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A273372
Squares ending in digit 1.
3
1, 81, 121, 361, 441, 841, 961, 1521, 1681, 2401, 2601, 3481, 3721, 4761, 5041, 6241, 6561, 7921, 8281, 9801, 10201, 11881, 12321, 14161, 14641, 16641, 17161, 19321, 19881, 22201, 22801, 25281, 25921, 28561, 29241, 32041, 32761, 35721, 36481, 39601, 40401
OFFSET
1,2
COMMENTS
Intersection of A000290 and A017281; also, union of A017282 and A017378. The square roots are in A017281 or in A017377 (numbers ending in 1 or 9, respectively). - David A. Corneth, May 22 2016
FORMULA
G.f.: x*(1 + 80*x + 38*x^2 + 80*x^3 + x^4) / ((1 + x)^2*(1 - x)^3).
a(n) = 10*A132356(n-1) + 1 = 5*(10*n+(-1)^n-5)*(2*n+(-1)^n-1)/4+1.
a(n) = (5*n - 5/2 + (3/2)*(-1)^n)^2 = 25*n^2 - 25*n + 17/2 + 15*n*(-1)^n - (15/2)*(-1)^n. - David A. Corneth, May 21 2016
a(n) = A090771(n)^2. - Michel Marcus, May 25 2016
Sum_{n>=1} 1/a(n) = Pi^2*(3+sqrt(5))/50. - Amiram Eldar, Feb 16 2023
MATHEMATICA
Table[5 (10 n + (-1)^n - 5) (2 n + (-1)^n - 1)/4 + 1, {n, 1, 50}]
PROG
(Magma) /* By definition: */ [n^2: n in [0..200] | Modexp(n, 2, 10) eq 1];
(Magma) [5*(10*n+(-1)^n-5)*(2*n+(-1)^n-1)/4+1: n in [1..50]];
(Ruby) p (1..(n + 1) / 2).inject([]){|s, i| s + [(10 * i - 9) ** 2, (10 * i - 1) ** 2]}[0..n - 1] # Seiichi Manyama, May 24 2016
(Python)
A273372_list = [(10*n+m)**2 for n in range(10**3) for m in (1, 9)] # Chai Wah Wu, May 24 2016
CROSSREFS
Cf. A017281 (numbers ending in 1), A017283 (cubes ending in 1).
Cf. similar sequences listed in A273373.
Sequence in context: A357168 A253261 A280587 * A366079 A053887 A119943
KEYWORD
nonn,base,easy
AUTHOR
Vincenzo Librandi, May 21 2016
EXTENSIONS
Edited by Bruno Berselli, May 24 2016
STATUS
approved