A180436 -id:A180436 - cp's OEIS frontend

A216446 Palindromic numbers which can be written as the sum of two or more consecutive squares.

5, 55, 77, 181, 313, 434, 505, 545, 595, 636, 818, 1001, 1111, 1441, 1771, 4334, 6446, 17371, 17871, 19691, 21712, 41214, 42924, 44444, 46564, 51015, 65756, 81818, 97679, 99199, 108801, 127721, 137731, 138831, 139931, 148841, 161161, 166661, 171171, 188881

Offset: 1

V. Raman, Sep 07 2012

			636 is in the sequence because it is a palindrome and 636 = 4^2+5^2+6^2+7^2+8^2+9^2+10^2+11^2+12^2.

V. Raman and Giovanni Resta, Table of n, a(n) for n = 1..10306 (terms < 10^18, first 298 terms from V. Raman)

Cf. A034705, A180436, A267600 (terms with more than one representation).

palQ[n_] := Block[{d = IntegerDigits@ n}, d == Reverse@ d]; upto = 10^6; Union[ Reap[ For[i=1, s=i^2 + (i+1)^2; s < upto, i++, For[j=i+1, s < upto, j++; s += j^2, If[palQ[s], Sow@ s]]]][[2, 1]]] (* Giovanni Resta, Jun 14 2018 *)
With[{nn=200},Select[Union[Flatten[Table[Total/@Partition[Range[nn]^2,n,1],{n,2,nn}]]],PalindromeQ]] (* Harvey P. Dale, Oct 17 2021 *)

Errors in previous b-file noticed by Riley Waugh, Jun 13 2018

A267600 Palindromic numbers which are sum of consecutive squares in more than one way.

Original entry on oeis.org

554455, 9343439, 923222222329

Offset: 1

Chai Wah Wu, Jan 18 2016

A subsequence of A180436.

a(4) > 10^18, if it exists. - Giovanni Resta, Jun 14 2018

			554455 = 9^2 + ... + 118^2 = 331^2 + ... + 335^2.
9343439 = 102^2 + ... + 307^2 = 657^2 + ... + 677^2.
923222222329 = 2967^2 + ... + 14087^2 = 42462^2 + ... + 42967^2.

Cf. A180436.

A364143 a(n) is the minimal number of consecutive squares needed to sum to A216446(n).

Original entry on oeis.org

2, 5, 3, 2, 2, 3, 10, 2, 7, 9, 12, 11, 6, 11, 14, 3, 11, 29, 14, 7, 23, 4, 49, 8, 24, 5, 17, 12, 38, 46, 27, 34, 6, 14, 22, 66, 11, 66, 14, 11, 6, 77, 36, 63, 96, 11, 50, 3, 19, 96, 52, 41, 66, 33, 11, 3, 14, 121, 66, 89, 34, 127, 51, 2, 86, 54, 181, 48, 8

Offset: 1

Darío Clavijo, Jul 10 2023

			a(8) = 7 is because 7 consecutive squares are needed to sum to A216446(8) = 595 = 6^2 + 7^2 + 8^2 + 9^2 + 10^2 + 11^2 + 12^2.

Project Euler, Problem 125: Palindromic Sums.

Cf. A216446, A034705, A180436, A267600.

is_palindrome = lambda n: str(n) == str(n)[::-1]
def g(L):
  L2, squares, D = L*L, [x*x for x in range(0, L + 1)], {}
  for i in range(1, L + 1):
    for j in range(i + 1, L + 1):
      candidate = sum(squares[i:j+1])
      if candidate < L2 and is_palindrome(candidate):
        if candidate in D:
          D[candidate]= min(D[candidate], j-i-1)
        else:
          D[candidate] = j-i+1
  return [D[k] for k in sorted(D.keys())]
print(g(1000))

cp's OEIS Frontend

A216446 Palindromic numbers which can be written as the sum of two or more consecutive squares.

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A267600 Palindromic numbers which are sum of consecutive squares in more than one way.

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A364143 a(n) is the minimal number of consecutive squares needed to sum to A216446(n).

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