A166525 -id:A166525 - cp's OEIS frontend

A166520 a(n) = (10n + 11(-1)^n + 5)/4.

1, 9, 6, 14, 11, 19, 16, 24, 21, 29, 26, 34, 31, 39, 36, 44, 41, 49, 46, 54, 51, 59, 56, 64, 61, 69, 66, 74, 71, 79, 76, 84, 81, 89, 86, 94, 91, 99, 96, 104, 101, 109, 106, 114, 111, 119, 116, 124, 121, 129, 126, 134, 131, 139, 136, 144, 141, 149, 146, 154, 151, 159

Offset: 1

Vincenzo Librandi, Oct 16 2009

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

Cf. A166519, A166521, A166522, A166523, A166524, A166525, A166526.

[5*n/2 + 11*(-1)^n/4 +5/4: n in [1..80]]; // Vincenzo Librandi, Sep 13 2013

Table[5 n / 2 + 11 (-1)^n / 4 + 5 / 4, {n, 100}] (* Vincenzo Librandi, Sep 13 2013 *)

def A166520(n): return (5*n -3 +11*((n+1)%2))//2
[A166520(n) for n in range(1,101)] # G. C. Greubel, Aug 03 2024

a(n) = 5*n - a(n-1) for n > 1, with a(1) = 1.
G.f. x*(1+8*x-4*x^2) / ( (1+x)*(1-x)^2 ). - R. J. Mathar, Mar 08 2011
From G. C. Greubel, May 16 2016: (Start)
E.g.f.: (5*(1 + 2*x)*exp(x) + 11*exp(-x) - 16)/4.
a(n) = a(n-1) + a(n-2) - a(n-3). (End)

A166521 a(n) = (6n + 7(-1)^n + 3)/2.

Original entry on oeis.org

1, 11, 7, 17, 13, 23, 19, 29, 25, 35, 31, 41, 37, 47, 43, 53, 49, 59, 55, 65, 61, 71, 67, 77, 73, 83, 79, 89, 85, 95, 91, 101, 97, 107, 103, 113, 109, 119, 115, 125, 121, 131, 127, 137, 133, 143, 139, 149, 145, 155, 151, 161, 157, 167, 163, 173, 169, 179, 175, 185

Offset: 1

Vincenzo Librandi, Oct 16 2009

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

Cf. A166519, A166520, A166522, A166523, A166524, A166525, A166526.

[(6*n+7*(-1)^n+3)/2: n in [1..80]]; // Vincenzo Librandi, Sep 13 2013

Table[(6 n + 7 (-1)^n + 3)/2, {n, 60}] (* Vincenzo Librandi, Sep 13 2013 *)
LinearRecurrence[{1,1,-1},{1,11,7},90] (* Harvey P. Dale, Apr 29 2018 *)

def A166521(n): return 3*n -2 +7*((n+1)%2)
[A166521(n) for n in range(1,101)] # G. C. Greubel, Aug 03 2024

a(n) = 6*n - a(n-1), for n > 1, with a(1) = 1.
G.f.: x*(1+10*x-5*x^2) / ((1+x)*(1-x)^2). - R. J. Mathar, Mar 08 2011
From G. C. Greubel, May 16 2016: (Start)
E.g.f.: (1/2)*(7*exp(-x) + 3*(1+2*x)*exp(x) -10).
a(n) = a(n-1) + a(n-2) - a(n-3). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = 1/5 + Pi/(2*sqrt(3)). - Amiram Eldar, Feb 24 2023

A166522 a(n) = 7*n - a(n-1), with a(1) = 1.

Original entry on oeis.org

1, 13, 8, 20, 15, 27, 22, 34, 29, 41, 36, 48, 43, 55, 50, 62, 57, 69, 64, 76, 71, 83, 78, 90, 85, 97, 92, 104, 99, 111, 106, 118, 113, 125, 120, 132, 127, 139, 134, 146, 141, 153, 148, 160, 155, 167, 162, 174, 169, 181, 176, 188, 183, 195, 190, 202, 197, 209, 204

Offset: 1

Vincenzo Librandi, Oct 16 2009

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

Cf. A166519, A166520, A166521, A166523, A166524, A166525, A166526.

