A010703 -id:A010703 - cp's OEIS frontend

A047461 Numbers that are congruent to {1, 4} mod 8.

1, 4, 9, 12, 17, 20, 25, 28, 33, 36, 41, 44, 49, 52, 57, 60, 65, 68, 73, 76, 81, 84, 89, 92, 97, 100, 105, 108, 113, 116, 121, 124, 129, 132, 137, 140, 145, 148, 153, 156, 161, 164, 169, 172, 177, 180, 185, 188, 193, 196, 201, 204, 209, 212, 217, 220, 225, 228, 233

Offset: 1

N. J. A. Sloane

Maximal number of squares that can be covered by a queen on an n X n chessboard. - Reinhard Zumkeller, Dec 15 2008

Muniru A Asiru, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A004526, A004770, A010703, A017077, A017113, A047398, A047452.

Cf. A047470, A047524, A047535, A047615, A047617, A153125, A342819.

Filtered([1..250], n->n mod 8=1 or n mod 8 =4); # Muniru A Asiru, Jul 23 2018

[4*n-3 - ((n+1) mod 2): n in [1..70]]; // G. C. Greubel, Mar 15 2024

seq(coeff(series(factorial(n)*((8-exp(-x)+(8*x-7)*exp(x))/2), x,n+1),x,n),n=1..60); # Muniru A Asiru, Jul 23 2018

Flatten[(#+{1,4})&/@(8Range[0,30])] (* or *) LinearRecurrence[ {1,1,-1},{1,4,9},60] (* Harvey P. Dale, Jun 18 2013 *)
CoefficientList[ Series[(4x^2 + 3x + 1)/((x + 1) (x - 1)^2), {x, 0, 58}], x] (* Robert G. Wilson v, Jul 24 2018 *)

makelist(4*n -(7 + (-1)^n)/2, n, 1, 100); /* Franck Maminirina Ramaharo, Jul 22 2018 */

def A047461(n): return (n-1<<2)|(n&1) # Chai Wah Wu, Mar 30 2024

[4*n-3 - ((n+1)%2) for n in range(1,71)] # G. C. Greubel, Mar 15 2024

From R. J. Mathar, Oct 29 2008: (Start)
G.f.: x*(1+3*x+4*x^2)/((1+x)*(1-x)^2).
a(n) = a(n-2) + 8.
a(n) + a(n+1) = A004770(n).
a(n+1) - a(n) = A010703(n). (End)
a(n) = 8*floor((n-1)/2) + 4 - 3*(n mod 2). - Reinhard Zumkeller, Dec 15 2008
a(n) = A153125(n,n). - Reinhard Zumkeller, Dec 20 2008
a(n) = 8*n - a(n-1) - 11 (with a(1)=1). - Vincenzo Librandi, Aug 06 2010
a(n) = 4*n - (7 + (-1)^n)/2. - Arkadiusz Wesolowski, Sep 18 2012
a(1)=1, a(2)=4, a(3)=9, a(n) = a(n-1) + a(n-2) - a(n-3). - Harvey P. Dale, Jun 18 2013
a(n) = 1 + A004526(n)*3 + A004526(n-1)*5. - Gregory R. Bryant, Apr 16 2014
From Franck Maminirina Ramaharo, Jul 22 2018: (Start)
a(n) = A047470(n) + 1.
E.g.f.: (8 - exp(-x) + (8*x - 7)*exp(x))/2. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)+1)*Pi/16 + log(2)/4 + sqrt(2)*arccoth(sqrt(2))/8. - Amiram Eldar, Dec 11 2021

A171498 a(n) = 4*3^n-1.

Original entry on oeis.org

3, 11, 35, 107, 323, 971, 2915, 8747, 26243, 78731, 236195, 708587, 2125763, 6377291, 19131875, 57395627, 172186883, 516560651, 1549681955, 4649045867, 13947137603, 41841412811, 125524238435, 376572715307, 1129718145923, 3389154437771, 10167463313315, 30502389939947

Offset: 0

Klaus Brockhaus, Dec 10 2009

Binomial transform of A171497.

