A131579 -id:A131579 - cp's OEIS frontend

A144468 Final digit of multiples of 7.

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

Offset: 0

Paul Curtz, Oct 09 2008

Period 10: repeat [0, 7, 4, 1, 8, 5, 2, 9, 6, 3]. The ten digits by jumps of three terms or a(3n). Also reverse of A131579(n+1), n =1..10, repeated.

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

Cf. A131579.

Table[Last[IntegerDigits[7n]], {n, 0, 98}] (* Alonso del Arte, Feb 06 2014 *)
Mod[7*Range[0,100],10] (* or *) PadRight[{},100,{0,7,4,1,8,5,2,9,6,3}] (* Harvey P. Dale, Jul 28 2017 *)

a(n)=7*n%10 \\ Charles R Greathouse IV, Jun 02 2011

a(n) = 7*n (mod 10) = -3*n (mod 10), n >= 0. See the name.

a(n+10) = a(n). - Wesley Ivan Hurt, Apr 16 2024

A172447 a(n) = (-1 + 52^(2n + 1) - 3*n)/9.

Original entry on oeis.org

1, 4, 17, 70, 283, 1136, 4549, 18202, 72815, 291268, 1165081, 4660334, 18641347, 74565400, 298261613, 1193046466, 4772185879, 19088743532, 76354974145, 305419896598, 1221679586411, 4886718345664, 19546873382677, 78187493530730, 312749974122943, 1250999896491796

Offset: 0

Paul Curtz, Feb 03 2010

a(n) mod 10 gives the 10-periodic sequence 1, 4, 7, 0, 3, 6, 9, 2, 5, 8 (and repeat, A131579 shifted, A144468 reversed) which contains all ten digits, that has a "palindromic" symmetry: 1 + 8 = 4 + 5 = 7 + 2 = 0 + 9 = 3 + 6 = 9.

The inverse binomial transform gives 1, 3, 10, 30, 90, ... (A062107 shifted). - R. J. Mathar, Feb 11 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-9,4).

Cf. A072197 (first differences).

[(-1+5*2^(2*n+1)-3*n)/9: n in [0..30]]; // Vincenzo Librandi, Aug 05 2011

LinearRecurrence[{6, -9, 4}, {1, 4, 17}, 30] (* Harvey P. Dale, Mar 25 2016 *)
((-1 + 5 2^(2# + 1) - 3#)/9  &) /@ Range[0, 29] (* Alonso del Arte, Apr 25 2020 *)

a(n)=(10*4^n-3*n)\9 \\ Charles R Greathouse IV, Jul 21 2015

val powerOf2: LazyList[BigInt] = LazyList.iterate(1: BigInt)(_ * 2)
(0 to 29).map(n => (-1 + 5 * powerOf2(2 * n + 1) - 3 * n)/9) // Alonso del Arte, Apr 25 2020

a(n) = 6*a(n - 1) - 9*a(n - 2) + 4*a(n - 3).
a(n + 1) - 4*a(n) = n.
a(n) = A172416(2n + 1).
G.f.: (1 - 2*x + 2*x^2)/((1 - 4*x) * (x - 1)^2). - R. J. Mathar, Feb 11 2010
E.g.f.: (10*exp(4*x) - (1 + 3*x)*exp(x))/9. - G. C. Greubel, Nov 02 2018

Definition replaced by closed formula by R. J. Mathar, Feb 11 2010

A017359 a(n) = (10*n + 7)^7.

Original entry on oeis.org

823543, 410338673, 10460353203, 94931877133, 506623120463, 1954897493193, 6060711605323, 16048523266853, 37725479487783, 80798284478113, 160578147647843, 300124211606973, 532875860165503

Offset: 0

N. J. A. Sloane

The penultimate digit of a(n) is period 10, repeat (4,7,0,3,6,9,2,5,8,1) = A131579(n+8). - Paul Curtz, Mar 07 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

[(10*n+7)^7: n in [0..20]]; // Vincenzo Librandi, Aug 14 2011

Table[(10*n + 7)^7, {n,0,20}] (* G. C. Greubel, Nov 08 2018 *)

vector(20, n, n--; (10*n + 7)^7) \\ G. C. Greubel, Nov 08 2018

From G. C. Greubel, Nov 08 2018: (Start)
G.f.: (823543 + 403750329*x + 7200703023*x^2 + 22692415945*x^3 + 17136675405*x^4 + 2902898547*x^5 + 62731021*x^6 + 2187*x^7)/(1-x)^8.
E.g.f.: (823543 +409515130*x +4820249700*x^2 +10796835000*x^3 + 7834050000*x^4 +2237900000*x^5 +259000000*x^6 + 10000000*x^7)*exp(x). (End)

A144459 a(n) = (3n+1)(5*n+1).

Original entry on oeis.org

1, 24, 77, 160, 273, 416, 589, 792, 1025, 1288, 1581, 1904, 2257, 2640, 3053, 3496, 3969, 4472, 5005, 5568, 6161, 6784, 7437, 8120, 8833, 9576, 10349, 11152, 11985, 12848, 13741, 14664, 15617, 16600, 17613, 18656, 19729, 20832, 21965, 23128, 24321, 25544, 26797, 28080

Offset: 0

Paul Curtz, Oct 08 2008

This appears in a "diagonal" scan through the numerators of the fractions of the hydrogen spectrum: A005563(4), A061037(9), A061039(13), etc.

a(n) mod 9 is a sequence of period length 9: repeat 1, 6, 5, 7, 3, 2, 4, 0, 8 (a permutation of A142069).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001622, A005563, A016777, A016861, A061037, A061039, A131579, A142069.

[(3*n+1)*(5*n+1): n in [0..40]]; // Vincenzo Librandi, Aug 07 2011

Table[(3n+1)(5n+1),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{1,24 ,77}, 50] (* Harvey P. Dale, Jul 16 2014 *)

a(n)=(3*n+1)*(5*n+1) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = A016777(n)*A016861(n).
a(n) mod 10 = A131579(n+7).
G.f.: (1+21*x+8*x^2) / (1-x)^3 . - R. J. Mathar, Jul 01 2011
a(0)=1, a(1)=24, a(2)=77, a(n) = 3*a(n-1) -3*a(n-2) +a(n-3). - Harvey P. Dale, May 02 2015
E.g.f.: (1 + 23*x + 15*x^2)*exp(x). - G. C. Greubel, Sep 20 2018
Sum_{n>=0} 1/a(n) = sqrt(2 + 3/sqrt(5) - sqrt(3 + 6/sqrt(5)))*Pi/(2*sqrt(6)) + sqrt(5)*log(phi)/4 + 5*log(5)/8 - 3*log(3)/4, where phi is the golden ratio (A001622). - Amiram Eldar, Sep 17 2023

cp's OEIS Frontend

A144468 Final digit of multiples of 7.

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A172447 a(n) = (-1 + 5*2^(2*n + 1) - 3*n)/9.

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A017359 a(n) = (10*n + 7)^7.

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Magma

Mathematica

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A144459 a(n) = (3*n+1)*(5*n+1).

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Author

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Programs

Magma

Mathematica

PARI

Formula

A172447 a(n) = (-1 + 52^(2n + 1) - 3*n)/9.

A144459 a(n) = (3n+1)(5*n+1).