A273374 Squares ending in digit 9.

9, 49, 169, 289, 529, 729, 1089, 1369, 1849, 2209, 2809, 3249, 3969, 4489, 5329, 5929, 6889, 7569, 8649, 9409, 10609, 11449, 12769, 13689, 15129, 16129, 17689, 18769, 20449, 21609, 23409, 24649, 26569, 27889, 29929, 31329, 33489, 34969, 37249, 38809

Offset: 1

Author

Vincenzo Librandi, May 21 2016

Keywords

• nonn
• base
• easy

Comments

A quasipolynomial of order two and degree two: a(n) = 25n^2 - 30n + 9 if n is even and 25n^2 - 20n + 4 if n is odd. - Charles R Greathouse IV, Nov 03 2021

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).

Crossrefs

Cf. A000290, A016754, A063226.

Cf. A017377 (numbers ending in 9), A017379 (cubes ending in 9).

Cf. similar sequences listed in A273373.

Programs

Magma

/* By definition: */ [n^2: n in [0..200] | Modexp(n,2,10) eq 9];

Magma

[6+(50*(n-1)*n-5*(2*n-1)*(-1)^n+1)/2: n in [1..50]];

Mathematica

Table[6 + (50 (n - 1) n - 5 (2 n - 1) (-1)^n + 1)/2, {n, 1, 50}]

PARI

a(n)=(5*n-3+n%2)^2 \\ Charles R Greathouse IV, Nov 03 2021

Formula

G.f.: x*(9 + 40*x + 102*x^2 + 40*x^3 + 9*x^4)/((1 + x)^2*(1 - x)^3).

a(n) = 6 + (50*(n-1)*n - 5*(2*n-1)*(-1)^n + 1)/2.

a(n) = A063226(n)^2. - Seiichi Manyama, May 25 2016

Sum_{n>=1} 1/a(n) = Pi^2*(3-sqrt(5))/50. - Amiram Eldar, Feb 16 2023

Extensions

Corrected and extended by Bruno Berselli, May 21 2016