Inverse binomial transform of A171499.

Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A010703, A036543, A171497, A171499.

NestList[3#+2&,3,30]  (* Harvey P. Dale, Feb 25 2011 *)

{m=25; v=concat([3], vector(m-1)); for(n=2, m, v[n]=3*v[n-1]+2); v}

a(n) = 3*a(n-1)+2 for n > 0; a(0) = 3.
G.f.: (3-x)/((1-x)*(1-3*x)).
a(n) = A036543(n) + 2. - Philippe Deléham, Apr 13 2013
E.g.f.: exp(x)*(4*exp(2*x) - 1). - Stefano Spezia, Aug 04 2024

a(25)-a(27) from Stefano Spezia, Aug 04 2024

A171499 a(n) = 6a(n-1) - 8a(n-2) for n > 1; a(0) = 3, a(1) = 14.

Original entry on oeis.org

3, 14, 60, 248, 1008, 4064, 16320, 65408, 261888, 1048064, 4193280, 16775168, 67104768, 268427264, 1073725440, 4294934528, 17179803648, 68719345664, 274877644800, 1099511103488, 4398045462528, 17592183947264, 70368739983360

Offset: 0

Klaus Brockhaus, Dec 10 2009

Binomial transform of A171498; second binomial transform of A171497; third binomial transform of A010703.
Related to sequences A001969 and A000069, sum of each group with exponent 1. - Eric Desbiaux, Jul 24 2013
a(n) in base 2 is n+2 1s followed by n 0s. - Hussam al-Homsi, Oct 12 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (6,-8).

Cf. A010703, A171472, A171473, A171476, A171497, A171498.

[4*4^n-2^n: n in [0..30]]; // Vincenzo Librandi, Jul 18 2011

(* This program shows how A171499 arises from the Vandermonde determinant of (1,2,4,...,2^(n-1)). *)
f[j_]:= 2^j - 1; z = 15;
v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
d[n_]:= Product[(i-1)!, {i,n}]
Table[v[n], {n,z}]                     (* A203303 *)
Table[v[n+1]/v[n], {n,z}]              (* A002884 *)
Table[v[n]*v[n+2]/(2*v[n+1])^2, {n,z}]  (* A171499 *)
(* Clark Kimberling, Jan 02 2011 *)
LinearRecurrence[{6,-8},{3,14},30] (* Harvey P. Dale, Sep 05 2021 *)

{m=23; v=concat([3, 14], vector(m-2)); for(n=3, m, v[n]=6*v[n-1]-8*v[n-2]); v}

[4^(n+1) -2^n for n in range(31)] # G. C. Greubel, Aug 31 2023

a(n) = 4*4^n - 2^n = 2^n * (2^(n+2) - 1).
G.f.: (3-4*x)/((1-2*x)*(1-4*x)).
a(n) = 4*a(n-1) + 2^n for n > 0. - Vincenzo Librandi, Jul 18 2011
a(n) = A171476(n+1)/2. - Hussam al-Homsi, Jun 06 2021
E.g.f.: 4*exp(4*x) - exp(2*x). - G. C. Greubel, Aug 31 2023

A171497 a(n) = 2*a(n-1) for n > 1; a(0) = 3, a(1) = 8.

Original entry on oeis.org

3, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592, 17179869184

Offset: 0

Klaus Brockhaus, Dec 10 2009

8*A000079 preceded by 3.
Binomial transform of A010703.
Inverse binomial transform of A171498.

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

Cf. A000079 (powers of 2), A010703 (repeat 3, 5), A171498.

Join[{3},NestList[2#&,8,30]] (* Harvey P. Dale, Dec 17 2011 *)

{m=30; v=concat([3, 8], vector(m-2)); for(n=3, m, v[n]=2*v[n-1]); v}

a(n) = 4*2^n for n > 0; a(0) = 3.
G.f.: (3+2*x)/(1-2*x).
E.g.f.: 4*exp(2*x) - 1. - Stefano Spezia, Aug 04 2024

a(30)-a(32) from Stefano Spezia, Aug 04 2024

A010726 Period 2: repeat (6,10).

