A017101 -id:A017101 - cp's OEIS frontend

A004769 Numbers whose binary expansion ends in 011.

11, 19, 27, 35, 43, 51, 59, 67, 75, 83, 91, 99, 107, 115, 123, 131, 139, 147, 155, 163, 171, 179, 187, 195, 203, 211, 219, 227, 235, 243, 251, 259, 267, 275, 283, 291, 299, 307, 315, 323, 331, 339, 347, 355, 363, 371, 379, 387, 395, 403, 411, 419, 427, 435, 443, 451, 459, 467, 475, 483, 491

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Essentially same as A017101.

[8*n+11: n in [0..60]]; // Vincenzo Librandi, Jul 12 2011

Table[8*n+11, {n,0,60}] (* G. C. Greubel, Oct 13 2018 *)
LinearRecurrence[{2,-1},{11,19},80] (* Harvey P. Dale, Aug 09 2023 *)
Select[Range[10,500],Take[IntegerDigits[#,2],-3]=={0,1,1}&] (* or *) Rest[FromDigits[#,2]&/@(Join[#,{0,1,1}]&/@Tuples[{0,1},6])] (* Harvey P. Dale, Mar 23 2025 *)

a(n)=8*n+11 \\ Charles R Greathouse IV, Jul 11 2016

a(n) = 8*n + 11. - Vincenzo Librandi, Jul 12 2011
From G. C. Greubel, Oct 13 2018: (Start)
a(n) = 2*a(n-1) - a(n-2).
G.f.: (11 - 3*x)/(1-x)^2.
E.g.f.: (8*x + 11)*exp(x). (End)

A047458 Numbers that are congruent to {0, 3, 4} mod 8.

Original entry on oeis.org

0, 3, 4, 8, 11, 12, 16, 19, 20, 24, 27, 28, 32, 35, 36, 40, 43, 44, 48, 51, 52, 56, 59, 60, 64, 67, 68, 72, 75, 76, 80, 83, 84, 88, 91, 92, 96, 99, 100, 104, 107, 108, 112, 115, 116, 120, 123, 124, 128, 131, 132, 136, 139, 140, 144, 147, 148, 152, 155, 156

Offset: 1

N. J. A. Sloane

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

Union of A008586 and A017101. - Michel Marcus, Jun 01 2017

[n : n in [0..150] | n mod 8 in [0, 3, 4]]; // Wesley Ivan Hurt, Jun 09 2016

A047458:=n->8*n/3-3-cos(2*n*Pi/3)-sin(2*n*Pi/3)/(3*sqrt(3)): seq(A047458(n), n=1..100); # Wesley Ivan Hurt, Jun 09 2016

Select[Range[0, 150], MemberQ[{0, 3, 4}, Mod[#, 8]] &] (* Wesley Ivan Hurt, Jun 09 2016 *)
LinearRecurrence[{1,0,1,-1},{0,3,4,8},90] (* Harvey P. Dale, May 31 2017 *)

G.f.: x^2*(3+x+4*x^2)/((1-x)^2*(1+x+x^2)). [Colin Barker, May 13 2012]
From Wesley Ivan Hurt, Jun 09 2016: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
a(n) = 8*n/3-3-cos(2*n*Pi/3)-sin(2*n*Pi/3)/(3*sqrt(3)).
a(3k) = 8k-4, a(3k-1) = 8k-5, a(3k-2) = 8k-8. (End)

A078688 Continued fraction expansion of e^(1/4).

Original entry on oeis.org

1, 3, 1, 1, 11, 1, 1, 19, 1, 1, 27, 1, 1, 35, 1, 1, 43, 1, 1, 51, 1, 1, 59, 1, 1, 67, 1, 1, 75, 1, 1, 83, 1, 1, 91, 1, 1, 99, 1, 1, 107, 1, 1, 115, 1, 1, 123, 1, 1, 131, 1, 1, 139, 1, 1, 147, 1, 1, 155, 1, 1, 163, 1, 1, 171, 1, 1, 179, 1, 1, 187

Offset: 0

Benoit Cloitre, Dec 17 2002

G. Xiao, Contfrac
K. Matthews, Finding the continued fraction of e^(l/m)

Cf. A017101, A058281.

ContinuedFraction[E^(1/4),80] (* Harvey P. Dale, Nov 19 2011 *)

a(4k+2) = 8k+3, otherwise a(i) = 1.

G.f.: (6x^4+2x)/(1-x^3)^2+1/(1-x). - Ralf Stephan, Mar 13 2003

A168378 a(n) = 3 + 8*floor(n/2).

