A266297 Numbers whose last digit is a square.

0, 1, 4, 9, 10, 11, 14, 19, 20, 21, 24, 29, 30, 31, 34, 39, 40, 41, 44, 49, 50, 51, 54, 59, 60, 61, 64, 69, 70, 71, 74, 79, 80, 81, 84, 89, 90, 91, 94, 99, 100, 101, 104, 109, 110, 111, 114, 119, 120, 121, 124, 129, 130, 131, 134, 139, 140, 141, 144, 149

Offset: 1

Author

Wesley Ivan Hurt, Dec 26 2015

Keywords

• nonn
• easy
• base

Comments

Numbers ending in 0, 1, 4 and 9.

Union of A008592, A017281, A017317 and A017377. - Hurt

None of these numbers are prime in Z[phi] (where phi = 1/2 + sqrt(5)/2 is the golden ratio), since the numbers in this sequence that are prime in Z can be expressed in the form (a - b sqrt(5))(a + b sqrt(5)). - Alonso del Arte, Dec 30 2015

Union of A197652 and A016897. - Wesley Ivan Hurt, Dec 31 2015

Union of A146763 and A090771. - Wesley Ivan Hurt, Jan 01 2016

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Crossrefs

Cf. A008592, A016897, A017281, A017317, A017377, A090771, A091084, A141158, A146763, A197652.

Programs

Magma

[(10*n-11+(-1)^n+(4+2*(-1)^n)*(-1)^((2*n-1+(-1)^n) div 4))/4: n in [1..60]]; // Vincenzo Librandi, Dec 27 2015

Maple

A266297:=n->(10*n-11+(-1)^n+(4+2*(-1)^n)*(-1)^((2*n-1+(-1)^n)/4))/4: seq(A266297(n), n=1..100);

Mathematica

Table[(10 n - 11 + (-1)^n + (4 + 2 (-1)^n)*(-1)^((2 n - 1 + (-1)^n)/4))/4, {n, 50}] (* G. C. Greubel, Dec 27 2015 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {0, 1, 4, 9, 10}, 60] (* Vincenzo Librandi, Dec 27 2015 *)
CoefficientList[Series[x*(1 + 3*x + 5*x^2 + x^3)/((x - 1)^2*(1 + x + x^2 + x^3)), {x, 0, 100}], x] (* Wesley Ivan Hurt, Dec 30 2015 *)
Flatten[Table[10n + {0, 1, 4, 9}, {n, 0, 19}]] (* Alonso del Arte, Dec 30 2015 *)
Select[Range[0,150],MemberQ[{0,1,4,9},Mod[#,10]]&] (* Harvey P. Dale, Jul 30 2019 *)

PARI

is(n) = issquare(n%10); \\ Altug Alkan, Dec 29 2015

Formula

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

a(n) = a(n - 1) + (n - 4) - a(n - 5) for n > 5.

a(n) = (10n - 11 + (-1)^n + (4 + 2(-1)^n) * (-1)^((2n - 1 + (-1)^n)/4))/4.

a(n+1) - a(n) = A091084(n+1) for n>0.

Sum_{n>=2} (-1)^n/a(n) = (14*sqrt(5)*arccoth(sqrt(5)) - 2*Pi*sqrt(1-2/sqrt(5)) + 16*log(2) + 5*log(5))/40. - Amiram Eldar, Jul 30 2024