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Numbers that are congruent to {1, 4, 11, 14} mod 15.
5

%I #29 Sep 08 2022 08:46:01

%S 1,4,11,14,16,19,26,29,31,34,41,44,46,49,56,59,61,64,71,74,76,79,86,

%T 89,91,94,101,104,106,109,116,119,121,124,131,134,136,139,146,149,151,

%U 154,161,164,166,169,176,179,181,184,191,194,196,199,206,209,211,214,221,224

%N Numbers that are congruent to {1, 4, 11, 14} mod 15.

%C The exponents in the q-series for A204220 are the squares of the numbers of this sequence.

%C The product of any two terms belongs to the sequence and therefore also a(n)^2, a(n)^3, a(n)^4, etc. - _Bruno Berselli_, Nov 28 2012

%H G. C. Greubel, <a href="/A204542/b204542.txt">Table of n, a(n) for n = 1..5000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).

%F G.f.: x * (1 + 3*x + 7*x^2 + 3*x^3 + x^4) / ((1-x) * (1-x^4)).

%F a(n) = -a(1-n), a(n) = 15 + a(n-4), a(n) = floor(15 * n / 4) - ((n + 1) mod 4) for all n in Z.

%F a(n) = (30*n+10*i^(n*(n+1))-3*(-1)^n+1)/8 -2, where i=sqrt(-1). - _Bruno Berselli_, Nov 28 2012

%F a(n) = a(n-1) + a(n-4) - a(n-5) for n>5. - _Wesley Ivan Hurt_, Jun 07 2016

%F E.g.f.: (4 - 5*(sin(x) - cos(x)) + 3*(5*x - 2)*sinh(x) + 3*(5*x - 3)*cosh(x))/4. - _Ilya Gutkovskiy_, Jun 07 2016

%F Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(2*(sqrt(5)+5))*Pi/15. - _Amiram Eldar_, Dec 30 2021

%e G.f. = x + 4*x^2 + 11*x^3 + 14*x^4 + 16*x^5 + 19*x^6 + 26*x^7 + 29*x^8 + 31*x^9 + ...

%p A204542:=n->floor(15 * n / 4) - ((n + 1) mod 4): seq(A204542(n), n=1..100); # _Wesley Ivan Hurt_, Jun 07 2016

%t Select[Range[250], MemberQ[{1,4,11,14}, Mod[#,15]]&] (* or *) LinearRecurrence[{1,0,0,1,-1}, {1,4,11,14,16}, 60] (* _Harvey P. Dale_, Apr 15 2015 *)

%o (PARI) {a(n) = (n * 15) \ 4 - (n + 1) % 4};

%o (PARI) {a(n) = if( n<1, -a(1-n), polcoeff( x * (1 + 3*x + 7*x^2 + 3*x^3 + x^4) / ((1-x) * (1-x^4)) + x * O(x^n), n))};

%o (Magma) [n : n in [0..100] | n mod 15 in [1, 4, 11, 14]]; // _Wesley Ivan Hurt_, Jun 07 2016

%Y Cf. A204220, A204221.

%K nonn,easy

%O 1,2

%A _Michael Somos_, Jan 16 2012