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

1,1

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).

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

Mathematica

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

Prog

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

Crossrefs

Cf. A000290, A016754, A063226.

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

Cf. similar sequences listed in A273373.

Sequence in context: A212503 A295019 A164343 * A020245 A082608 A058031

Adjacent sequences: A273371 A273372 A273373 * A273375 A273376 A273377

Keyword

nonn,base,easy

Author

Vincenzo Librandi, May 21 2016

Extensions

Corrected and extended by Bruno Berselli, May 21 2016

Status

approved