A273374 -id:A273374 - cp's OEIS frontend

A273373 Squares ending in digit 6.

16, 36, 196, 256, 576, 676, 1156, 1296, 1936, 2116, 2916, 3136, 4096, 4356, 5476, 5776, 7056, 7396, 8836, 9216, 10816, 11236, 12996, 13456, 15376, 15876, 17956, 18496, 20736, 21316, 23716, 24336, 26896, 27556, 30276, 30976, 33856, 34596, 37636, 38416, 41616

Offset: 1

Vincenzo Librandi, May 21 2016

These are the only squares whose second last digit is odd. This implies that the only squares whose last two digits are the same are those ending with 0 or 4; those ending with 1, 5, and 9 are paired with even second last digits. - Waldemar Puszkarz, May 24 2016

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).

Cf. A000290, A047221, A090773.
Cf. A017341 (numbers ending in 6), A017343 (cubes ending in 6).
Cf. squares with last digit k: A017270 (k=0), A273372 (k=1), A273375 (k=4), A017330 (k=5), this sequence (k=6), A273374 (k=9).

/* By definition: */ [n^2: n in [0..200] | Modexp(n,2,10) eq 6];

[(10*n - 3*(-1)^n - 5)^2/4: n in [1..50]];

seq(seq((10*i+j)^2,j=[4,6]),i=0..20); # Robert Israel, May 24 2016

Table[(10 n - 3 (-1)^n - 5)^2/4, {n, 1, 50}]
CoefficientList[Series[4 (4 + 5 x + 32 x^2 + 5 x^3 + 4 x^4) / ((1 + x)^2 (1 - x)^3), {x, 0, 50}], x]
Select[Range[250]^2,Mod[#,10]==6&] (* Harvey P. Dale, May 31 2020 *)

G.f.: 4*x*(4 + 5*x + 32*x^2 + 5*x^3 + 4*x^4)/((1 + x)^2*(1 - x)^3).
a(n) = 4*A047221(n)^2 = (10*n - 3*(-1)^n - 5)^2/4.
a(n) = A090773(n)^2. - Michel Marcus, May 25 2016
Sum_{n>=1} 1/a(n) = 2*Pi^2/(25*(5+sqrt(5))). - Amiram Eldar, Feb 16 2023

Corrected and extended by Bruno Berselli, May 23 2016

A348491 Positive numbers whose square starts and ends with exactly one 9.

Original entry on oeis.org

3, 97, 303, 307, 313, 953, 957, 963, 967, 973, 977, 983, 987, 993, 3003, 3007, 3013, 3017, 3023, 3027, 3033, 3037, 3043, 3047, 3053, 3057, 3063, 3067, 3073, 3077, 3083, 3087, 3093, 3097, 3103, 3107, 3113, 3117, 3123, 3127, 3133, 3137, 3143, 9487, 9493, 9497, 9503, 9507, 9513, 9517

Offset: 1

Bernard Schott, Nov 02 2021

When a square ends with 9, it ends with only one 9.
From Marius A. Burtea, Nov 02 2021 : (Start)
The sequence is infinite because the numbers 303, 3003, 30003, ..., 3*(10^k + 1), k >= 2, are terms with squares 91809, 9018009, 900180009, 90001800009, ... 9*(10^(2*k) + 2*10^k + 1), k >= 2.
Numbers 97, 967, 9667, 96667, 966667, ..., (29*10^n + 1) / 3, k >= 1, are terms and have no digits 0, because their squares are 9409, 935089, 93450889, 9344508889, 934445088889, ...
Also 963, 9663, 96663, 966663, 9666663, 96666663, ... (29*10^k - 11) / 3, k >= 2, are terms and have no digits 0, because their squares are 927369, 93373569, 9343735569, 934437355569, 93444373555569, 9344443735555569, ... (End)

			97^2 = 9409, hence 97 is a term.
997^2 = 994009, hence  997 is not a term.

Subsequence of A305719, A063226, and A045863.
Cf. A017377, A045863, A273374 (squares ending with 9).
Similar to: A348487 (k=1), A348488 (k=4), A348489 (k=5), A348490 (k=6), this sequence (k=9).

[3] cat [n:n in [4..9600]|Intseq(n*n)[1] eq 9 and Intseq(n*n)[#Intseq(n*n)] eq 9]; // Marius A. Burtea, Nov 02 2021

Join[{3}, Select[Range[10, 10^4], (d = IntegerDigits[#^2])[[1]] == d[[-1]] == 9 && d[[2]] != 9 &]] (* Amiram Eldar, Nov 02 2021 *)

isok(k) = my(d=digits(sqr(k))); (d[1]==9) && (d[#d]==9) && if (#d>2, (d[2]!=9) && (d[#d-1]!=9), 1); \\ Michel Marcus, Nov 03 2021

list(lim)=my(v=List([3])); for(d=2, 2*#digits(lim\=1), my(s=sqrtint(9*10^(d-1)-1)+1); s+=[3,2,1,0,3,2,1,0,5,4][s%10+1]; forstep(n=s, min(sqrtint(10^d-10^(d-2)-1), lim), if(s%10==3, [4,6], [6,4]), listput(v, n))); Vec(v) \\ Charles R Greathouse IV, Nov 03 2021

from itertools import count, takewhile
def ok(n):
  s = str(n*n); return len(s.rstrip("9")) == len(s.lstrip("9")) == len(s)-1
def aupto(N):
  r = takewhile(lambda x: x<=N, (10*i+d for i in count(0) for d in [3, 7]))
  return [k for k in r if ok(k)]
print(aupto(9517)) # Michael S. Branicky, Nov 02 2021

A152752 Terms of A118962 that are == 9 (mod 10), in the order of appearance.

Original entry on oeis.org

9, 9, 49, 49, 49, 49, 169, 169, 169, 169, 289, 169, 289, 169, 289, 169, 289, 169, 289, 169, 289, 529, 289, 529, 529, 289, 529, 289, 729, 529, 289, 729, 529, 289, 529, 729, 289, 529, 729, 529, 1089, 529, 729, 1089, 529, 729, 529, 1089, 1369, 529, 729, 1089

Offset: 1

Paul Curtz, Dec 12 2008

nonn

Cf. A118962, A273374.

Edited by N. J. A. Sloane, Dec 13 2008

a(11) corrected and more terms from Georg Fischer, Oct 29 2024

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A273373 Squares ending in digit 6.

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A348491 Positive numbers whose square starts and ends with exactly one 9.

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A152752 Terms of A118962 that are == 9 (mod 10), in the order of appearance.

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