A165402 -id:A165402 - cp's OEIS frontend

A045877 Rotating digits of a(n)^2 right once still yields a square.

1, 2, 3, 16, 21, 31, 129, 221, 247, 258, 1062, 1593, 1964, 2221, 13516, 17287, 18516, 19821, 22221, 28064, 29631, 103764, 182362, 222221, 273543, 1246713, 1509437, 1635219, 1856538, 2222221, 2253804, 2749249, 2784807, 11619096, 11949507

Offset: 1

Erich Friedman

Squares resulting in leading zeros excluded.

(2*10^k-11)/9 are terms, i.e. A165402 is a subsequence. - Chai Wah Wu, Apr 23 2022

			13516^2 = 18268225{6} -> {6}18268225 = 24865^2.

Chai Wah Wu, Table of n, a(n) for n = 1..1838
Eric Weisstein's World of Mathematics, Square Number

Cf. A000290, A045878, A035126, A035128, A165402.

from itertools import count, islice
from sympy.solvers.diophantine.diophantine import diop_DN
def A045877_gen(): # generator of terms
    for l in count(0):
        l1, l2 = 10**(l+1), 10**l
        yield from sorted(set(abs(x) for z in (diop_DN(10,m*(1-l1)) for m in range(10)) for x, y in z if l1 >= x**2 >= l2))
A045877_list = list(islice(A045877_gen(),30)) # Chai Wah Wu, Apr 23 2022

More terms from Patrick De Geest, Nov 15 1998

A238303 Triangle T(n,k), 0<=k<=n, read by rows given by T(n,0) = 1, T(n,k) = 2 if k>0.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Philippe Deléham, Feb 24 2014

Row sums are A005408(n).
Diagonals sums are A109613(n).
Sum_{k=0..n} T(n,k)*x^k = A033999(n), A000012(n), A005408(n), A036563(n+2), A058481(n+1), A083584(n), A137410(n), A233325(n), A233326(n), A233328(n), A211866(n+1), A165402(n+1) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively.
Sum_{k=0..n} T(n,k)*x^(n-k) = A151575(n), A000012(n), A040000(n), A005408(n), A033484(n), A048473(n), A020989(n), A057651(n), A061801(n), A238275(n), A238276(n), A138894(n), A090843(n), A199023(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 respectively.
Sum_{k=0..n} T(n,k)^x = A000027(n+1), A005408(n), A016813(n), A017077(n) for x = 0, 1, 2, 3 respectively.
Sum_{k=0..n} k*T(n,k) = A002378(n).
Sum_{k=0..n} A000045(k)*T(n,k) = A019274(n+2).
Sum_{k=0..n} A000142(k)*T(n,k) = A066237(n+1).

			Triangle begins:
1;
1, 2;
1, 2, 2;
1, 2, 2, 2;
1, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2;
...

Antti Karttunen, Table of n, a(n) for n = 0..22154; the first 210 rows of triangle

Cf. Diagonals: A040000.
Cf. Columns: A000012, A007395.
First differences of A001614.

A238303(n) = (2-ispolygonal(n,3)); \\ Antti Karttunen, Jan 19 2025

T(n,0) = A000012(n) = 1, T(n+k,k) = A007395(n) = 2 for k>0.

Data section extended to a(104) by Antti Karttunen, Jan 19 2025

A309603 Digits of the 10-adic integer (-11/9)^(1/3).

Original entry on oeis.org

1, 4, 1, 3, 3, 4, 9, 2, 3, 9, 4, 8, 9, 1, 0, 5, 1, 4, 1, 4, 3, 7, 7, 4, 7, 8, 1, 2, 3, 0, 0, 1, 7, 6, 1, 8, 9, 4, 1, 4, 2, 9, 9, 0, 3, 2, 5, 7, 9, 3, 3, 2, 2, 8, 7, 5, 8, 2, 0, 8, 7, 4, 5, 1, 2, 2, 6, 5, 5, 0, 8, 2, 3, 0, 3, 2, 9, 2, 5, 8, 6, 6, 3, 0, 2, 5, 3, 0, 1, 1, 4, 0, 9, 9, 8, 4, 5, 9, 4, 5

Offset: 0

Seiichi Manyama, Aug 09 2019

nonn

			       1^3 == 1      (mod 10).
      41^3 == 21     (mod 10^2).
     141^3 == 221    (mod 10^3).
    3141^3 == 2221   (mod 10^4).
   33141^3 == 22221  (mod 10^5).
  433141^3 == 222221 (mod 10^6).

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A165402, A309600, A309645.

N=100; Vecrev(digits(lift(chinese(Mod((-11/9+O(2^N))^(1/3), 2^N), Mod((-11/9+O(5^N))^(1/3), 5^N)))), N)

def A309603(n)
  ary = [1]
  a = 1
  n.times{|i|
    b = (a + 7 * (9 * a ** 3 + 11)) % (10 ** (i + 2))
    ary << (b - a) / (10 ** (i + 1))
    a = b
  }
  ary
end
p A309603(100)

Define the sequence {b(n)} by the recurrence b(0) = 0 and b(1) = 1, b(n) = b(n-1) + 7 * (9 * b(n-1)^3 + 11) mod 10^n for n > 1, then a(n) = (b(n+1) - b(n))/10^n.

