A017005 -id:A017005 - cp's OEIS frontend

A168333 a(n) = (14n + 7(-1)^n + 1)/4.

2, 9, 9, 16, 16, 23, 23, 30, 30, 37, 37, 44, 44, 51, 51, 58, 58, 65, 65, 72, 72, 79, 79, 86, 86, 93, 93, 100, 100, 107, 107, 114, 114, 121, 121, 128, 128, 135, 135, 142, 142, 149, 149, 156, 156, 163, 163, 170, 170, 177, 177, 184, 184, 191, 191, 198, 198, 205, 205

Offset: 1

Vincenzo Librandi, Nov 23 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. A017005, A168212, A168331, A168337, A168374.

[n eq 1 select 2 else 7*n-Self(n-1)-3: n in [1..70]]; // Vincenzo Librandi, Sep 17 2013

CoefficientList[Series[(2 + 7 x - 2 x^2)/((1 + x) (x - 1)^2), {x, 0, 70}], x] (* Vincenzo Librandi, Sep 17 2013 *)
LinearRecurrence[{1,1,-1},{2,9,9},70] (* Harvey P. Dale, Mar 13 2014 *)

a(n) = 7*n - a(n-1) - 3, with n>1, a(1)=2.
G.f.: x*(2 + 7*x - 2*x^2)/((1+x)*(x-1)^2). - Vincenzo Librandi, Sep 17 2013
a(n) = a(n-1) +a(n-2) -a(n-3). - Vincenzo Librandi, Sep 17 2013
a(n) = A168331(n) - 1 = A168337(n) + 1 = A168212(n) - 2 = A168374(n) + 2. - Bruno Berselli, Sep 17 2013
E.g.f.: (1/4)*(7 - 8*exp(x) + (14*x + 1)*exp(2*x))*exp(-x). - G. C. Greubel, Jul 18 2016

New definition by Bruno Berselli, Sep 17 2013

A017010 a(n) = (7*n+2)^6.

Original entry on oeis.org

64, 531441, 16777216, 148035889, 729000000, 2565726409, 7256313856, 17596287801, 38068692544, 75418890625, 139314069504, 243087455521, 404567235136, 646990183449, 1000000000000, 1500730351849, 2194972623936, 3138428376721, 4398046511104, 6053445140625

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A001014, A017005.

[(7*n+2)^6: n in [0..25]]; // Vincenzo Librandi, Jul 13 2011

A017010:=n->(7*n+2)^6: seq(A017010(n), n=0..40); # Wesley Ivan Hurt, Apr 22 2016

(7 Range[0, 30] + 2)^6 (* Wesley Ivan Hurt, Apr 22 2016 *)

lista(nn) = for(n=0, nn, print1((7*n+2)^6, ", ")); \\ Altug Alkan, Apr 24 2016

From Wesley Ivan Hurt, Apr 22 2016: (Start)
G.f.: (64 + 530993*x + 13058473*x^2 + 41753398*x^3 + 26472118*x^4 + 2876609*x^5 + 15625*x^6)/(1 - x)^7.
a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7).
a(n) = A001014(A017005(n)). (End)
E.g.f.: (64 + 531377*x + 7857199*x^2 + 16549750*x^3 + 9808085*x^4 + 1966419*x^5 + 117649*x^6)*exp(x). - Ilya Gutkovskiy, Apr 23 2016

A047310 Numbers that are congruent to {0, 1, 3, 4, 5, 6} mod 7.

Original entry on oeis.org

0, 1, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78

Offset: 1

N. J. A. Sloane

Complement of A017005. - Michel Marcus, Sep 08 2015

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

Cf. A017005 (7n+2).

[n+Floor((n-3)/6): n in [1..100]]; // Wesley Ivan Hurt, Sep 08 2015

[n: n in [0..100] | n mod 7 in [0,1,3,4,5,6]]; // Vincenzo Librandi, Sep 10 2015

A047310:=n->n+floor((n-3)/6): seq(A047310(n), n=1..100); # Wesley Ivan Hurt, Sep 08 2015

Table[n+Floor[(n-3)/6], {n, 100}] (* Wesley Ivan Hurt, Sep 08 2015 *)
LinearRecurrence[{1, 0, 0, 0, 0, 1, -1}, {0, 1, 3, 4, 5, 6, 7}, 70] (* Vincenzo Librandi, Sep 10 2015 *)
Select[Range[0,100],MemberQ[{0,1,3,4,5,6},Mod[#,7]]&] (* Harvey P. Dale, Dec 02 2024 *)

