A273373 Squares ending in digit 6.

16, 36, 196, 256, 576, 676, 1156, 1296, 1936, 2116, 2916, 3136, 4096, 4356, 5476, 5776, 7056, 7396, 8836, 9216, 10816, 11236, 12996, 13456, 15376, 15876, 17956, 18496, 20736, 21316, 23716, 24336, 26896, 27556, 30276, 30976, 33856, 34596, 37636, 38416, 41616

Offset

1,1

Comments

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

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

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

Formula

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

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

a(n) = A090773(n)^2. - Michel Marcus, May 25 2016

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

Maple

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

Mathematica

Table[(10 n - 3 (-1)^n - 5)^2/4, {n, 1, 50}]
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]
Select[Range[250]^2, Mod[#, 10]==6&] (* Harvey P. Dale, May 31 2020 *)

Prog

(Magma) /* By definition: */ [n^2: n in [0..200] | Modexp(n, 2, 10) eq 6];
(Magma) [(10*n - 3*(-1)^n - 5)^2/4: n in [1..50]];

Crossrefs

Cf. A000290, A047221, A090773.

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

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).

Sequence in context: A076956 A075369 A318425 * A264546 A218523 A300535

Adjacent sequences: A273370 A273371 A273372 * A273374 A273375 A273376

Keyword

nonn,base,easy

Author

Vincenzo Librandi, May 21 2016

Extensions

Corrected and extended by Bruno Berselli, May 23 2016

Status

approved