A166522:= func< n | ( 7*n -5 +17*((n+1) mod 2) )/2 >;
[A166522(n): n in [1..100]]; // G. C. Greubel, Aug 03 2024

RecurrenceTable[{a[1]==1,a[n]==7n-a[n-1]},a,{n,60}] (* or *) LinearRecurrence[{1,1,-1},{1,13,8},60] (* Harvey P. Dale, Jun 07 2012 *)

def A166522(n): return ( 7*n -5 +17*((n+1)%2) )//2
[A166522(n) for n in range(1,101)] # G. C. Greubel, Aug 03 2024

G.f.: x*(1+12*x-6*x^2) / ( (1+x)*(1-x)^2 ). - R. J. Mathar, Mar 08 2011
a(n) = a(n-1) + a(n-2) - a(n-3), a(1)=1, a(2)=13, a(3)=8. - Harvey P. Dale, Jun 07 2012
E.g.f.: (1/4)*(17*exp(-x) + 7*(1 + 2*x)*exp(x) - 24). - G. C. Greubel, May 16 2016
a(n) = (1/4)*(14*n + 7 + 17*(-1)^n). - G. C. Greubel, Aug 03 2024

A166523 a(n) = 8*n - a(n-1), with n>1, a(1)=1.

Original entry on oeis.org

1, 15, 9, 23, 17, 31, 25, 39, 33, 47, 41, 55, 49, 63, 57, 71, 65, 79, 73, 87, 81, 95, 89, 103, 97, 111, 105, 119, 113, 127, 121, 135, 129, 143, 137, 151, 145, 159, 153, 167, 161, 175, 169, 183, 177, 191, 185, 199, 193, 207, 201, 215, 209, 223, 217, 231, 225, 239, 233

Offset: 1

Vincenzo Librandi, Oct 16 2009

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

Cf. A166519, A166520, A166521, A166522, A166524, A166525, A166526.

[n eq 1 select 1 else 8*n-Self(n-1): n in [1..70]]; // Vincenzo Librandi, Sep 13 2013

CoefficientList[Series[(1 +14 x -7 x^2)/((1+x) (1-x)^2), {x,0,60}], x] (* Vincenzo Librandi, Sep 13 2013 *)

def A166523(n): return 4*n - 3 + 10*((n+1)%2)
[A166523(n) for n in range(1,101)] # G. C. Greubel, Aug 03 2024

G.f.: x*(1+14*x-7*x^2) / ( (1+x)*(1-x)^2 ). - R. J. Mathar, Mar 08 2011
From G. C. Greubel, May 16 2016: (Start)
E.g.f.: 5*exp(-x) + 2*(1 + 2*x)*exp(x) - 7.
a(n) = a(n-1) + a(n-2) - a(n-3). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = 1/7 + (1/8 + 1/(4*sqrt(2)))*Pi. - Amiram Eldar, Feb 24 2023
a(n) = 4*n + 2 + 5*(-1)^n. - G. C. Greubel, Aug 03 2024

A166524 a(n) = 9*n - a(n-1), with n>1, a(1)=1.

Original entry on oeis.org

1, 17, 10, 26, 19, 35, 28, 44, 37, 53, 46, 62, 55, 71, 64, 80, 73, 89, 82, 98, 91, 107, 100, 116, 109, 125, 118, 134, 127, 143, 136, 152, 145, 161, 154, 170, 163, 179, 172, 188, 181, 197, 190, 206, 199, 215, 208, 224, 217, 233, 226, 242, 235, 251, 244, 260, 253

Offset: 1

Vincenzo Librandi, Oct 16 2009

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

Cf. A166519, A166520, A166521, A166522, A166523, A166525, A166526.

[n eq 1 select 1 else 9*n-Self(n-1): n in [1..80]]; // Vincenzo Librandi, Sep 13 2013

CoefficientList[Series[(1 +16 x -8 x^2)/((1+x) (1-x)^2), {x,0,80}], x] (* Vincenzo Librandi, Sep 13 2013 *)
LinearRecurrence[{1,1,-1},{1,17,10},60] (* Harvey P. Dale, Dec 24 2014 *)

def A166524(n): return (9*n - 7 + 23*((n+1)%2))//2
[A166524(n) for n in range(1, 101)] # G. C. Greubel, Aug 04 2024

G.f.: x*(1+16*x-8*x^2) / ( (1+x)*(1-x)^2 ). - R. J. Mathar, Mar 08 2011
From G. C. Greubel, May 16 2016: (Start)
E.g.f.: (1/4)*(23*exp(-x) + 9*(1 + 2*x)*exp(x) - 32).
a(n) = a(n-1) + a(n-2) - a(n-3). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = 1/8 + cot(Pi/9)*Pi/9. - Amiram Eldar, Feb 24 2023
a(n) = (1/4)*(18*n + 9 + 23*(-1)^n). - G. C. Greubel, Aug 04 2024

A166526 a(n) = 12*n - a(n-1), with n>1, a(1)=1.