Original entry on oeis.org

6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10

Offset: 0

N. J. A. Sloane

From Klaus Brockhaus, Dec 10 2009: (Start)
Interleaving of A010722 and A010692.
Also continued fraction expansion of 3 + 4*sqrt(15)/5.
Binomial transform of 6 followed by A122803 without initial terms 1,-2.
Inverse binomial transform of A171494. (End)

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

Equals 2*A010703. Cf. A010722 (all 6's sequence), A010692 (all 10's sequence), A122803 (powers of -2), A171494. - Klaus Brockhaus, Dec 10 2009

&cat[ [6, 10]: n in [1..42] ]; // Klaus Brockhaus, Dec 10 2009

LinearRecurrence[{0,1},{6,10},90] (* or *) PadRight[{},90,{6,10}] (* Harvey P. Dale, Mar 07 2015 *)

a(n)=6+n%2*4 \\ Charles R Greathouse IV, Dec 21 2011

a(n) = -2*(-1)^n + 8. - Paolo P. Lava, Oct 27 2006
From Klaus Brockhaus, Dec 10 2009: (Start)
a(n) = a(n-2) for n > 1; a(0) = 6, a(1) = 10.
G.f.: 2*(3+5*x)/((1-x)*(1+x)). (End)

A176103 Decimal expansion of (15+sqrt(285))/10.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 10 2010

Continued fraction expansion of (15+sqrt(285))/10 is A010703.

			(15+sqrt(285))/10 = 3.18819430161341321831...

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

Cf. A176104 (decimal expansion of sqrt(285)), A010703 (repeat 3, 5).

RealDigits[(15+Sqrt[285])/10,10,120][[1]] (* Harvey P. Dale, Aug 04 2020 *)

A176318 Decimal expansion of (15 + sqrt(285))/6.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 15 2010

Continued fraction expansion of (15+sqrt(285))/6 is A010703 preceded by 5.

			(15+sqrt(285))/6 = 5.31365716935568869718...

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A176104 (decimal expansion of sqrt(285)), A010703 (repeat 3, 5).

SetDefaultRealField(RealField(120)); (15 + Sqrt(285))/6; // G. C. Greubel, Nov 26 2019

evalf( (15+sqrt(285))/6, 120); # G. C. Greubel, Nov 26 2019

RealDigits[(15+Sqrt[285])/6,10,120][[1]] (* Harvey P. Dale, May 07 2017 *)

default(realprecision, 120); (15+sqrt(285))/6 \\ G. C. Greubel, Nov 26 2019

numerical_approx((15+sqrt(285))/6, digits=120) # G. C. Greubel, Nov 26 2019

A267317 a(n) = final digit of 2^n-1.

Original entry on oeis.org

0, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5

Offset: 0

Ilya Gutkovskiy, Jan 13 2016

Decimal expansion of 25/1818.

Period 4: repeat [1, 3, 7, 5] for n > 0.

Eric Weisstein's World of Mathematics, Mersenne Number
Index entries for linear recurrences with constant coefficients, signature (1,-1,1).

Cf. A000225, A000689, A010688, A010703, A010879, A080172.

[0] cat &cat[[1, 3, 7, 5]^^25]; // Bruno Berselli, Jan 13 2016

A267317:=n->(2^n-1) mod 10: seq(A267317(n), n=0..150); # Wesley Ivan Hurt, Jun 15 2016

