A253195 - OEIS

%I #22 Sep 07 2022 10:25:12

%S 5,8,14,17,23,26,32,35,41,44,50,53,59,62,68,71,77,80,86,89,95,98,104,

%T 107,113,116,122,125,131,134,140,143,149,152,158,161,167,170,176,179,

%U 185,188,194,197,203,206,212,215,221,224,230,233,239,242,248,251

%N Numbers congruent to 5 or 8 mod 9.

%C These numbers cannot be written as the sum of two triangular numbers.

%H Arkadiusz Wesolowski, <a href="/A253195/b253195.txt">Table of n, a(n) for n = 1..1000</a>

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

%F a(n) = a(n-1) + a(n-2) - a(n-3), n >= 4.

%F G.f.: x*(5 + 3*x + x^2)/((1 + x)*(1 - x)^2).

%F a(n) = a(n-2) + 9.

%F a(n) = 9*n - a(n-1) - 5.

%F a(n) = 4*n + 2*ceiling(n/2) - floor(n/2) - 1.

%F a(n) = (9*n - (3/2)*(1 + (- 1)^n) + 1)/2.

%F E.g.f.: 1 + ((18*x - 1)*exp(x) - 3*exp(-x))/4. - _David Lovler_, Sep 06 2022

%t LinearRecurrence[{1, 1, -1}, {5, 8, 14}, 56]

%t Select[Range[300],MemberQ[{5,8},Mod[#,9]]&] (* _Harvey P. Dale_, Mar 17 2020 *)

%o (Magma) [n: n in [0..251] | n mod 9 in {5, 8}];

%o (PARI) Vec(x*(5 + 3*x + x^2)/((1 + x)*(1 - x)^2) + O(x^80)) \\ _Michel Marcus_, Mar 25 2015

%Y Subsequence of A014132 and of A020757.

%K nonn,easy

%O 1,1

%A _Arkadiusz Wesolowski_, Mar 24 2015