jean-François Alcover, Oct 22 2019 *)

a(n) = 1 + Sum_{j=1..g} 2^(2j-1) if n = 2g+2, = 1 + 4 Sum_{j=1..g} C(2g+1, 2j) 5^(j-1) if n = 2g+1.
From Chai Wah Wu, Jun 19 2016: (Start)
a(n) = 17*a(n-2) - 80*a(n-4) + 128*a(n-6) - 64*a(n-8) for n > 10.
G.f.: x^3*(-64*x^7 + 96*x^5 - 40*x^3 - 4*x^2 + 3*x + 1)/(64*x^8 - 128*x^6 + 80*x^4 - 17*x^2 + 1). (End)

More terms from Emeric Deutsch, Jun 22 2005

A113049 Triangle of sums of Jacobsthal numbers related to binomial(4n,n)/(3n+1) mod 4.

Original entry on oeis.org

3, 11, 13, 43, 45, 53, 171, 173, 181, 213, 683, 685, 693, 725, 853, 2731, 2733, 2741, 2773, 2901, 3413, 10923, 10925, 10933, 10965, 11093, 11605, 13653, 43691, 43693, 43701, 43733, 43861, 44373, 46421, 54613, 174763, 174765, 174773, 174805, 174933

Offset: 0

Paul Barry, Oct 11 2005

First column is A007583(n+1). Main diagonal is A072197. Conjecture: A113048(n)=2 only if n=T(i,j).

			Rows begin
3;
11, 13;
43, 45, 53;
171, 173, 181, 213;
683, 685, 693, 725, 683;
2731,2733, 2741, 2773, 2901, 3413;

T(n, k)=if(k<=n, J(2n+3)+2J(2k), 0), J(n)=A001045(n).

A165275 Table read by antidiagonals: T(n, k) is the k-th number with n-1 odd-power summands in its base 2 representation.

Original entry on oeis.org

1, 4, 2, 5, 3, 10, 16, 6, 11, 42, 17, 7, 14, 43, 170, 20, 8, 15, 46, 171, 682, 21, 9, 26, 47, 174, 683, 2730, 64, 12, 27, 58, 175, 686, 2731, 10922, 65, 13, 30, 59, 186, 687, 2734, 10923, 43690, 68, 18, 31, 62, 187, 698, 2735, 10926, 43691, 174762, 69, 19, 34

Offset: 1

Clark Kimberling, Sep 12 2009

For n>=0, row n is the ordered sequence of positive integers m such that the number of odd powers of 2 in the base 2 representation of m is n.
Every positive integer occurs exactly once in the array, so that as a sequence it is a permutation of the positive integers.
For even powers, see A165274. For the number of even powers of 2 in the base 2 representation of n, see A139351; for odd, see A139352.
Essentially, (Row 0)=A000695, (Column 1)=A020988, also possibly (Column 2)=A007583.
It appears that, for n>=3, a(t(n)) = 4*a(t(n-1))+2, where t(n) is the n-th triangular number t(n)=n(n+1)/2 (A000217). [John W. Layman, Sep 15 2009]

			Northwest corner:
  1....4....5...16...17...20...21...64
  2....3....6....7....8....9...12...13
  10..11...14...26...27...30...31...34
  42..43...46...47...58...59...62...63
Examples:
20 = 16 + 4 = 2^4 + 2^2, so that 20 is in row 0.
13 = 8 + 4 + 1 = 2^3 + 2^2 + 2^0, so that 13 is in row 1.

Cf. A139351, A139352, A165274, A165276, A165277, A165278, A165279.

Cf. A000217.

f[n_] := Total[(Reverse@IntegerDigits[n, 2])[[2 ;; -1 ;; 2]]]; T = GatherBy[ SortBy[Range[10^5], f], f]; Table[Table[T[[n - k + 1, k]], {k, n, 1, -1}], {n, 1, Length[T]}] // Flatten (* Amiram Eldar, Feb 04 2020*)

a(27) corrected and a(28)-a(54) added by John W. Layman, Sep 15 2009

More terms from Amiram Eldar, Feb 04 2020

A210985 Number of segments needed to draw the toothpick structure of A139250 as it is after 2^n stages.

Original entry on oeis.org

1, 3, 7, 19, 63, 235, 919, 3651, 14575

Offset: 0

Omar E. Pol, Sep 11 2012

It appears that this is also the partial sums of A039301.

It appears that this is also 1 together with A160128.

			For n = 3, after 2^3 stages the toothpick structure of A139250 contains 43 toothpicks (A139250(2^3) = 43). However the toothpick structure can be essentially represented by 19 segments (A139252(2^3) = 19), so a(3) = 19.

David Applegate, The movie version
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A007583, A039301, A139250, A139252, A147614, A160128.

a(n) = A139252(2^n).

cp's OEIS Frontend

A239130 Smallest positive integer solution x = a(n) of (3^4)*x - 2^n*y = 1 for n >= 0.

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A321358 a(n) = (2*4^n + 7)/3.

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A362783 Square array A(n,k) = (n^(2*k + 1) + 1)/(n + 1), n >= 0, k >= 0, read by antidiagonals.

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A377626 Cogrowth sequence of the 12-element group A4 = .

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A087449 a(n) = n * 4^(n-1) + (2*4^n + 1) / 3.

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A103279 Array read by antidiagonals, generated by the matrix M = [1,1,1;1,N,1;1,1,1].

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A107733 Column 2 of the array in A107735.

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A113049 Triangle of sums of Jacobsthal numbers related to binomial(4n,n)/(3n+1) mod 4.

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A239130 Smallest positive integer solution x = a(n) of (3^4)x - 2^ny = 1 for n >= 0.