A050519 -id:A050519

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A249674

a(n) = 30*n.

+10
26

0, 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330, 360, 390, 420, 450, 480, 510, 540, 570, 600, 630, 660, 690, 720, 750, 780, 810, 840, 870, 900, 930, 960, 990, 1020, 1050, 1080, 1110, 1140, 1170, 1200, 1230, 1260, 1290, 1320, 1350, 1380, 1410, 1440

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Numbers divisible by 2, 3 and 5. - Robert Israel, Nov 19 2014

a(n) is the maximum score of a 10-pin n-frame bowling game and the maximum score of an n-pin 10-frame bowling game, given the rules: a strike is worth the number of pins in each frame plus the number of pins knocked down by the next two balls (except in the last frame), a spare is worth the number of pins in each frame plus the number of pins knocked down by the next ball (except in the last frame), and if a strike or spare is earned in the last frame then the player must continue to throw balls until they have thrown 3 balls in the last frame. - Iain Fox, Mar 02 2018

LINKS

Iain Fox, Table of n, a(n) for n = 0..10000

Eric Weisstein's World of Mathematics, Bowling

Wikipedia, Ten-pin bowling

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

FORMULA

G.f.: 30*x/(x-1)^2; a(n) = 2*a(n-1) - a(n-2). - Wesley Ivan Hurt, Nov 18 2014

a(n) = 2*A008597(n) = 3*A008592(n) = 5*A008588(n) = 6*A008587(n) = 10*A008585(n) = 15*A005843(n). - Omar E. Pol, Nov 24 2014

EXAMPLE

a(7) = 7 * 30 = 210.

MAPLE

A249674:=n->30*n: seq(A249674(n), n=0..50); # Wesley Ivan Hurt, Nov 18 2014

MATHEMATICA

30*Range[0, 59] (* Alonso del Arte, Nov 18 2014 *)

PROG

(Magma) [30*n : n in [0..50]]; // Wesley Ivan Hurt, Nov 18 2014

(PARI) vector(100, n, 30*(n-1)) \\ Derek Orr, Nov 18 2014

(PARI) first(n) = Vec(30*x/(x-1)^2 + O(x^n), -n) \\ Iain Fox, Mar 02 2018

CROSSREFS

Cf. A008585, A008592, A050519.

KEYWORD

nonn,easy

AUTHOR

Kaylan Purisima, Nov 03 2014

STATUS

approved

A050518

An arithmetic progression of at least 6 terms having the same value of phi starts at these numbers.

+10
3

583200, 1166400, 1749600, 2332800, 2916000, 3499200, 4082400, 4665600, 5248800, 5832000, 6415200, 6998400, 7581600, 8164800, 8748000, 9331200, 9914400, 10497600, 11080800, 11664000, 12247200, 12830400, 13413600, 13996800, 14580000, 15163200, 15746400

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

From Mauro Fiorentini, Apr 12 2015 (Start):

The following are all the terms between 13413600 and 10^9 with increment <= 1000:

13996800, 14580000, 15163200, 15746400, 16329600, 16912800, 17496000, 18079200, 18662400, 19245600, 65621220, 85731240, 131242440, 165488430, 171462480, 196863660, 257193720, 262484880, 330976860, 342924960, 496465290, 504932430, 544924830, 661953720, 827442150, 892306830, 992930580.

(End)

If phi is constant on the arithmetic progression A = [x, x+d, ..., x+m*d], and k is an integer such that each prime factor of k divides either all members of A or no members of A, then phi is also constant on the arithmetic progression k*A = [x*k, x*k+d*k, ..., x*k+m*(d*k)]. - Robert Israel, Apr 12 2015

The a.p. of 7 terms starting at 1158419010 with increment 210 have the same value of phi. - Robert Israel, Apr 15 2015

a(n) = 583200*n for n <= 112, but a(113) = 65621220. - Robert Israel, May 10 2015

LINKS

Robert Israel, Table of n, a(n) for n = 1..114 (all the terms <= 6.6*10^7).

Tanya Khovanova, Non Recursions

Eric Weisstein's World of Mathematics, Totient function.

MAPLE

N:= 10^7: # to get all terms <= N

with(numtheory):

Res:= NULL:

phis:= {seq(phi(i), i=2..N)}:

for m in phis do

S:= convert(invphi(m), set);

if nops(S) < 6 then next fi;

for d from 0 to 4 do

Sd[d]:= select(t-> (t mod 5 = d), S, d);

nd:= nops(Sd[d]);

for i0 from 1 to nd-1 do

s0:= Sd[d][i0];

if s0 > N then break fi;

for i5 from i0+1 to nd do

s5:= Sd[d][i5];

incr:= (s5 - s0)/5;

if {s0+incr, s0+2*incr, s0+3*incr, s0+4*incr} subset S then

Res:= Res, [s0, incr];

od;

od:

sort([Res], (s, t)->s[1]<t[1]); # gives both A050518 and A050519 entries

map2(op, 1, %); # Robert Israel, Apr 16 2015

CROSSREFS

Cf. A000010, A050495, A050496, A050497, A050515-A050520.

The increments are in A050519. The values of phi are in A050520.

KEYWORD

nonn

AUTHOR

Jud McCranie, Dec 28 1999

STATUS

approved

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