G.f.: x^2*(1+2*x+x^2+x^3+x^4+x^5) / ( (1+x)*(1+x+x^2)*(x^2-x+1)*(x-1)^2 ). - R. J. Mathar, Oct 25 2011
From Wesley Ivan Hurt, Sep 08 2015: (Start)
a(n) = a(n-1)+a(n-6)-a(n-7) for n>7.
a(n) = n + floor((n-3)/6). (End)
From Wesley Ivan Hurt, Jun 15 2016: (Start)
a(n) = (42*n-33-3*cos(n*Pi)+4*sqrt(3)*cos((1-4*n)*Pi/6)-12*sin((1+2*n)*Pi/6))/36.
a(6k) = 7k-1, a(6k-1) = 7k-2, a(6k-2) = 7k-3, a(6k-3) = 7k-4, a(6k-4) = 7k-6, a(6k-5) = 7k-7. (End)

More terms from Vincenzo Librandi, Sep 10 2015

A047370 Numbers that are congruent to {2, 3, 5} mod 7.

Original entry on oeis.org

2, 3, 5, 9, 10, 12, 16, 17, 19, 23, 24, 26, 30, 31, 33, 37, 38, 40, 44, 45, 47, 51, 52, 54, 58, 59, 61, 65, 66, 68, 72, 73, 75, 79, 80, 82, 86, 87, 89, 93, 94, 96, 100, 101, 103, 107, 108, 110, 114, 115, 117, 121, 122, 124, 128, 129, 131, 135, 136, 138, 142

Offset: 1

N. J. A. Sloane, Dec 11 1999

Union of A017005, A017017 and A017041. - Michel Marcus, May 25 2014

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

Cf. A017005, A017017, A017041.

[7*Floor((n-1)/3)+2^((n-1) mod 3)+1: n in [1..50]]; // Wesley Ivan Hurt, May 25 2014

A047370:=n->7*floor((n-1)/3) + 2^((n-1) mod 3)+1; seq(A047370(n), n=1..50); # Wesley Ivan Hurt, May 25 2014

Select[Range[200], MemberQ[{2,3,5}, Mod[#,7]]&] (* or *) LinearRecurrence[ {1,0,1,-1}, {2,3,5,9}, 60] (* Harvey P. Dale, Apr 29 2013 *)
Table[7*Floor[(n - 1)/3] + 2^Mod[n - 1, 3] + 1, {n, 50}] (* Wesley Ivan Hurt, May 25 2014 *)

x='x + O('x^50); Vec(x*(2+x+2*x^2+2*x^3)/((1+x+x^2)*(x-1)^2)) \\ G. C. Greubel, Feb 21 2017

G.f.: x*(2+x+2*x^2+2*x^3)/((1+x+x^2)*(x-1)^2). - R. J. Mathar, Dec 04 2011
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4, with a(1)=2, a(2)=3, a(3)=5, a(4)=9. - Harvey P. Dale, Apr 29 2013
a(n) = 7*floor((n-1)/3)+2^((n-1) mod 3)+1. - Gary Detlefs, May 25 2014
a(n) = (1/9)*(21*n+4*sqrt(3)*sin((2*Pi*n)/3)-6*cos((2*Pi*n)/3)-12). - Alexander R. Povolotsky, May 25 2014
a(3k) = 7k-2, a(3k-1) = 7k-4, a(3k-2) = 7k-5. - Wesley Ivan Hurt, Jun 10 2016

More terms from Wesley Ivan Hurt, May 25 2014

A105452 Numerator of (7 n -1)/3.

Original entry on oeis.org

2, 13, 20, 9, 34, 41, 16, 55, 62, 23, 76, 83, 30, 97, 104, 37, 118, 125, 44, 139, 146, 51, 160, 167, 58, 181, 188, 65, 202, 209, 72, 223, 230, 79, 244, 251, 86, 265, 272, 93, 286, 293, 100, 307, 314, 107, 328, 335, 114, 349

Offset: 1

Zak Seidov, May 02 2005

Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

a(3*n+1) = 7*n+2 = A017005(n), a(3*n+2) = 2*n+13, a(3*n+3) = 21*n+20.
From Chai Wah Wu, Sep 24 2020: (Start)
a(n) = 2*a(n-3) - a(n-6) for n > 6.
G.f.: x*(x^5 + 8*x^4 + 5*x^3 + 20*x^2 + 13*x + 2)/(x^6 - 2*x^3 + 1). (End)

A115020 Count backwards from 100 in steps of 7.

Original entry on oeis.org

100, 93, 86, 79, 72, 65, 58, 51, 44, 37, 30, 23, 16, 9, 2

Offset: 0

Robert Happelberg (roberthappelberg(AT)yahoo.com), Feb 23 2006

Sometimes used to gauge the concentration ability of a patient with Alzheimer's disease. The combination of a simple arithmetic operation with an unusual decrement forces the patient to rely on short-term memory for the computation. Most patients would stop when they get to 2, but the psychiatrist might be satisfied with the first five or so terms. In this capacity the sequence is mentioned a few times in the movie "Safe House" (1998) in which Mace Sowell (Patrick Stewart) tries to memorize the sequence and is asked it by the psychiatrist (Hector Elizondo). In real life, other decrements can be used; Fish (1996) suggests 3 and 4.