Mathematica
```
Table[Mod[2^n - 1, 10], {n, 0, 120}]
```

a(n) = if(n==0, 0, if(n%4==0, 5, if(n%4==1, 1, if(n%4==2, 3, if(n%4==3, 7))))) \\ Felix Fröhlich, Jan 19 2016

a(n) = lift(Mod(2^n-1, 10)) \\ Felix Fröhlich, Jan 19 2016

G.f.: x*(1 + 2*x + 5*x^2)/(1 - x + x^2 - x^3).
a(n) = A010879(A000225(n)).
a(n) = A000689(n) - 1.
a(n) = (1+(-1)^n)*(-1)^(n*(n-1)/2)/2 + 3*(1-(-1)^n)*(-1)^(n*(n+1)/2)/2 + 4 for n > 0, a(0) = 0. [Bruno Berselli, Jan 13 2016]
From Wesley Ivan Hurt, Jun 15 2016: (Start)
a(n) = a(n-4) for n>4.
a(2k+2) = A010703(k), a(2k+1) = A010688(k). (End)
From Wesley Ivan Hurt, Jul 06 2016: (Start)
a(n) = a(n-1) - a(n-2) + a(n-3) for n > 3.
a(n) = 4 + cos(n*Pi/2) - 3*sin(n*Pi/2) for n > 0. (End)
E.g.f.: -5 + cos(x) - 3*sin(x) + 4*exp(x). - Ilya Gutkovskiy, Jul 06 2016

A306278 Numbers congruent to 2 or 11 mod 14.

Original entry on oeis.org

2, 11, 16, 25, 30, 39, 44, 53, 58, 67, 72, 81, 86, 95, 100, 109, 114, 123, 128, 137, 142, 151, 156, 165, 170, 179, 184, 193, 198, 207, 212, 221, 226, 235, 240, 249, 254, 263, 268, 277, 282, 291, 296, 305, 310, 319, 324, 333, 338, 347, 352, 361, 366, 375, 380, 389, 394

Offset: 1

Davis Smith, Feb 02 2019

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

Cf. A007310, A010703, A020639, A047470, A091999, A273669, A306277, A306289.

Primes greater than 2 in this sequence: A045471.

Maple
```
seq(seq(14*i+j, j=[2, 11]), i=0..28);
```

Flatten[Table[{14n + 2, 14n + 11}, {n, 0, 28}]]
LinearRecurrence[{1,1,-1},{2,11,16},60] (* Harvey P. Dale, Jan 16 2023 *)

for(n=2, 394, if((n%14==2) || (n%14==11), print1(n, ", ")))

PARI
```
for(n=1,57,print1(7*n-4+(-1)^n,", "))
```

for(n=1,500,if(n%14==2,print1(n,", "));if(n%14==11,print1(n,", "))) \\ Jinyuan Wang, Feb 03 2019

Vec(x*(2 + 9*x + 3*x^2) / ((1 - x)^2*(1 + x)) + O(x^40)) \\ Colin Barker, Mar 14 2019

upto(n) = forstep(i = 2, n, [9, 5], print1(i", ")) \\ David A. Corneth, Mar 27 2019

a(n) = 7*n - A010703(n).
a(n) = 7*n - 4 + (-1)^n.
a(n) = a(n - 1) + a(n - 2) - a(n - 3) for n > 3.
A007310(a(n) + 1) = 7*A007310(n)
From Jinyuan Wang, Feb 03 2019: (Start)
For odd number k, a(k) = 7*k - 5.
For even number k, a(k) = 7*k - 3.
(End)
G.f.: x*(2 + 9*x + 3*x^2) / ((1 - x)^2*(1 + x)). - Colin Barker, Mar 14 2019
E.g.f.: 3 + (7*x - 4)*exp(x) + exp(-x). - David Lovler, Sep 07 2022

cp's OEIS Frontend

A047461 Numbers that are congruent to {1, 4} mod 8.

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A171498 a(n) = 4*3^n-1.

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A171499 a(n) = 6*a(n-1) - 8*a(n-2) for n > 1; a(0) = 3, a(1) = 14.

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A171497 a(n) = 2*a(n-1) for n > 1; a(0) = 3, a(1) = 8.

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A010726 Period 2: repeat (6,10).

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A176103 Decimal expansion of (15+sqrt(285))/10.

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A176318 Decimal expansion of (15 + sqrt(285))/6.

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A171499 a(n) = 6a(n-1) - 8a(n-2) for n > 1; a(0) = 3, a(1) = 14.