A175887 - OEIS

%I #44 Nov 27 2024 07:25:00

%S 1,14,16,29,31,44,46,59,61,74,76,89,91,104,106,119,121,134,136,149,

%T 151,164,166,179,181,194,196,209,211,224,226,239,241,254,256,269,271,

%U 284,286,299,301,314,316,329,331,344,346,359,361,374,376,389,391,404

%N Numbers that are congruent to {1, 14} mod 15.

%C Cf. property described by Gary Detlefs in A113801: more generally, these numbers are of the form (2*h*n+(h-4)*(-1)^n-h)/4 (h, n natural numbers), therefore ((2*h*n+(h-4)*(-1)^n-h)/4)^2-1==0 (mod h); in this case, a(n)^2-1==0 (mod 15).

%H Bruno Berselli, <a href="/A175887/b175887.txt">Table of n, a(n) for n = 1..10000</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).

%F G.f.: x*(1+13*x+x^2)/((1+x)*(1-x)^2).

%F a(n) = (30*n+11*(-1)^n-15)/4.

%F a(n) = -a(-n+1) = a(n-1)+a(n-2)-a(n-3).

%F a(n) = 15*A000217(n-1) -2*sum(a(i), i=1..n-1) +1 for n>1.

%F a(n) = A047209(A047225(n+1)).

%F Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/15)*cot(Pi/15) = A019693 * A019976 / 10. - _Amiram Eldar_, Dec 04 2021

%F E.g.f.: 1 + ((30*x - 15)*exp(x) + 11*exp(-x))/4. - _David Lovler_, Sep 05 2022

%F From _Amiram Eldar_, Nov 25 2024: (Start)

%F Product_{n>=1} (1 - (-1)^n/a(n)) = (Pi/15)*cosec(Pi/15).

%F Product_{n>=2} (1 + (-1)^n/a(n)) = 2*cos(Pi/15). (End)

%t Select[Range[1, 450], MemberQ[{1,14}, Mod[#, 15]]&]

%t CoefficientList[Series[(1 + 13 x + x^2) / ((1 + x) (1 - x)^2), {x, 0, 55}], x] (* _Vincenzo Librandi_, Aug 19 2013 *)

%o (Magma) [n: n in [1..450] | n mod 15 in [1,14]];

%o (Haskell)

%o a175887 n = a175887_list !! (n-1)

%o a175887_list = 1 : 14 : map (+ 15) a175887_list

%o -- _Reinhard Zumkeller_, Jan 07 2012

%o (Magma) [(30*n+11*(-1)^n-15)/4: n in [1..55]]; // _Vincenzo Librandi_, Aug 19 2013

%o (PARI) a(n)=(30*n+11*(-1)^n-15)/4 \\ _Charles R Greathouse IV_, Sep 28 2015

%Y Cf. A000217, A019693, A019976, A113801, A175886, A091998, A175885, A090771, A056020, A047522, A047336, A007310, A047209, A005408, A001651.

%Y Cf. A132240 (primes).

%K nonn,easy

%O 1,2

%A _Bruno Berselli_, Oct 08 2010 - Nov 17 2010