A317633 -id:A317633 - cp's OEIS frontend

A090772 Numbers that are congruent to {2, 8} mod 10.

2, 8, 12, 18, 22, 28, 32, 38, 42, 48, 52, 58, 62, 68, 72, 78, 82, 88, 92, 98, 102, 108, 112, 118, 122, 128, 132, 138, 142, 148, 152, 158, 162, 168, 172, 178, 182, 188, 192, 198, 202, 208, 212, 218, 222, 228, 232, 238, 242, 248, 252, 258, 262, 268, 272, 278, 282

Offset: 1

Giovanni Teofilatto, Feb 07 2004

Their square ends in the digit 4. - Kausthub Gudipati, Sep 08 2011

10*a(n) = 20, 80, 120, 180, 220, ... are the only numbers written in French ending in "vingt(s)". - Paul Curtz, Aug 02 2018

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A019952, A047209, A090298, A179290, A226294, A317633.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(2*x*(1+3*x+x^2)/((1+x)*(1-x)^2))); // G. C. Greubel, Aug 08 2018

Union@ Flatten@ Outer[Plus, {2, 8}, 10 Range[0, 28]] (* or *)
CoefficientList[Series[2 (1 + 3x + x^2)/((1 + x) (1 - x)^2), {x, 0, 57}], x] (* Michael De Vlieger, Aug 02 2018 *)
LinearRecurrence[{1, 1, -1}, {2, 8, 12}, 61] (* Robert G. Wilson v, Aug 08 2018 *)

is(n) = #setintersect([2, 8], [n%10]) > 0 \\ Felix Fröhlich, Aug 02 2018

Vec(2*x*(1+3*x+x^2)/((1+x)*(1-x)^2) + O(x^60)) \\ Felix Fröhlich, Aug 02 2018

a(n) = 2 * A047209(n).
a(n) = 10*n - a(n-1) - 10 (with a(1)=2). - Vincenzo Librandi, Nov 16 2010
G.f.: 2*x*(1+3*x+x^2)/((1+x)*(1-x)^2). - Bruno Berselli, Sep 08 2011
a(1) = 2. For n > 1, a(n) = a(n-1) + A226294(n). - Felix Fröhlich, Aug 02 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(1+2/sqrt(5))*Pi/10. - Amiram Eldar, Dec 28 2021
E.g.f.: 2 + ((10*x - 5)*exp(x) + exp(-x))/2. - David Lovler, Sep 03 2022
From Amiram Eldar, Nov 23 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = tan(3*Pi/10) (A019952).
Product_{n>=1} (1 + (-1)^n/a(n)) = cosec(2*Pi/5)/2 (= A179290 / 2). (End)

Edited and extended by Ray Chandler, Feb 10 2004

A317657 Numbers congruent to {15, 75, 95} mod 100.

Original entry on oeis.org

15, 75, 95, 115, 175, 195, 215, 275, 295, 315, 375, 395, 415, 475, 495, 515, 575, 595, 615, 675, 695, 715, 775, 795, 815, 875, 895, 915, 975, 995, 1015, 1075, 1095, 1115, 1175, 1195, 1215, 1275, 1295, 1315, 1375, 1395, 1415, 1475, 1495, 1515

Offset: 1

Paul Curtz, Aug 03 2018

Numbers written in French ending in "quinze".

a(n) = 5 * (3, 15, 19, 23, 35, 39, 43, 55, 59, ... ).

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

Cf. A008587, A008854, A047205, A090772, A290781, A317633.

Filtered([0..1520], n->n mod 100=15 or n mod 100=75 or n mod 100=95); # Muniru A Asiru, Aug 29 2018

select(n->modp(n,100)=15 or modp(n,100)=75 or modp(n,100)=95,[$0..1520]); # Muniru A Asiru, Aug 29 2018

Rest@ CoefficientList[Series[(5 x (x^3 + 4 x^2 + 12 x + 3))/((x^2 + x + 1) (x - 1)^2), {x, 0, 46}], x] (* Michael De Vlieger, Aug 05 2018 *)
Table[100*n/3 - 80*Sin[2*n*Pi/3]/(3*Sqrt[3]) - 5,{n,1,46}] (* Stefano Spezia, Aug 29 2018 *)

a(n) = 10*A317633(n) + 5.
a(n) = a(n-3) + 100, a(1) = 15, a(2) = 75, a(3) = 95.
From Franck Maminirina Ramaharo, Aug 05 2018: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4), n>4.
a(n) = A290781(A047205(n)).
a(n) = 20*A008854(n+1) - 5.
a(n) = 100*n/3 - 80*sin(2*n*Pi/3)/(3*sqrt(3)) - 5.
G.f.: (5*x*(x^3 + 4*x^2 + 12*x + 3))/((x^2 + x + 1)*(x - 1)^2).
E.g.f.: 100*x*exp(x)/3 - 80*sin(sqrt(3)*x/2)/(exp(x/2)*(3*sqrt(3)))-5*exp(x).
(End)

cp's OEIS Frontend

A090772 Numbers that are congruent to {2, 8} mod 10.

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