Original entry on oeis.org

3, 11, 11, 19, 19, 27, 27, 35, 35, 43, 43, 51, 51, 59, 59, 67, 67, 75, 75, 83, 83, 91, 91, 99, 99, 107, 107, 115, 115, 123, 123, 131, 131, 139, 139, 147, 147, 155, 155, 163, 163, 171, 171, 179, 179, 187, 187, 195, 195, 203, 203, 211, 211, 219, 219, 227, 227, 235, 235

Offset: 1

Vincenzo Librandi, Nov 24 2009

More generally, the sequences generated by the recursive relation b(n) = h*n - b(n-1) + k, with b(1)=c and h, k, c, prefixed integers, have the closed form b(n) = (2*h*n + (3*h + 2*k - 4*c)*(-1)^n + h + 2*k)/4. Also, if 2*c = h+k, then b(n) = c + h*floor(n/2); if 2*c = 2*h+k, then b(n) = c + h*floor((n-1)/2); if 2*c = k, b(n) = c + h*floor((n+1)/2). - Bruno Berselli, Sep 18 2013

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

Cf. A017101, A168381, A168398.

[3+8*Floor(n/2): n in [1..70]]; // Vincenzo Librandi, Sep 18 2013

Table[ 3 + 8*floor(n/2), {n,60}] (* Bruno Berselli, Sep 18 2013 *)
CoefficientList[Series[(3 + 8 x - 3 x^2)/((1 + x) (x - 1)^2), {x, 0, 70}], x] (* Vincenzo Librandi, Sep 18 2013 *)
LinearRecurrence[{1,1,-1},{3,11,11},80] (* Harvey P. Dale, Oct 05 2022 *)

a(n) = 8*n - a(n-1) - 2, with n>1, a(1)=3.
G.f.: x*(3 + 8*x - 3*x^2)/((1+x)*(x-1)^2). - Vincenzo Librandi, Sep 18 2013
a(n) = a(n-1) +a(n-2) -a(n-3). - Vincenzo Librandi, Sep 18 2013
a(n) = 4*n + 2*(-1)^n + 1. - Bruno Berselli, Sep 18 2013
a(n) = A168381(n) + 1 = A168398(n) - 1. - Bruno Berselli, Sep 18 2013
E.g.f.: (4*x + 3)*cosh(x) + (4*x - 1)*sinh(x) - 3. - G. C. Greubel, Jul 19 2016

New definition by Vincenzo Librandi, Sep 18 2013

A177065 a(n) = (8n+3)(8*n+5).

Original entry on oeis.org

15, 143, 399, 783, 1295, 1935, 2703, 3599, 4623, 5775, 7055, 8463, 9999, 11663, 13455, 15375, 17423, 19599, 21903, 24335, 26895, 29583, 32399, 35343, 38415, 41615, 44943, 48399, 51983, 55695, 59535, 63503, 67599, 71823, 76175, 80655, 85263, 89999, 94863, 99855

Offset: 0

Vincenzo Librandi, May 31 2010

Cf. comment of Reinhard Zumkeller in A177059: in general, (h*n+h-k)*(h*n+k) = h^2*A002061(n+1) + (h-k)*k - h^2; therefore a(n) = 64*A002061(n+1) - 49. - Bruno Berselli, Aug 24 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A002061, A004770, A016754, A017101, A125169, A177059.

[(8*n+3)*(8*n+5): n in [0..50]]; // Vincenzo Librandi, Apr 08 2013

A177065:=n->(8*n+3)*(8*n+5): seq(A177065(n), n=0..100); # Wesley Ivan Hurt, Apr 24 2017

Table[(8n+3)(8n+5),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{15,143,399},40] (* Harvey P. Dale, Mar 13 2013 *)
CoefficientList[Series[(15 + 98 x + 15 x^2)/(1-x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Apr 08 2013 *)

a(n)=(8*n+3)*(8*n+5) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 128*n + a(n-1) with n > 0, a(0)=15.
a(n) = A125169(A016754(n) - 1). - Reinhard Zumkeller, Jul 05 2010
a(0)=15, a(1)=143, a(2)=399, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Mar 13 2013
G.f.: (15+98*x+15*x^2)/(1-x)^3. - Vincenzo Librandi, Apr 08 2013
From Amiram Eldar, Feb 19 2023: (Start)
a(n) = A017101(n)*A004770(n).
Sum_{n>=0} 1/a(n) = (sqrt(2)-1)*Pi/16.
Sum_{n>=0} (-1)^n/a(n) = (cos(Pi/8) * log(tan(3*Pi/16)) + sin(Pi/8) * log(cot(Pi/16)))/4.
Product_{n>=0} (1 - 1/a(n)) = sec(Pi/8)*cos(Pi/(4*sqrt(2))).
Product_{n>=0} (1 + 1/a(n)) = sec(Pi/8). (End)
E.g.f.: exp(x)*(15 + 64*x*(2 + x)). - Elmo R. Oliveira, Oct 25 2024

Edited by N. J. A. Sloane, Jun 22 2010

A234011 The sums of 2 consecutive odious numbers (A000069).