Original entry on oeis.org

1, 23, 13, 35, 25, 47, 37, 59, 49, 71, 61, 83, 73, 95, 85, 107, 97, 119, 109, 131, 121, 143, 133, 155, 145, 167, 157, 179, 169, 191, 181, 203, 193, 215, 205, 227, 217, 239, 229, 251, 241, 263, 253, 275, 265, 287, 277, 299, 289, 311, 301, 323, 313, 335, 325, 347

Offset: 1

Vincenzo Librandi, Oct 16 2009

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

Cf. A166519, A166520, A166521, A166522, A166523, A166524, A166525.

[n eq 1 select 1 else 12*n-Self(n-1): n in [1..80]]; // Vincenzo Librandi, Sep 13 2013

RecurrenceTable[{a[1]==1,a[n]==12n-a[n-1]},a[n],{n,60}] (* or *) LinearRecurrence[{1,1,-1},{1,23,13},60] (* Harvey P. Dale, Aug 10 2011 *)
CoefficientList[Series[(1 + 22 x - 11 x^2) / ((x - 1)^2 (1 + x)), {x, 0, 60}], x] (* Vincenzo Librandi, Sep 13 2013 *)

def A166526(n): return 6*n - 5 + 16*((n+1)%2)
[A166526(n) for n in range(1, 101)] # G. C. Greubel, Aug 04 2024

From Harvey P. Dale, Aug 10 2011: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3), a(1)=1, a(2)=23, a(3)=13, for n>3.
G.f.: x*(1+22*x-11*x^2)/((x-1)^2*(1+x)). (End)
E.g.f.: 8*exp(-x) + 3*(1 + 2*x)*exp(x) - 11. - G. C. Greubel, May 16 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = 1/11 + (1/6 + 1/(4*sqrt(3)))*Pi. - Amiram Eldar, Feb 24 2023
a(n) = 6*n + 3 + 8*(-1)^n. - G. C. Greubel, Aug 04 2024

A166517 a(n) = (3 + 5(-1)^n + 6n)/4.

Original entry on oeis.org

2, 1, 5, 4, 8, 7, 11, 10, 14, 13, 17, 16, 20, 19, 23, 22, 26, 25, 29, 28, 32, 31, 35, 34, 38, 37, 41, 40, 44, 43, 47, 46, 50, 49, 53, 52, 56, 55, 59, 58, 62, 61, 65, 64, 68, 67, 71, 70, 74, 73, 77, 76, 80, 79, 83, 82, 86, 85, 89, 88, 92, 91, 95, 94, 98, 97, 101, 100, 104, 103, 107

Offset: 0

Vincenzo Librandi, Oct 16 2009

A sequence defined by a(1)=1, a(n)=k*n-a(n-1), k a constant parameter, has recurrence a(n)= 3*a(n-1) -3*a(n-2) +a(n-3). Its generating function is x*(1+2*(k-1)*x+(1-k)*x^2)/((1+x)*(1-x)^2). The closed form is a(n) = k*n/2+k/4+(-1)^n*(3*k/4-1). This applies with k=3 to this sequence here, and for example to sequences A165033, and A166519-A166525. - R. J. Mathar, Oct 17 2009
From Paul Curtz, Feb 20 2010: (Start)
Also: A001651, terms swapped by pairs.
a(n) mod 9 defines a period-6 sequence which is a permutation of A141425. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A001651, A010685, A010716, A016777, A016945, A073010.

[(3 +5*(-1)^n+6*n)/4: n in [0..80]]; // Vincenzo Librandi, Sep 13 2013

CoefficientList[Series[(2 x^2 - x + 2)/((1 + x) (x - 1)^2), {x, 0, 80}], x] (* Harvey P. Dale, Mar 25 2011 *)
Table[(3 + 5 (-1)^n + 6 n) / 4, {n, 0, 100}] (* Vincenzo Librandi, Sep 13 2013 *)

a(n) = 3*n - a(n-1).
From Paul Curtz, Feb 20 2010: (Start)
a(n+1)-a(n) = (-1)^(n+1)*A010685(n).
Second differences: |a(n+2)-2*a(n+1)+a(n)| = A010716(n).
a(2*n) + a(2*n+1) = A016945(n) = 6*n+3.
a(2*n) = A016945(n).
a(2*n+1) = A016777(n). (End)
G.f. ( 2-x+2*x^2 ) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Mar 08 2011
E.g.f.: (1/4)*exp(-x)*(5 + 3*exp(2*x) + 6*x*exp(2*x)). - G. C. Greubel, May 15 2016
Sum_{n>=0} (-1)^(n+1)/a(n) = Pi/(3*sqrt(3)) (A073010). - Amiram Eldar, Feb 24 2023

a(0)=2 added by Paul Curtz, Feb 20 2010

A166539 a(n) = (10n + 7(-1)^n + 5)/4.

