A176059 -id:A176059 - cp's OEIS frontend

A047216 Numbers that are congruent to {1, 2} mod 5.

1, 2, 6, 7, 11, 12, 16, 17, 21, 22, 26, 27, 31, 32, 36, 37, 41, 42, 46, 47, 51, 52, 56, 57, 61, 62, 66, 67, 71, 72, 76, 77, 81, 82, 86, 87, 91, 92, 96, 97, 101, 102, 106, 107, 111, 112, 116, 117, 121, 122, 126, 127, 131, 132, 136, 137, 141, 142, 146, 147

Offset: 1

N. J. A. Sloane

Equivalently, numbers ending in 1, 2, 6 and 7. - Bruno Berselli, Sep 04 2018

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

Cf. A001622, A047209, A047211, A047212, A176059.

[(10*n-3*(-1)^n-9)/4 : n in [1..100]]; // Wesley Ivan Hurt, Dec 29 2016

A047216:=n->(10*n-3*(-1)^n-9)/4: seq(A047216(n), n=1..100); # Wesley Ivan Hurt, Dec 29 2016

Select[Range[0, 200], MemberQ[{1, 2}, Mod[#, 5]] &] (* Vladimir Joseph Stephan Orlovsky, Feb 12 2012 *)

a(n)=(n-1)\2*5+2-n%2 \\ Charles R Greathouse IV, Dec 22 2011

a(n) = 5*n-a(n-1)-7 for n>1, with a(1)=1. - Vincenzo Librandi, Aug 05 2010
G.f.: x*(1+x+3*x^2) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011
From Bruno Berselli, Mar 10 2012: (Start)
a(n) = (10*n-3*(-1)^n-9)/4.
a(n) = - A176059(n) + Sum_{i=0..n-1} A176059(i). (End)
From Wesley Ivan Hurt, Dec 29 2016: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3) for n > 3.
a(2*k) = 5*k-3, a(2*k-1) = 5*k-4. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(2-2*sqrt(5)/5)*Pi/10 + log(phi)/sqrt(5), where phi is the golden ratio (A001622). - Amiram Eldar, Dec 07 2021
E.g.f.: 3 + ((5*x - 9/2)*exp(x) - (3/2)*exp(-x))/2. - David Lovler, Aug 23 2022

A026532 Ratios of successive terms are 3, 2, 3, 2, 3, 2, 3, 2, ...

Original entry on oeis.org

1, 3, 6, 18, 36, 108, 216, 648, 1296, 3888, 7776, 23328, 46656, 139968, 279936, 839808, 1679616, 5038848, 10077696, 30233088, 60466176, 181398528, 362797056, 1088391168, 2176782336, 6530347008, 13060694016, 39182082048, 78364164096, 235092492288, 470184984576

Offset: 1

Clark Kimberling

Preface the series with a 1: (1, 1, 3, 6, 18, 36, ...); then the next term in the series = (1, 1, 3, 6, ...) dot (1, 2, 1, 2, ...). Example: 36 = (1, 1, 3, 6, 18) dot (1, 2, 1, 2, 1) = (1 + 2 + 3 + 12 + 18). - Gary W. Adamson, Apr 18 2009

Partial products of A176059. - Reinhard Zumkeller, Apr 04 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..700
Sean A. Irvine, Walks on Graphs.
José L. Ramírez, Bi-periodic incomplete Fibonacci sequences, Annales Mathematicae et Informaticae 42 (2013), 83-92. See the 1st column of Table 1 on p. 85.
Index entries for linear recurrences with constant coefficients, signature (0,6).

Cf. A026519, A026520, A026521, A026522, A026523, A026524, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026533, A026534, A027262, A027263, A027264, A027265, A027266.
Cf. A026534, A026549, A176059, A208131.
Cf. A038730, A038792, and A134511 for incomplete Fibonacci sequences, and A324242 for incomplete Lucas sequences.

a026532 n = a026532_list !! (n-1)
a026532_list = scanl (*) 1 $ a176059_list
-- Reinhard Zumkeller, Apr 04 2012

[(1/4)*(3-(-1)^n)*6^Floor(n/2) : n in [1..30]]; // Vincenzo Librandi, Jun 08 2011

FoldList[(2 + Boole[EvenQ@ #2]) #1 &, Range@ 28] (* or *)
CoefficientList[Series[x*(1+3x)/(1-6x^2), {x,0,31}], x] (* Michael De Vlieger, Aug 02 2017 *)
LinearRecurrence[{0,6},{1,3},30] (* Harvey P. Dale, Jul 11 2018 *)

a(n)=if(n%2,3,1)*6^(n\2) \\ Charles R Greathouse IV, Jul 02 2013

def a(n): return (3 if n%2 else 1)*6**(n//2)
print([a(n) for n in range(31)]) # Indranil Ghosh, Aug 02 2017