Original entry on oeis.org

3, 6, 11, 15, 19, 24, 27, 30, 35, 40, 43, 47, 51, 54, 59, 63, 67, 72, 75, 79, 83, 86, 91, 96, 99, 102, 107, 111, 115, 120, 123, 126, 131, 136, 139, 143, 147, 150, 155, 160, 163, 166, 171, 175, 179, 184, 187, 191, 195, 198, 203, 207, 211, 216, 219, 222, 227, 232, 235, 239, 243, 246

Offset: 1

Gerasimov Sergey, Dec 27 2013

The union of A131323(k) and (A225822(m)+(-1)^m).
All even numbers in this sequence are evil numbers (A001969).
It seems that A233388(n) = a(A091785(n)).

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A000069, A003159 (indices of odd numbers in A234011), A036554 (indices of even numbers in A234011), A131323 (odd sums of 2 successive odious or 2 successive evil numbers), A233388 (odious numbers in A234011), A234431 (sums of 2 consecutive evil numbers), A017101, A091785, A225822, A227930, A233388.

Plus @@@ Partition[Select[Range[125], OddQ[DigitCount[#, 2][[1]]] &], 2, 1] (* Amiram Eldar, Jul 24 2023 *)

a(n)=4*n-hammingweight(n-1)%2-hammingweight(n)%2 \\ Charles R Greathouse IV, Dec 29 2013

(define (A234011 n) (+ (A000069 n) (A000069 (+ n 1)))) ;; Antti Karttunen, Dec 29 2013

a(n) = A000069(n) + A000069(n + 1).
4n - 2 <= a(n) <= 4n. - Charles R Greathouse IV, Dec 29 2013
a(2n+1) = 8n + 3 = A017101(n). - Ralf Stephan, Dec 31 2013

Terms recomputed and checked by Antti Karttunen, Dec 29 2013

A354938 Row 8 of A354940: Numbers k for which A345992(k) = 8, divided by 8.

Original entry on oeis.org

3, 9, 11, 17, 19, 25, 27, 33, 41, 43, 49, 57, 59, 67, 73, 81, 83, 89, 97, 105, 107, 113, 121, 129, 131, 137, 139, 145, 161, 163, 169, 177, 179, 185, 193, 201, 209, 211, 217, 225, 227, 233, 241, 243, 249, 251, 257, 281, 283, 289, 297, 305, 307, 313, 321, 329, 331, 337, 345, 347, 353, 361, 377, 379, 393, 401, 409, 417

Offset: 1

Antti Karttunen, Jun 15 2022

nonn

Apparently, all terms are either of the form 8k+1 (in A017077) or 8k+3 (in A017101).

Row 8 of A354940.

Cf. A017077, A017101, A344005, A345992.

q[n_] := Module[{m = 1}, While[!Divisible[m*(m + 1), 8*n], m++]; GCD[8*n, m] == 8]; Select[Range[420], q] (* Amiram Eldar, Jun 17 2022 *)

PARI
```
A354938(n) = A354940sq(8,n);
```

A047400 Numbers that are congruent to {1, 3, 6} mod 8.

Original entry on oeis.org

1, 3, 6, 9, 11, 14, 17, 19, 22, 25, 27, 30, 33, 35, 38, 41, 43, 46, 49, 51, 54, 57, 59, 62, 65, 67, 70, 73, 75, 78, 81, 83, 86, 89, 91, 94, 97, 99, 102, 105, 107, 110, 113, 115, 118, 121, 123, 126, 129, 131, 134, 137, 139, 142, 145, 147, 150, 153, 155, 158

Offset: 1

N. J. A. Sloane

Union of A017077, A017101 and A017137. - R. J. Mathar, Apr 14 2008

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

Cf. A004773.

Cf. A017077, A017101, A017137.

