A047270 -id:A047270 - cp's OEIS frontend

A047250 Numbers that are congruent to {0, 3, 4, 5} (mod 6).

0, 3, 4, 5, 6, 9, 10, 11, 12, 15, 16, 17, 18, 21, 22, 23, 24, 27, 28, 29, 30, 33, 34, 35, 36, 39, 40, 41, 42, 45, 46, 47, 48, 51, 52, 53, 54, 57, 58, 59, 60, 63, 64, 65, 66, 69, 70, 71, 72, 75, 76, 77, 78, 81, 82, 83, 84, 87, 88, 89, 90, 93, 94, 95, 96, 99

Offset: 1

N. J. A. Sloane

The sequence is the interleaving of A047233 with A047270. - Guenther Schrack, Feb 15 2019

Guenther Schrack, Table of n, a(n) for n = 1..10010
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A047233, A047246, A047270.

Complement: A047239.

[n : n in [0..150] | n mod 6 in [0, 3, 4, 5]]; // Wesley Ivan Hurt, Jun 02 2016

A047250:=n->(6*n-3+I^(2*n)-(1+I)*I^(-n)-(1-I)*I^n)/4: seq(A047250(n), n=1..100); # Wesley Ivan Hurt, Jun 02 2016

Select[Range[0,100], MemberQ[{0,3,4,5}, Mod[#,6]]&] (* or *) LinearRecurrence[{1,0,0,1,-1}, {0,3,4,5,6}, 60] (* Harvey P. Dale, Apr 01 2013 *)

my(x='x+O('x^70)); concat([0], Vec(x^2*(3+x+x^2+x^3)/((1+x)*(1+x^2)*(1-x)^2))) \\ G. C. Greubel, Feb 16 2019

a=(x^2*(3+x+x^2+x^3)/((1+x)*(1+x^2)*(1-x)^2)).series(x, 72).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 16 2019

G.f.: x^2*(3+x+x^2+x^3)/((1+x)*(1+x^2)*(1-x)^2). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, Jun 02 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (6*n - 3 + i^(2*n) - (1+i)*i^(-n) - (1-i)*i^n)/4 where i=sqrt(-1).
a(2*k) = A047270(k), a(2*k-1) = A047233(k). (End)
E.g.f.: (2 - sin(x) - cos(x) + (3*x - 2)*sinh(x) + (3*x - 1)*cosh(x))/2. - Ilya Gutkovskiy, Jun 02 2016
From Guenther Schrack, Feb 15 2019: (Start)
a(n) = (6*n - 3 + (-1)^n - 2*(-1)^(n*(n-1)/2))/4.
a(n) = a(n-4) + 6, a(1)=0, a(2)=3, a(3)=4, a(4)=5, for n > 4.
a(-n) = -A047246(n+2). (End)
Sum_{n>=2} (-1)^n/a(n) = 2*log(2)/3 - Pi/(6*sqrt(3)). - Amiram Eldar, Dec 17 2021

A084060 a(n) = 1/2 + (1-6n)(-1)^n/2.

Original entry on oeis.org

1, 3, -5, 9, -11, 15, -17, 21, -23, 27, -29, 33, -35, 39, -41, 45, -47, 51, -53, 57, -59, 63, -65, 69, -71, 75, -77, 81, -83, 87, -89, 93, -95, 99, -101, 105, -107, 111, -113, 117, -119, 123, -125, 129, -131, 135, -137, 141, -143, 147, -149, 153, -155, 159, -161, 165, -167, 171, -173, 177, -179

Offset: 0

Paul Barry, May 11 2003

abs(a(n+1)) = A047270(n).

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

Cf. A032766, A084056.

List([0..60], n-> (1 + (1-6*n)*(-1)^n)/2); # G. C. Greubel, Jan 03 2020

[1/2+(1-6*n)*(-1)^n/2: n in [0..60]]; // Vincenzo Librandi, Oct 26 2011

seq( (1 + (1-6*n)*(-1)^n)/2, n=0..60); # G. C. Greubel, Jan 03 2020

Table[(1 + (1-6*n)*(-1)^n)/2, {n,0,60}] (* G. C. Greubel, Jan 03 2020 *)
LinearRecurrence[{-1,1,1},{1,3,-5},100] (* Harvey P. Dale, Mar 05 2023 *)

vector(61, n, (1 - (7-6*n)*(-1)^n)/2) \\ G. C. Greubel, Jan 03 2020

[(1 + (1-6*n)*(-1)^n)/2 for n in (0..60)] # G. C. Greubel, Jan 03 2020

Unsigned version is sum of alternate terms of A032766 (numbers congruent to {0,1,3} mod 4): (1, 3, 4, 6, 7, 9, 10, 12, ...) such that a(n) = A032766(n-1) + A032766(n+1). - Gary W. Adamson, Sep 13 2007
G.f.: (1 + 4*x - 3*x^2 )/( (1-x)*(1+x)^2 ). - R. J. Mathar, Oct 25 2011
E.g.f.: (1+3*x)*cosh(x) - 3*x*sinh(x). - G. C. Greubel, Jan 03 2020

A154253 Primes of the form 9n^2-8n+2.