[(1/2)*6^((n-2)/2)*(3*(1+(-1)^n) + sqrt(6)*(1-(-1)^n)) for n in (1..30)] # G. C. Greubel, Dec 21 2021

a(n) = T(n, 0) + T(n, 1) + ... + T(n, 2n-2), T given by A026519.
From Benoit Cloitre, Nov 14 2003: (Start)
a(n) = (1/2)*(5+(-1)^n)*a(n-1) for n>1, a(1) = 1.
a(n) = (1/4)*(3-(-1)^n)*6^floor(n/2).  (End)
From Ralf Stephan, Feb 03 2004: (Start)
G.f.: x*(1+3*x)/(1-6*x^2).
a(n+2) = 6*a(n). (End)
a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011
a(n) = (1/2)*6^((n-2)/2)*(3*(1+(-1)^n) + sqrt(6)*(1-(-1)^n)). - G. C. Greubel, Dec 21 2021
Sum_{n>=1} 1/a(n) = 8/5. - Amiram Eldar, Feb 13 2023

New definition from Ralf Stephan, Dec 01 2004

Offset changed from 0 to 1 by Vincenzo Librandi, Jun 08 2011

A247387 Base-n state complexity of partitioned deterministic finite automaton (PDFA) for the periodic sequence (1234567)*.

Original entry on oeis.org

21, 42, 21, 42, 14, 8, 7, 21, 42, 21, 42, 14, 8, 7, 21, 42, 21, 42, 14, 8, 7, 21, 42, 21, 42, 14, 8, 7, 21, 42, 21, 42, 14, 8, 7, 21, 42, 21, 42, 14, 8, 7, 21, 42, 21, 42, 14, 8, 7, 21, 42, 21, 42, 14, 8, 7, 21, 42, 21, 42, 14, 8, 7, 21, 42, 21, 42, 14, 8, 7

Offset: 2

Vincenzo Librandi, Sep 16 2014

Klaus Sutner and Sam Tetruashvili, Inferring automatic sequences (see table on the p. 5).
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,1).

Cf. A176059, A217515, A217516, A217517, A217518.

&cat[[21, 42, 21, 42, 14, 8,7]: n in [0..30]];

CoefficientList[Series[(21 + 42 x + 21 x^2 + 42 x^3 + 14 x^4 + 8 x^5 + 7 x^6)/(1 - x^7), {x, 0, 40}], x]

G.f.: x^2*(21 + 42*x + 21*x^2 + 42*x^3 + 14*x^4 + 8*x^5 + 7*x^6)/(1-x^7).

A247391 Base-n state complexity of partitioned deterministic finite automaton (PDFA) for the periodic sequence (1234567891011).

Original entry on oeis.org

110, 55, 55, 55, 110, 110, 110, 55, 22, 12, 11, 110, 55, 55, 55, 110, 110, 110, 55, 22, 12, 11, 110, 55, 55, 55, 110, 110, 110, 55, 22, 12, 11, 110, 55, 55, 55, 110, 110, 110, 55, 22, 12, 11, 110, 55, 55, 55, 110, 110, 110, 55, 22, 12, 11, 110, 55, 55, 55

Offset: 2

Vincenzo Librandi, Sep 17 2014

Klaus Sutner and Sam Tetruashvili, Inferring automatic sequences (see table on the p. 5).
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,1).

Cf. A176059, A217515, A217516, A217517, A217518, A247387, A247390.

Cf. A010880.

&cat[[110,55,55,55,110,110,110,55,22,12,11]: n in [0..10]];

CoefficientList[Series[(110 + 55 x + 55 x^2 + 55 x^3 + 110 x^4 + 110 x^5 + 110 x^6 + 55 x^7 + 22 x^8 + 12 x^9 + 11 x^10)/(1-x^11), {x, 0, 60}], x]

G.f.: x^2*(110 + 55*x + 55*x^2 + 55*x^3 + 110*x^4 + 110*x^5 + 110*x^6 + 55*x^7 + 22*x^8 + 12*x^9 + 11*x^10)/(1-x^11).

a(n) = (1283*m^10 - 64570*m^9 + 1396065*m^8 - 16960020*m^7 + 127065939*m^6 - 605936100*m^5 + 1828078285*m^4 - 3335483030*m^3 + 3289569228*m^2 - 1288120680*m + 5443200)/453600 where m = (n mod 11). - Luce ETIENNE, Nov 04 2018

A247390 Base-n state complexity of partitioned deterministic finite automaton (PDFA) for the periodic sequence (12345678910)*.