[n: n in [1..300] | n mod 8 in [1, 3, 6]]; // Vincenzo Librandi, Mar 27 2011

A047400:=n->2*(12*n-9+sqrt(3)*sin(2*n*Pi/3))/9: seq(A047400(n), n=1..100); # Wesley Ivan Hurt, Jun 10 2016

Select[Range[0, 150], MemberQ[{1, 3, 6}, Mod[#, 8]] &] (* Wesley Ivan Hurt, Jun 10 2016 *)

a(n) = {x=8*floor((n-1)/3);if(n%3==1,x=x+1);if(n%3==2,x=x+3);if(n%3==0,x=x+6);x} \\ Michael B. Porter, Oct 02 2009

a(n) = A004773(n-1) + A004773(n). - Gary W. Adamson, Sep 13 2007
G.f.: x*(1+x)*(2x^2+x+1)/((-1+x)^2*(x^2+x+1)). a(n) = a(n-3)+8 for n>3. - R. J. Mathar, Apr 14 2008
From Wesley Ivan Hurt, Jun 10 2016: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
a(n) = 2*(12*n-9+sqrt(3)*sin(2*n*Pi/3))/9.
a(3k) = 8k-2, a(3k-1) = 8k-5, a(3k-2) = 8k-7. (End)

A047573 Numbers that are congruent to {0, 1, 2, 4, 5, 6, 7} mod 8.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81

Offset: 1

N. J. A. Sloane

Complement of A017101. - Michel Marcus, Sep 13 2015

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

Cf. A017101 (8n+3).

[n+Floor((n-4)/7) : n in [1..100]]; // Wesley Ivan Hurt, Sep 12 2015

I:=[0,1,2,4,5,6,7,8]; [n le 8 select I[n] else Self(n-1) + Self(n-7) - Self(n-8): n in [1..80]]; // Vincenzo Librandi, Sep 13 2015

for n from 0 to 200 do if n mod 8 <> 3 then printf(`%d,`,n) fi: od:

Table[n+Floor[(n-4)/7], {n, 100}] (* Wesley Ivan Hurt, Sep 12 2015 *)
LinearRecurrence[{1, 0, 0, 0, 0, 0, 1, -1}, {0, 1, 2, 4, 5, 6, 7, 8}, 80] (* Vincenzo Librandi, Sep 13 2015 *)
DeleteCases[Range[0,100],?(Mod[#,8]==3&)] (* _Harvey P. Dale, Oct 05 2020 *)

def A047573(n):
    a, b = divmod(n-1,7)
    return (0,1,2,4,5,6,7)[b]+(a<<3) # Chai Wah Wu, Jan 27 2023

G.f.: x^2*(x^6+x^5+x^4+x^3+2*x^2+x+1)/((x-1)^2*(x^6+x^5+x^4+x^3+x^2+x+1)). [Colin Barker, Jun 22 2012]
From Wesley Ivan Hurt, Sep 12 2015: (Start)
a(n) = a(n-1) + a(n-7) - a(n-8) for n>8.
a(n) = n + floor((n-4)/7). (End)
From Wesley Ivan Hurt, Jul 21 2016: (Start)
a(n) = a(n-7) + 8 for n>7.
a(n) = (56*n - 49 + (n mod 7) + ((n+1) mod 7) + ((n+2) mod 7) - 6*((n+3) mod 7) + ((n+4) mod 7) + ((n+5) mod 7) + ((n+6) mod 7))/49.
a(7*k) = 8*k-1, a(7*k-1) = 8*k-2, a(7*k-2) = 8*k-3, a(7*k-3) = 8*k-4, a(7*k-4) = 8*k-6, a(7*k-5) = 8*k-7, a(7*k-6) = 8*k-8. (End)

More terms from James Sellers, Feb 19 2001

A137192 Lucky numbers (A000959) which are congruent to 3 mod 8.

Original entry on oeis.org

3, 43, 51, 67, 75, 99, 115, 163, 171, 195, 211, 219, 235, 259, 267, 283, 307, 331, 339, 427, 451, 475, 483, 579, 619, 643, 651, 699, 723, 739, 787, 819, 867, 883, 931, 979, 1011, 1107, 1123, 1147, 1155, 1179, 1203, 1219, 1251, 1275, 1291, 1323, 1339, 1387, 1395, 1419

Offset: 1

N. J. A. Sloane, Mar 07 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A000959 and A017101.

cp's OEIS Frontend

A004769 Numbers whose binary expansion ends in 011.

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A047458 Numbers that are congruent to {0, 3, 4} mod 8.

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A078688 Continued fraction expansion of e^(1/4).

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A168378 a(n) = 3 + 8*floor(n/2).

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A177065 a(n) = (8*n+3)*(8*n+5).

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A234011 The sums of 2 consecutive odious numbers (A000069).

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A354938 Row 8 of A354940: Numbers k for which A345992(k) = 8, divided by 8.

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A177065 a(n) = (8n+3)(8*n+5).