Original entry on oeis.org

2, 3, 59, 659, 1907, 2467, 3803, 9539, 19507, 23003, 24859, 30859, 37507, 42299, 52747, 58403, 61339, 67427, 98387, 122267, 126499, 139627, 162947, 182899, 209459, 214987, 232003, 243707, 280547, 347707, 362003, 383987, 429899, 478403

Offset: 1

Vincenzo Librandi, Jan 05 2009

Primes in A154254.
Primes generated by n = 1, 3, 9, 15, 17, 21, 33, 47, 51, 53, 59,...
(All but the first of these are either 3 (mod 6) or 5 (mod 6), as in A047270, because otherwise 9n^2-8n+2 is a multiple of 2 or 3. R. J. Mathar, Jul 17 2012).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

[a: n in [0..250] | IsPrime(a) where a is 9*n^2-8*n+2]; // Vincenzo Librandi, Jul 16 2012

Select[Table[9n^2-8n+2,{n,0,1500}],PrimeQ] (* Vincenzo Librandi, Jul 16 2012 *)

A269112 a(n) = (3(n-1)n + (-1)^((n-1)*n/2) + 5)/2.

Original entry on oeis.org

3, 5, 11, 21, 33, 47, 65, 87, 111, 137, 167, 201, 237, 275, 317, 363, 411, 461, 515, 573, 633, 695, 761, 831, 903, 977, 1055, 1137, 1221, 1307, 1397, 1491, 1587, 1685, 1787, 1893, 2001, 2111, 2225, 2343, 2463, 2585, 2711, 2841, 2973, 3107, 3245, 3387, 3531, 3677, 3827

Offset: 1

Mikk Heidemaa, Feb 19 2016

Bruno Berselli, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

Cf. A005408, A047270.

I:=[3,5]; [n le 2 select I[n] else 3*(n-1)^2-3*(n-1)- Self(n-2)+8: n in [1..50]]; // Vincenzo Librandi, Feb 22 2016

Table[(3 (n - 1) n + (-1)^((n - 1) n/2) + 5)/2, {n, 100}]
LinearRecurrence[{3, -4, 4, -3, 1}, {3, 5, 11, 21, 33}, 100]
CoefficientList[Series[(3-4*x+8*x^2-4*x^3+3*x^4)/((1+x^2)*(1-x)^3), {x, 0, 100}], x]

Vec((3-4*x+8*x^2-4*x^3+3*x^4)/((1+x^2)*(1-x)^3) + O(x^60)) \\ Michel Marcus, Feb 22 2016

[(3*(n-1)*n+(-1)^((n-1)*n/2)+5)/2 for n in (1..50)] # Bruno Berselli, Feb 23 2016

G.f.: x*(3 - 4*x + 8*x^2 - 4*x^3 + 3*x^4)/((1 + x^2)*(1 - x)^3).
a(n) = (3*n^2 - 3*n + cos(n*Pi/2) + sin(n*Pi/2) + 5)/2.
a(n) = 3*(n-1)^2 - 3*(n-1) - a(n-2) + 8 for n>2.
a(n) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 3*a(n-4) + a(n-5) for n>5. - Colin Barker, Feb 22 2016
a(n) = A047270(A000124(n-1)). - Bruno Berselli, Feb 23 2016

Edited by Bruno Berselli, Feb 23 2016

A058301 Number of solutions to c(0)F(0) + ... + c(n)F(n) = 0, where c(i) = +-1 for i >= 0, number of (+1)'s >= number of (-1)'s, F(i) = A000045(i) = Fibonacci numbers.

Original entry on oeis.org

1, 0, 2, 3, 0, 6, 4, 0, 8, 11, 0, 22, 16, 0, 32, 42, 0, 84, 64, 0, 128, 165, 0, 330, 256, 0, 512, 654, 0, 1308, 1024, 0, 2048, 2605, 0, 5210, 4096, 0, 8192, 10398, 0, 20796, 16384, 0, 32768, 41550, 0, 83100, 65536, 0, 131072, 166116, 0, 332232, 262144, 0

Offset: 0

Naohiro Nomoto, Dec 08 2000

nonn

			a(3) = 3 because +0+1+1-2 = -0+1+1-2 = +0-1-1+2 = 0;
a(5) = 6 because +0+1-1-2-3+5 = +0-1+1-2-3+5 = +0+1-1+2+3-5 = -0+1-1+2+3-5 = +0-1+1+2+3-5 = -0-1+1+2+3-5 = 0.

Cf. A000045, A002083, A047270, A047238.

a(3n+1) = 0, a(A047270(n)) = A002083(n+5), a(A047238(n)) = 2^n.

More terms from Sean A. Irvine, Aug 02 2022

cp's OEIS Frontend

A047250 Numbers that are congruent to {0, 3, 4, 5} (mod 6).

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A084060 a(n) = 1/2 + (1-6*n)*(-1)^n/2.

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A154253 Primes of the form 9n^2-8n+2.

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A269112 a(n) = (3*(n-1)*n + (-1)^((n-1)*n/2) + 5)/2.

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A058301 Number of solutions to c(0)F(0) + ... + c(n)F(n) = 0, where c(i) = +-1 for i >= 0, number of (+1)'s >= number of (-1)'s, F(i) = A000045(i) = Fibonacci numbers.

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Formula

Extensions

A084060 a(n) = 1/2 + (1-6n)(-1)^n/2.

A269112 a(n) = (3(n-1)n + (-1)^((n-1)*n/2) + 5)/2.