Original entry on oeis.org

41, 40, 21, 11, 11, 40, 41, 20, 11, 10, 41, 40, 21, 11, 11, 40, 41, 20, 11, 10, 41, 40, 21, 11, 11, 40, 41, 20, 11, 10, 41, 40, 21, 11, 11, 40, 41, 20, 11, 10, 41, 40, 21, 11, 11, 40, 41, 20, 11, 10, 41, 40, 21, 11, 11, 40, 41, 20, 11, 10, 41, 40, 21, 11, 11, 40

Offset: 2

Vincenzo Librandi, Sep 16 2014

Klaus Sutner and Sam Tetruashvili, Inferring automatic sequences (see table on the p. 5).

Cf. A176059, A217515 - A217518, A247387.

&cat[[41,40,21,11,11,40,41,20,11,10]: n in [0..10]];

CoefficientList[Series [(41 + 40 x + 21 x^2 + 11 x^3 + 11 x^4 + 40 x^5 + 41 x^6 + 20 x^7 + 11 x^8 + 10 x^9)/(1 - x^10), {x, 0, 40}], x]

G.f.: x^2*(41 + 40*x + 21*x^2 + 11*x^3 + 11*x^4 + 40*x^5 + 41*x^6 + 20*x^7 + 11*x^8 + 10*x^9) / (1-x^10).

A368179 Square array read by ascending antidiagonals: row n is the trajectory of n under the A006368 map.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 2, 2, 1, 0, 5, 6, 3, 3, 1, 0, 6, 4, 9, 2, 2, 1, 0, 7, 9, 6, 7, 3, 3, 1, 0, 8, 5, 7, 9, 5, 2, 2, 1, 0, 9, 12, 4, 5, 7, 4, 3, 3, 1, 0, 10, 7, 18, 6, 4, 5, 6, 2, 2, 1, 0, 11, 15, 5, 27, 9, 6, 4, 9, 3, 3, 1, 0, 12, 8, 11, 4, 20, 7, 9, 6, 7, 2, 2, 1, 0

Offset: 0

Paolo Xausa, Dec 15 2023

			Array begins:
  [ 0]   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, ... = A000004
  [ 1]   1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ... = A000012
  [ 2]   2,  3,  2,  3,  2,  3,  2,  3,  2,  3,  2, ... = A010693
  [ 3]   3,  2,  3,  2,  3,  2,  3,  2,  3,  2,  3, ... = A176059
  [ 4]   4,  6,  9,  7,  5,  4,  6,  9,  7,  5,  4, ... = A180853
  [ 5]   5,  4,  6,  9,  7,  5,  4,  6,  9,  7,  5, ... = A180853 (shifted)
  [ 6]   6,  9,  7,  5,  4,  6,  9,  7,  5,  4,  6, ... = A180853 (shifted)
  [ 7]   7,  5,  4,  6,  9,  7,  5,  4,  6,  9,  7, ... = A180853 (shifted)
  [ 8]   8, 12, 18, 27, 20, 30, 45, 34, 51, 38, 57, ... = A028393
  [ 9]   9,  7,  5,  4,  6,  9,  7,  5,  4,  6,  9, ... = A180853 (shifted)
  [10]  10, 15, 11,  8, 12, 18, 27, 20, 30, 45, 34, ... = A180864 (shifted)
  ...    |   |   |
      A001477|A168221
             |
          A006368

Paolo Xausa, Table of n, a(n) for n = 0..11324 (antidiagonals 1..150 of the array, flattened).
Index entries for sequences related to 3x+1 (or Collatz) problem

Rows: A000004, A000012, A010693,  A028393, A028395, A094332, A176059, A180853, A180864, A182205, A217218, A223086, A223087, A223088.
Columns: A001477, A006368, A168221.
Main diagonal: A368180.

A006368[n_]:=If[OddQ[n],Floor[(3n+2)/4],3n/2];
A368179list[dmax_]:=With[{a=Reverse[Table[NestList[A006368,n-1,dmax-n],{n,dmax}]]},Array[Diagonal[a,#]&,dmax,1-dmax]];
A368179list[15] (* Generates 15 antidiagonals *)

A226294 Period 2: repeat [6, 4].