A322925 Expansion of x(1 + 2x + 10x^2)/((1 - x^2)(1 - 10*x^2)).

Original entry on oeis.org

0, 1, 2, 21, 22, 221, 222, 2221, 2222, 22221, 22222, 222221, 222222, 2222221, 2222222, 22222221, 22222222, 222222221, 222222222, 2222222221, 2222222222, 22222222221, 22222222222, 222222222221, 222222222222, 2222222222221, 2222222222222, 22222222222221

Offset: 0

Vincenzo Librandi, Mar 16 2019

Muniru A Asiru, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,11,0,-10).

Bisections give: A002276 (even part), A165402 (odd part).

a:=[0,1,2,21];; for n in [5..30] do a[n]:=11*a[n-2]-10*a[n-4]; od; Print(a); # Muniru A Asiru, Apr 10 2019

I:=[0,1,2,21]; [n le 4 select I[n] else 11*Self(n-2)-10*Self(n-4): n in [1..30]];

seq(coeff(series(x*(1+2*x+10*x^2)/((1-x^2)*(1-10*x^2)),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Apr 10 2019

CoefficientList[Series[x (1 + 2 x + 10 x^2)/((1 - x^2) (1 - 10 x^2)), {x, 0, 33}], x]
LinearRecurrence[{0,11,0,-10},{0,1,2,21},30] (* Harvey P. Dale, Mar 02 2021 *)

G.f.: x*(1 + 2*x + 10*x^2)/((1 - x^2)*(1 - 10*x^2)).
a(n) = 11*a(n-2) - 10* a(n-4).
a(n) = 2*(10^n - 1)/9 for n even; a(n) = (2*10^n - 11)/9 otherwise.
a(n) = (2/9)*10^floor((n + 1)/2) + (-1)^n/2 - 13/18. - Bruno Berselli, Mar 16 2019

A238339 Square number array read by ascending antidiagonals: T(1,k) = 2k + 1, and T(n,k) = (2n^(k+1)-n-1)/(n-1) otherwise.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 13, 7, 1, 1, 9, 25, 29, 9, 1, 1, 11, 41, 79, 61, 11, 1, 1, 13, 61, 169, 241, 125, 13, 1, 1, 15, 85, 311, 681, 727, 253, 15, 1, 1, 17, 113, 517, 1561, 2729, 2185, 509, 17, 1, 1, 19, 145, 799, 3109, 7811, 10921, 6559, 1021, 19, 1

Offset: 0

Philippe Deléham, Feb 24 2014

			Square array begins:
1..1...1.....1......1.......1........1........1...
1..3...5.....7......9......11.......13.......15...
1..5..13....29.....61.....125......253......509...
1..7..25....79....241.....727.....2185.....6559...
1..9..41...169....681....2729....10921....43689...
1.11..61...311...1561....7811....39061...195311...
1.13..85...517...3109...18661...111973...671845...
1.15.113...799...5601...39215...274513..1921599...
1.17.145..1169...9361...74897...599185..4793489...
1.19.181..1639..14761..132859..1195741.10761679...
1.21.221..2221..22221..222221..2222221.22222221...

Cf. A238303.

T:= proc(n, k); if n=1 then 2*k+1 else (2*n^(k+1)-n-1)/(n-1) fi end:
seq(seq(T(n-k, k), k=0..n), n=0..10); # Georg Fischer, Oct 14 2023

T(0,k) = A000012(k) = 1;
T(1,k) = A005408(k) = 2k+1;
T(2,k) = A036563(k+2);
T(3,k) = A058481(k+1);
T(4,k) = A083584(k);
T(5,k) = A137410(k);
T(6,k) = A233325(k);
T(7,k) = A233326(k);
T(8,k) = A233328(k);
T(9,k) = A211866(k+1);
T(10,k) = A165402(k+1);
T(n,0) = A000012(n) = 1;
T(n,1) = A005408(n) = 2*n+1;
T(n,2) = A001844(n) = 2*n^2 + 2*n + 1.

Definition amended by Georg Fischer, Oct 14 2023

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A045877 Rotating digits of a(n)^2 right once still yields a square.

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A238303 Triangle T(n,k), 0<=k<=n, read by rows given by T(n,0) = 1, T(n,k) = 2 if k>0.

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A309603 Digits of the 10-adic integer (-11/9)^(1/3).

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A322925 Expansion of x*(1 + 2*x + 10*x^2)/((1 - x^2)*(1 - 10*x^2)).

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A238339 Square number array read by ascending antidiagonals: T(1,k) = 2*k + 1, and T(n,k) = (2*n^(k+1)-n-1)/(n-1) otherwise.

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Author

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Examples

Crossrefs

Programs

Maple

Formula

Extensions

A322925 Expansion of x(1 + 2x + 10x^2)/((1 - x^2)(1 - 10*x^2)).

A238339 Square number array read by ascending antidiagonals: T(1,k) = 2k + 1, and T(n,k) = (2n^(k+1)-n-1)/(n-1) otherwise.