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A273372 Squares ending in digit 1.

%I A273372 #66 Aug 17 2025 18:01:03
%S A273372 1,81,121,361,441,841,961,1521,1681,2401,2601,3481,3721,4761,5041,
%T A273372 6241,6561,7921,8281,9801,10201,11881,12321,14161,14641,16641,17161,
%U A273372 19321,19881,22201,22801,25281,25921,28561,29241,32041,32761,35721,36481,39601,40401
%N A273372 Squares ending in digit 1.
%C A273372 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
%H A273372 Seiichi Manyama, <a href="/A273372/b273372.txt">Table of n, a(n) for n = 1..10000</a>
%H A273372 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A273372 G.f.: x*(1 + 80*x + 38*x^2 + 80*x^3 + x^4) / ((1 + x)^2*(1 - x)^3).
%F A273372 a(n) = 10*A132356(n-1) + 1 = 5*(10*n+(-1)^n-5)*(2*n+(-1)^n-1)/4+1.
%F A273372 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
%F A273372 a(n) = A090771(n)^2. - _Michel Marcus_, May 25 2016
%F A273372 Sum_{n>=1} 1/a(n) = Pi^2*(3+sqrt(5))/50. - _Amiram Eldar_, Feb 16 2023
%t A273372 Table[5 (10 n + (-1)^n - 5) (2 n + (-1)^n - 1)/4 + 1, {n, 1, 50}]
%o A273372 (Magma) /* By definition: */ [n^2: n in [0..200] | Modexp(n,2,10) eq 1];
%o A273372 (Magma) [5*(10*n+(-1)^n-5)*(2*n+(-1)^n-1)/4+1: n in [1..50]];
%o A273372 (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
%o A273372 (Python)
%o A273372 A273372_list = [(10*n+m)**2 for n in range(10**3) for m in (1,9)] # _Chai Wah Wu_, May 24 2016
%Y A273372 Cf. A000290, A090771, A132356.
%Y A273372 Cf. A017281 (numbers ending in 1), A017283 (cubes ending in 1).
%Y A273372 Cf. similar sequences listed in A273373.
%K A273372 nonn,base,easy
%O A273372 1,2
%A A273372 _Vincenzo Librandi_, May 21 2016
%E A273372 Edited by _Bruno Berselli_, May 24 2016