Original entry on oeis.org

6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6, 4, 6

Offset: 0

Richard R. Forberg, Jun 02 2013

A two number repeating sequence for constructing a summation sequence from negative to positive infinity containing all primes except 2 and 5.
Essentially the same as A168428, A101432 and A010711.
NOTE: This sequence has a shift in the starting value at index 0 relative to A010711. It is used here for the purpose stated with positive and negative indices making the formula in A010711 non-applicable.
This infinitely repeating sequence, a(n), of two numbers (6,4) starting with a(0) = 6, allows for the creation of an infinite summation sequence, s(n), extending from negative to positive infinity, using the formula below in parallel with how the same is done in A226276 using a different repeating sequence. Letting "s(n+)" be the set positive s(n) values, and "s(n-)" be the absolute value of the set of negative s(n) values, the following applies:
s(n+) includes all numbers with last digits of 1 and 7.
s(n-) includes all numbers with last digits of 3 and 9.
Therefore, s(n) includes all primes (except 2 and 5) without duplication.
This is one of only two such repeating patterns that accomplish this goal relative to the primes, while excluding all numbers with a last digit of 5. The other is (8,4,4,4) but with a different split between which primes occur as positive vs. negative numbers. See A226276 for details. Both patterns have the same density of primes relative to all s(n), and both, presumably, have the same average density of primes as positive vs. negative values of s(n).

			s(1) = 7, s(2) = 11, s(3) = 17, s(4) = 21, s(5) = 27, s(6) = 31;
s(-1) = -3, s(-2) = -9, s(-3) = -13, s(-4) = -19, s(-5) = -23, s(-6) = -29;

Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A010711, A101432, A168428, A226276.

&cat [[6, 4]^^50]; // Wesley Ivan Hurt, Jul 18 2016

A226294:=n->5+(-1)^n: seq(A226294(n), n=0..100); # Wesley Ivan Hurt, Mar 24 2014

Table[5+(-1)^n, {n,0,100}] (* Wesley Ivan Hurt, Mar 24 2014 *)

a(n)=6-n%2*2 \\ Charles R Greathouse IV, Jul 17 2016

a(n) = 5+(-1)^n = 2*A176059(n).
To generate the summation sequence s(n), start with s(0) = 1, and a(0) = 6.
For positive values of s(n):  s(n+1) = s(n) + a(n)
For negative values of s(n):  s(n-1) = s(n) - a(n-1). n is negative here.
See example values for s(n) below, for both positive and negative indices.
G.f.: ( 6+4*x ) / ( (1-x)*(1+x) ). - R. J. Mathar, Jun 12 2013
a(n) = a(n-2) for n>1. - Wesley Ivan Hurt, Jul 18 2016

A052616 Expansion of e.g.f. (3+2*x)/(1-x^2).

Original entry on oeis.org

3, 2, 6, 12, 72, 240, 2160, 10080, 120960, 725760, 10886400, 79833600, 1437004800, 12454041600, 261534873600, 2615348736000, 62768369664000, 711374856192000, 19207121117184000, 243290200817664000, 7298706024529920000, 102181884343418880000, 3372002183332823040000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 561.

Cf. A176059.

spec := [S,{S=Union(Sequence(Z),Sequence(Z),Sequence(Prod(Z,Z)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

With[{nn=30},CoefficientList[Series[(3+2x)/(1-x^2),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Dec 12 2021 *)

E.g.f.: (2*x+3)/(1-x^2).
Recurrence: {a(1)=2, a(0)=3, (-2-n^2-3*n)*a(n) + a(n+2) = 0}.
a(n) = Sum(1/2*(3*_alpha+2)*_alpha^(-1-n), _alpha=RootOf(-1+_Z^2))*n!.
a(n) = 3n! if n is even, 2n! otherwise.
a(n) = n!*A176059(n). - R. J. Mathar, Jun 03 2022
Sum_{n>=0} 1/a(n) = (5*e^2-1)/(12*e) = cosh(1)/3 + sinh(1)/2. - Amiram Eldar, Feb 02 2023

A176058 Decimal expansion of (3+sqrt(15))/2.

Original entry on oeis.org

3, 4, 3, 6, 4, 9, 1, 6, 7, 3, 1, 0, 3, 7, 0, 8, 4, 4, 2, 5, 8, 9, 6, 3, 2, 6, 9, 9, 8, 9, 1, 1, 9, 9, 8, 0, 5, 4, 1, 6, 4, 6, 0, 8, 5, 2, 6, 4, 5, 7, 9, 5, 4, 1, 3, 2, 9, 3, 7, 8, 6, 8, 8, 3, 0, 5, 6, 7, 4, 1, 5, 4, 5, 9, 6, 8, 4, 8, 9, 5, 1, 6, 7, 5, 9, 6, 4, 3, 6, 8, 8, 4, 2, 9, 3, 3, 6, 7, 5, 8, 9, 5, 8, 1, 5

Offset: 1

Klaus Brockhaus, Apr 07 2010

Continued fraction expansion of (3+sqrt(15))/2 is A176059.