In the next-to-last episode of "Boston Legal," Denny Crane (William Shatner) is asked this sequence by a doctor while in an MRI machine. He gets confused at 86, following it with 81. (Both Shatner and Stewart have played captain of the Enterprise on TV.) - Wilfredo Lopez (chakotay147138274(AT)yahoo.com), Dec 03 2008

Sharon Fish, "Alzheimer's: Caring for Your Loved One, Caring for Yourself", Wheaton, Illinois: Harold Shaw Publishers, 1996, p. 34

Max Hayman, Two minute clinical test for measurement of intellectual impairment in psychiatric disorders, Arch. Neurpsych. 47:3 (1942), pp. 454-464.
Tanya Khovanova, Non Recursions
Robert Thomas Manning, The serial sevens test, Arch Intern Med. 142:6 (1982), p. 1192.
Wikipedia, Serial sevens

First fifteen terms of A017005 backwards.

Range[100, 2, -7] (* Harvey P. Dale, Apr 11 2014 *)

a(n)=100-7*n \\ Charles R Greathouse IV, Feb 25 2019

(0 to 14).map(100 - 7 * ) // _Alonso del Arte, May 09 2019

a(n) = 100 - 7n.

A137184 Lucky numbers (A000959) which are congruent to 2 mod 7.

Original entry on oeis.org

9, 37, 51, 79, 93, 135, 163, 205, 219, 261, 289, 303, 331, 415, 429, 541, 583, 639, 723, 933, 961, 975, 1087, 1101, 1339, 1395, 1563, 1675, 1731, 1773, 1801, 1857, 1899, 1941, 1983, 2053, 2067, 2095, 2221, 2277, 2403, 2445, 2473, 2557, 2571, 2599, 2697, 2725

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 A017005.

A177060 a(n) = (7n+2)(7n+5) = 49n^2 + 49*n + 10.

Original entry on oeis.org

10, 108, 304, 598, 990, 1480, 2068, 2754, 3538, 4420, 5400, 6478, 7654, 8928, 10300, 11770, 13338, 15004, 16768, 18630, 20590, 22648, 24804, 27058, 29410, 31860, 34408, 37054, 39798, 42640, 45580, 48618, 51754, 54988, 58320, 61750, 65278, 68904, 72628, 76450

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) = 49*A002061(n+1) - 39. - Bruno Berselli, Aug 24 2010

			For n=1, a(1) = 98 + 10 = 108.
For n=2, a(2) = 98*2 + 108 = 304.
For n=3, a(3) = 98*3 + 304 = 598.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A002061, A017005, A017041, A177059.

a[n_] := (7*n + 2)*(7*n + 5); Array[a, 40, 0] (* Amiram Eldar, Feb 19 2023 *)

a(n)=49*n*(n+1)+10 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 98*n + a(n-1) with a(0) = 10.
From Amiram Eldar, Feb 19 2023: (Start)
a(n) = A017005(n)*A017041(n).
Sum_{n>=0} 1/a(n) = tan(3*Pi/14)*Pi/21.
Product_{n>=0} (1 - 1/a(n)) = sec(3*Pi/14)*cos(sqrt(13)*Pi/14).
Product_{n>=0} (1 + 1/a(n)) = sec(3*Pi/14)*cos(sqrt(5)*Pi/14). (End)
From Elmo R. Oliveira, Oct 24 2024: (Start)
G.f.: 2*(5 + 39*x + 5*x^2)/(1 - x)^3.
E.g.f.: exp(x)*(10 + 49*x*(2 + x)).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A239796 a(n) = 7n^2 + 2n - 15.

Original entry on oeis.org

-6, 17, 54, 105, 170, 249, 342, 449, 570, 705, 854, 1017, 1194, 1385, 1590, 1809, 2042, 2289, 2550, 2825, 3114, 3417, 3734, 4065, 4410, 4769, 5142, 5529, 5930, 6345, 6774, 7217, 7674, 8145, 8630, 9129, 9642, 10169, 10710, 11265, 11834, 12417, 13014, 13625, 14250, 14889, 15542, 16209, 16890

Offset: 1

Katherine Guo, Mar 26 2014

Follows the integer values from 1 on the parabola: 7*n^2 + 2*n - 15.
Real roots: (-1 +- sqrt(106))/7. - Wesley Ivan Hurt, Mar 26 2014
The first in the family of parabolas of the form: prime(k+3)*n^2 + prime(k)*n - prime(k+1)*prime(k+2), where k >= 1 (k=1 gives a(n)). - Wesley Ivan Hurt, Mar 26 2014

