A243091 -id:A243091 - cp's OEIS frontend

A090566 a(1) = 1; thereafter a(n) = smallest number > a(n-1) such that the concatenation of a(n-1) and a(n) is a square.

1, 6, 25, 281, 961, 6201, 59409, 187600, 730641, 4429444, 28600025, 85336064, 468650384, 4590568025, 23901253604, 36922256164, 228378872384, 519390415729, 3999576229761, 22053449580964, 52752598923921, 67153745961316, 346596997521321, 2205389504844676, 32117901134901281

Offset: 1

Amarnath Murthy, Dec 11 2003

From David W. Wilson, Nov 22 2015: (Start)
I used the following algorithm to extend the sequence:
x = a(n);
d = number of digits in x;
p = (10^d + 1)*x;                         #concat(x, x)
q = (floor(sqrt(p)) + 1)^2;               #smallest square > p
if (q < (10^d)(x + 1))
        a(n+1) = q mod (10^d);            #last d digits of q
else
        p = (10^d)*(10x + 1);             #concat(x, 10^d)
        q = (floor(sqrt(p - 1)) + 1)^2;   #smallest square >= p
        a(n+1) = q mod (10^(d + 1));      #last d+1 digits of q.
(End)

David W. Wilson and Robert G. Wilson v, Table of n, a(n) for n = 1..1840

See A082209 for another version. Cf. A243091.

A[1]:= 1:
for n from 2 to 100 do
  x:= A[n-1];
  d:= ilog10(x)+1;
  for dp from d while not assigned(A[n]) do
    if dp = d then
      ymin:= x+1
    else
      ymin:= 10^(dp-1)
    fi;
    zmin:= 10^dp*x + ymin;
    r:= isqrt(zmin);
    if r^2 < zmin then z:= (r+1)^2
    else z:= r^2
    fi;
    if z <= 10^dp*x + 10^dp - 1 then
        A[n]:= z - 10^dp*x;
    fi
  od
od:
seq(A[i],i=1..100); # Robert Israel, Nov 22 2015

a[1] = 1; a[n_] := Block[{x = a[n - 1], d = 1 + Floor@ Log10@ a[n - 1]}, q = (Floor@ Sqrt[(10^d + 1) x] + 1)^2; If[q < (10^d) (x + 1), Mod[q, 10^d], Mod[(Floor@ Sqrt[(10^d)(10x + 1) -1] + 1)^2, 10^(d + 1)]]]; Array[a, 25] (* after the algorithm of David W. Wilson, Robert G. Wilson v, Nov 22 2015 *)

A090566(n,show=0,a=1)={for(i=2,n,show&&print1(a","); a=A243091(a));a} \\ Use 2nd optional arg to print out intermediate values, 3rd optional arg to use another starting value. - M. F. Hasler, Nov 22 2015, revised version based on A243091: Nov 24 2015

a(n+1) = A243091(a(n)). - M. F. Hasler, Nov 24 2015

Corrected and extended by David W. Wilson, Nov 20 2015

A265155 Integers which are unique starting points for the algorithm described in A090566.

Original entry on oeis.org

1, 2, 4, 8, 10, 11, 14, 15, 16, 17, 18, 19, 21, 22, 23

Offset: 1

Robert G. Wilson v, Dec 02 2015

Consider the family of integer sequences generated from a starting value b(1) and the rule that each subsequent term is the smallest number greater than the previous term such that the concatenation of the two is a square. Then using
  b(1) = a(1) =  1 yields {1, 6, 25, 281, 961, ...}    (A090566),
  b(1) = a(2) =  2 yields {2, 5, 29, 241, 1809, ...}   (A265147),
  b(1) = a(3) =  4 yields {4, 9, 61, 504, 4516, ...}   (A265148),
  b(1) = a(4) =  8 yields {8, 41, 209, 764, 5225, ...} (A265149),
  b(1) = a(5) = 10 yields {10, 24, 336, 400, 689, ...} (A265150),
  b(1) = a(6) = 11 yields {11, 56, 169, 744, 769, ...} (A265151),
  ...

			The complement of {a(n)} is {3, 5, 6, 7, 9, 12, 13, 20, ...}; using any of these values as b(1) yields a sequence that quickly merges into one of the sequences obtained using a value from {a(n)} as b(1):
  b(1) =  3 -> {3, 6, 25, 281, 961, ...},    which quickly merges into A090566
    (as does the result of using b(1) = 6 or 12 or 20 ...);
  b(1) =  5 -> {5, 29, 241, 1809, ...},      which quickly merges into A265147
    (as does the result of using b(1) = 7 ...);
  b(1) =  9 -> {9, 61, 504, 4516, ...},      which quickly merges into A265148;
  b(1) = 13 -> {13, 69, 169, 744, 769, ...}, which quickly merges into A265151.

Cf. A090566, A243091, A265147, A265148, A265149, A265150, A265151, A265152, A265153, A265154.

(* See the Mmca coding in A090566 or A265147-A265154. *)

A082210 Square root of the squares arising in A082209.

Original entry on oeis.org

1, 4, 8, 7, 31, 248, 71, 4, 8, 7, 31, 248, 71, 4, 8, 7, 31, 248, 71, 4, 8, 7, 31, 248, 71, 4, 8, 7, 31, 248, 71, 4, 8, 7, 31, 248, 71, 4, 8, 7, 31, 248, 71, 4, 8, 7, 31, 248, 71, 4, 8, 7, 31, 248, 71, 4, 8, 7, 31, 248, 71, 4, 8, 7, 31, 248, 71, 4, 8, 7, 31, 248, 71, 4, 8, 7, 31, 248

Offset: 1

Amarnath Murthy, Apr 10 2003

The squares are given in A090567. - M. F. Hasler, Nov 24 2015

			a(3) = 31 = 961^(1/2) where A082209(3) = 9 and A082209(4) = 61.

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 1).

Cf. A082209.

Join[{1},LinearRecurrence[{0, 0, 0, 0, 0, 1},{4, 8, 7, 31, 248, 71},77]] (* Ray Chandler, Aug 26 2015 *)

A082210(n)=n<2||return([248,71,4,8,7,31][n%6+1]) \\ M. F. Hasler, Nov 24 2015

a(n) = {A082209(n) concatenated with A082209(n+1)}^(1/2).
Periodic with period 6 from n=2 onward. - Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 20 2003
a(n) = sqrt(A090567(n)) with A090567(n) = A082209(n) concat A243091(A082209(n)). - M. F. Hasler, Nov 24 2015

Corrected and extended by Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 20 2003

Definition corrected and minor edits by M. F. Hasler, Nov 24 2015

cp's OEIS Frontend

A090566 a(1) = 1; thereafter a(n) = smallest number > a(n-1) such that the concatenation of a(n-1) and a(n) is a square.

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A265155 Integers which are unique starting points for the algorithm described in A090566.

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A082210 Square root of the squares arising in A082209.

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Mathematica

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