			(3+sqrt(15))/2 = 3.43649167310370844258...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A010472 (decimal expansion of sqrt(15)), A176059 (repeat 3, 2).

First[RealDigits[(3 + Sqrt[15])/2, 10, 100]] (* Paolo Xausa, Jun 18 2024 *)

Equals 1.5 + A088543. [R. J. Mathar, Apr 12 2010]

A368227 Square array read by ascending antidiagonals: row n is the trajectory of n under the A006369 map.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 2, 2, 1, 0, 5, 5, 3, 3, 1, 0, 6, 7, 7, 2, 2, 1, 0, 7, 4, 9, 9, 3, 3, 1, 0, 8, 9, 5, 6, 6, 2, 2, 1, 0, 9, 11, 6, 7, 4, 4, 3, 3, 1, 0, 10, 6, 15, 4, 9, 5, 5, 2, 2, 1, 0, 11, 13, 4, 10, 5, 6, 7, 7, 3, 3, 1, 0, 12, 15, 17, 5, 13, 7, 4, 9, 9, 2, 2, 1, 0

Offset: 0

Paolo Xausa, Dec 18 2023

			Array begins:
  [ 0]   0,  0,  0,  0,  0,  0,  0,  0,  0,   0,  0, ... = A000004
  [ 1]   1,  1,  1,  1,  1,  1,  1,  1,  1,   1,  1, ... = A000012
  [ 2]   2,  3,  2,  3,  2,  3,  2,  3,  2,   3,  2, ... = A010693
  [ 3]   3,  2,  3,  2,  3,  2,  3,  2,  3,   2,  3, ... = A176059
  [ 4]   4,  5,  7,  9,  6,  4,  5,  7,  9,   6,  4, ... = A094328
  [ 5]   5,  7,  9,  6,  4,  5,  7,  9,  6,   4,  5, ... = A094328 (shifted)
  [ 6]   6,  4,  5,  7,  9,  6,  4,  5,  7,   9,  6, ... = A094328 (shifted)
  [ 7]   7,  9,  6,  4,  5,  7,  9,  6,  4,   5,  7, ... = A094328 (shifted)
  [ 8]   8, 11, 15, 10, 13, 17, 23, 31, 41,  55, 73, ... = A028394
  [ 9]   9,  6,  4,  5,  7,  9,  6,  4,  5,   7,  9, ... = A094328 (shifted)
  [10]  10, 13, 17, 23, 31, 41, 55, 73, 97, 129, 86, ... = A028394 (shifted)
  ...    |   |   |
      A001477|A168222
          A006369

Paolo Xausa, Table of n, a(n) for n = 0..11324 (antidiagonals 1..150 of the array, flattened).
Index entries for sequences related to 3x+1 (or Collatz) problem

Rows: A000004, A000012, A010693, A028394, A028396, A094328, A094329, A176059, A185589, A185590, A217729, A223083, A223084, A223085.
Columns: A001477, A006369, A168222.
Main diagonal: A368228.
Cf. A368179.

A006369[n_]:=If[Divisible[n,3],2n/3,Round[4n/3]];
A368227list[dmax_]:=With[{a=Reverse[Table[NestList[A006369,n-1,dmax-n],{n,dmax}]]},Array[Diagonal[a,#]&,dmax,1-dmax]];
A368227list[15] (* Generates 15 antidiagonals *)

cp's OEIS Frontend

A047216 Numbers that are congruent to {1, 2} mod 5.

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A026532 Ratios of successive terms are 3, 2, 3, 2, 3, 2, 3, 2, ...

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A247387 Base-n state complexity of partitioned deterministic finite automaton (PDFA) for the periodic sequence (1234567)*.

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A247391 Base-n state complexity of partitioned deterministic finite automaton (PDFA) for the periodic sequence (1234567891011).

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A247390 Base-n state complexity of partitioned deterministic finite automaton (PDFA) for the periodic sequence (12345678910)*.

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A368179 Square array read by ascending antidiagonals: row n is the trajectory of n under the A006368 map.

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A226294 Period 2: repeat [6, 4].

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