A273373 - OEIS

%I #44 Feb 16 2023 05:08:17

%S 16,36,196,256,576,676,1156,1296,1936,2116,2916,3136,4096,4356,5476,

%T 5776,7056,7396,8836,9216,10816,11236,12996,13456,15376,15876,17956,

%U 18496,20736,21316,23716,24336,26896,27556,30276,30976,33856,34596,37636,38416,41616

%N Squares ending in digit 6.

%C These are the only squares whose second last digit is odd. This implies that the only squares whose last two digits are the same are those ending with 0 or 4; those ending with 1, 5, and 9 are paired with even second last digits. - _Waldemar Puszkarz_, May 24 2016

%H Seiichi Manyama, <a href="/A273373/b273373.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).

%F G.f.: 4*x*(4 + 5*x + 32*x^2 + 5*x^3 + 4*x^4)/((1 + x)^2*(1 - x)^3).

%F a(n) = 4*A047221(n)^2 = (10*n - 3*(-1)^n - 5)^2/4.

%F a(n) = A090773(n)^2. - _Michel Marcus_, May 25 2016

%F Sum_{n>=1} 1/a(n) = 2*Pi^2/(25*(5+sqrt(5))). - _Amiram Eldar_, Feb 16 2023

%p seq(seq((10*i+j)^2,j=[4,6]),i=0..20); # _Robert Israel_, May 24 2016

%t Table[(10 n - 3 (-1)^n - 5)^2/4, {n, 1, 50}]

%t CoefficientList[Series[4 (4 + 5 x + 32 x^2 + 5 x^3 + 4 x^4) / ((1 + x)^2 (1 - x)^3), {x, 0, 50}], x]

%t Select[Range[250]^2,Mod[#,10]==6&] (* _Harvey P. Dale_, May 31 2020 *)

%o (Magma) /* By definition: */ [n^2: n in [0..200] | Modexp(n,2,10) eq 6];

%o (Magma) [(10*n - 3*(-1)^n - 5)^2/4: n in [1..50]];

%Y Cf. A000290, A047221, A090773.

%Y Cf. A017341 (numbers ending in 6), A017343 (cubes ending in 6).

%Y Cf. squares with last digit k: A017270 (k=0), A273372 (k=1), A273375 (k=4), A017330 (k=5), this sequence (k=6), A273374 (k=9).

%K nonn,base,easy

%O 1,1

%A _Vincenzo Librandi_, May 21 2016

%E Corrected and extended by _Bruno Berselli_, May 23 2016