			For n=3, a(3) = 7*3^2 + 2*3 - 15 = 54; for n=6, a(6) = 7*6^2 + 2*6 - 15 = 249.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Vi Hart, Doodling in Math Class: Connecting Dots (2012) [Video]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

[7*n^2+2*n-15: n in [1..50]]; // Bruno Berselli, Mar 27 2014

A239796:=n->7*n^2 +2*n - 15; seq(A239796(n), n=1..50); # Wesley Ivan Hurt, Mar 26 2014

Table[7 n^2 + 2 n - 15, {n, 50}] (* Wesley Ivan Hurt, Mar 26 2014 *)
CoefficientList[Series[(6 - 35 x + 15 x^2)/(x - 1)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Mar 29 2014 *)

a(n)=7*n^2 + 2*n - 15 \\ Charles R Greathouse IV, Jan 21 2016

a(n) = n * A017005(n) - 15. - Wesley Ivan Hurt, Mar 26 2014

G.f.: -x*(6 - 35*x + 15*x^2)/(1 - x)^3. - Bruno Berselli, Mar 27 2014

A253671 a(n) = floor(A000111(n)/A000111(n-1)).

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 8, 9, 10, 10, 11, 12, 12, 13, 14, 14, 15, 15, 16, 17, 17, 18, 19, 19, 20, 21, 21, 22, 22, 23, 24, 24, 25, 26, 26, 27, 28, 28, 29, 29, 30, 31, 31, 32, 33, 33, 34, 35, 35, 36, 36, 37, 38, 38, 39, 40, 40

Offset: 1

Paul Curtz, Jan 08 2015, with the help of Jean-François Alcover

1, 2, 3, 4, ... first appear at n = 1, 3, 5, 7, 8, 10, 11, 13, ... . a(500) = 318.
Numbers appearing only once: interleave 4+7*n, 6+7*n, 9+7*n = 4, 6, 9, 11, 13, 16, ... .
This is a nondecreasing sequence.
The ratio a(n)/n asymptotically tends to 7/11 = 0.6363... - Jean-François Alcover, Jul 21 2015

			Floor of 1/1, 1/1, 2/1, 5/2, 16/5, 61/16, ... .
1=1*1+0, 1=1*1+0, 2=2*1+0, 5=2*2+1, 16=3*5+1, 61=3*16+13, 272=4*61+28, ... .

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

Cf. A000111, A010727, A017005, A017029, A017053.

max = 500; ee = Table[2^n*EulerE[n, 1] + EulerE[n] - 1, {n, 0, max}]; A000111 = Table[Differences[ee, n] // First // Abs, {n, 0, max}]; Table[Quotient[A000111[[n + 1]], A000111[[n]]], {n, 1, max}] (* Jean-François Alcover, Jan 08 2015 *)

Vec(x*(x^14-x^13+x^12-x^11+x^10+x^9+x^7+x^6+x^4+x^2+1)/((x-1)^2*(x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1)) + O(x^100)) \\ Colin Barker, Jan 22 2015

# requires python 3.2 or higher
from itertools import accumulate
A253671_list, blist, l1, l2 = [1], [1], 1, 1
for n in range(10**2):
    blist = list(reversed(list(accumulate(reversed(blist))))) + [0] if n % 2 else [0]+list(accumulate(blist))
    l2, l1 = l1, sum(blist)
    A253671_list.append(l1//l2) # Chai Wah Wu, Jan 29 2015

a(n+2) = a(n+1) + (0, 1, 0, followed by a sequence of period 11: repeat 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1).
a(n+12) = a(n+1) + (6, 7, 6, followed by 7's = A010727).
a(n) = a(n-1) + a(n-11) - a(n-12) for n>15. - Colin Barker, Jan 22 2015
G.f.: x*(x^14-x^13+x^12-x^11+x^10+x^9+x^7+x^6+x^4+x^2+1) / ((x-1)^2*(x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1)). - Colin Barker, Jan 22 2015

cp's OEIS Frontend

A168333 a(n) = (14*n + 7*(-1)^n + 1)/4.

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A017010 a(n) = (7*n+2)^6.

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A047310 Numbers that are congruent to {0, 1, 3, 4, 5, 6} mod 7.

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A047370 Numbers that are congruent to {2, 3, 5} mod 7.

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Magma

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A105452 Numerator of (7 n -1)/3.

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A115020 Count backwards from 100 in steps of 7.

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A137184 Lucky numbers (A000959) which are congruent to 2 mod 7.

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A177060 a(n) = (7*n+2)*(7*n+5) = 49*n^2 + 49*n + 10.

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A168333 a(n) = (14n + 7(-1)^n + 1)/4.

A177060 a(n) = (7n+2)(7n+5) = 49n^2 + 49*n + 10.

A239796 a(n) = 7n^2 + 2n - 15.