Original entry on oeis.org

2, 8, 7, 13, 12, 18, 17, 23, 22, 28, 27, 33, 32, 38, 37, 43, 42, 48, 47, 53, 52, 58, 57, 63, 62, 68, 67, 73, 72, 78, 77, 83, 82, 88, 87, 93, 92, 98, 97, 103, 102, 108, 107, 113, 112, 118, 117, 123, 122, 128, 127, 133, 132, 138, 137, 143, 142, 148, 147, 153, 152, 158

Offset: 1

Vincenzo Librandi, Oct 16 2009

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

Cf. A166519, A166520, A166521, A166522, A166523, A166524, A166525, A166526.

[5*n/2 + (5+7*(-1)^n)/4: n in [1..70]]; // Vincenzo Librandi, May 15 2013

RecurrenceTable[{a[1]==2,a[n]==5n-a[n-1]},a[n],{n,70}] (* or *) LinearRecurrence[{1,1,-1},{2,8,7},70] (* Harvey P. Dale, Jun 29 2011 *)

def A166539(n): return (5*n - 1 + 7*((n+1)%2))//2
[A166539(n) for n in range(1, 101)] # G. C. Greubel, Aug 04 2024

a(n) = 5*n - a(n-1), n>=2.
From Harvey P. Dale, Jun 29 2011: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3), n>=4.
G.f.: x*(2+3*x*(2-x))/((1-x)^2*(1+x)). (End)
From G. C. Greubel, May 16 2016: (Start)
E.g.f.: (1/4)*(5*(1 + 2*x)*exp(x) + 7*exp(-x) - 12).
a(n) = a(n-1) + a(n-2) - a(n-3). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = 1/3 + sqrt((5-2*sqrt(5))/5)*Pi/5. - Amiram Eldar, Feb 24 2023
a(n) = A166520(n) - (-1)^n. - G. C. Greubel, Aug 04 2024

A339141 a(n) = reverse(10*n - a(n-1)), with n>1, a(1) = 1.

Original entry on oeis.org

1, 91, -16, 65, -51, 111, -14, 49, 14, 68, 24, 69, 16, 421, -172, 233, -36, 612, -224, 424, -412, 236, -6, 642, -293, 355, -58, 833, -345, 546, -632, 259, 17, 323, 72, 882, -215, 595, -502, 209, 102, 813, -383, 328, 221, 932, -264, 447, 34, 664, -451

Offset: 1

Clément Vovard, Nov 25 2020

Note that for x<0, reverse(x) is defined by -1*reverse(-x).

Starting the sequence with other numbers also gives similar-looking graphs.

			For n = 2, 10*n = 10*2 = 20, 20 - a(n-1) = 20 - 1 = 19, reverse(19) = 91.
For n = 3, 10*n = 10*3 = 30, 30 - a(3-1) = 30 - 91 = -61, reverse(-61) = -16.

Clément Vovard, Table of n, a(n) for n = 1..10000

Cf. A004086, A166525.

a:= proc(n) option remember; `if`(n<2, n, (p-> signum(p)* (f->
      parse(cat(f[-i]$i=1..length(f))))(""||(abs(p))))(10*n-a(n-1)))
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Jan 06 2021

nmax=51; a[1]=1; a[n_]:=Sign[10n-a[n-1]]IntegerReverse[10n-a[n-1]]; Table[a[n],{n,nmax}] (* Stefano Spezia, Dec 05 2020 *)

rev(n) = sign(n)*fromdigits(Vecrev(digits(n)));
a(n) = if (n==1, 1, rev(10*n-a(n-1))); \\ Michel Marcus, Dec 05 2020

a(n) = reverse(10*n - a(n-1)) where reverse means reverse the order of the digits.

cp's OEIS Frontend

A166520 a(n) = (10*n + 11*(-1)^n + 5)/4.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A166521 a(n) = (6*n + 7*(-1)^n + 3)/2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A166522 a(n) = 7*n - a(n-1), with a(1) = 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A166523 a(n) = 8*n - a(n-1), with n>1, a(1)=1.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A166524 a(n) = 9*n - a(n-1), with n>1, a(1)=1.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A166526 a(n) = 12*n - a(n-1), with n>1, a(1)=1.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A166517 a(n) = (3 + 5*(-1)^n + 6*n)/4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A166539 a(n) = (10*n + 7*(-1)^n + 5)/4.

A166520 a(n) = (10n + 11(-1)^n + 5)/4.

A166521 a(n) = (6n + 7(-1)^n + 3)/2.

A166517 a(n) = (3 + 5(-1)^n + 6n)/4.

A166539 a(n) = (10n + 7(-1)^n + 5)/4.