Palindromic numbers are numbers which read the same from
left to right (forwards)
as from the right to left (backwards)
Here are a few random examples : 535, 3773, 246191642
Palindromic Sums of Cubes of Consecutive Integers
General formulae ( A*x3 + B*x2 + C*x + D )
SOCU2 = 2*x^3 + 3*x^2 + 3*x + 1
SOCU3 = 3*x^3 + 9*x^2 + 15*x + 9
SOCU4 = 4*x^3 + 18*x^2 + 42*x + 36
SOCU5 = 5*x^3 + 30*x^2 + 90*x + 100
SOCU6 = 6*x^3 + 45*x^2 + 165*x + 225
SOCU7 = 7*x^3 + 63*x^2 + 273*x + 441
SOCU8 = 8*x^3 + 84*x^2 + 420*x + 784
SOCU9 = 9*x^3 + 108*x^2 + 612*x + 1296
The 'A' parameters are given by A001477 → a(n) = n+1 ; The nonnegative integers.
The 'B' parameters are given by A045943 → a(n) = 3*n*(n+1)/2 ; Triangular matchstick numbers.
The 'C' parameters are given by A059270 → a(n) = n*(n+1)*(2*n+1)/2 ; a(n) is both the sum of n+1 consecutive integers and the sum of the n immediately higher consecutive integers.
The 'D' parameters are given by A000537 → a(n) = (n*(n+1)/2)^2 ; Sum of first n cubes; or n-th triangular number squared.
In case of e.g. SOCU9 with 9 terms (m=9) substitute n in the above formulas with m1 and we get the following formulae according to the number of terms:
'A' = m
'B' = (3*m^23*m)/2
'C' = (2*m^33*m^2+m)/2
'D' = ((m^2m)/2)^2
Sums of TWO cubed consecutives of the form x^3 + (x+1)^3 can only start or end with a 1, 5 or 9.
1 can be followed by any number : 10, 11, 12, 13, 14, 15, 16, 17, 18 or 19
5 can be followed by any number : 50, 51, 52, 53, 54, 55, 56, 57, 58 or 59
9 can be followed by any number : 90, 91, 92, 93, 94, 95, 96, 97, 98 or 99
There are so far only palindromic sums of cubes of type [SOCU2] of lengths 1 and 4.
Sums of THREE cubed consecutives of the form 3*x^3 + 9*x^2 + 15*x + 9 can only start or end with a 0, 1, 4, 5, 6 or 9.
Alas, my palindromes may not have leading 0's! So the zero option must not be investigated.
1 can be followed by any number : 10, 11, 12, 13, 14, 15, 16, 17, 18 or 19
4 can only be followed by an even number : 40, 42, 44, 46 or 48
5 can be followed by any number : 50, 51, 52, 53, 54, 55, 56, 57, 58 or 59
6 can only be followed by an odd number : 61, 63, 65, 67 or 69
9 can be followed by any number : 90, 91, 92, 93, 94, 95, 96, 97, 98 or 99
There are so far only palindromic sums of cubes of type [SOCU3] of lengths 1, 2 and 5.
Sums of FOUR cubed consecutives of the form 4*x^3 + 18*x^2 + 42*x + 36 can only start or end with a 0, 2, 4, 6 or 8.
Alas, my palindromes may not have leading 0's! So the zero option must not be investigated.
2 can only be followed by an odd number : 21, 23, 25, 27 or 29
4 can only be followed by an even number : 40, 42, 44, 46 or 48
6 can only be followed by an odd number : 61, 63, 65, 67 or 69
8 can only be followed by an even number : 80, 82, 84, 86 or 88
There is so far only one palindromic sum of cubes of type [SOCU4] of length 6.
Sums of FIVE cubed consecutives of the form 5*x^3 + 30*x^2 + 90*x + 100 can only start or end with a 0 or 5.
Alas, my palindromes may not have leading 0's! So the zero option must not be investigated.
5 can be followed by any number : 50, 51, 52, 53, 54, 55, 56, 57, 58 or 59
There is so far no palindromic sum of cubes found of type [SOCU5].
Sums of SIX cubed consecutives of the form 6*x^3 + 45*x^2 + 165*x + 225 can only start or end with a 1, 3, 5, 7 or 9.
1 can be followed by any number : 10, 11, 12, 13, 14, 15, 16, 17, 18 or 19
3 can be followed by any number : 30, 31, 32, 33, 34, 35, 36, 37, 38 or 39
5 can only be followed by 2 or 7 : 52 or 57
7 can be followed by any number : 70, 71, 72, 73, 74, 75, 76, 77, 78 or 79
9 can be followed by any number : 90, 91, 92, 93, 94, 95, 96, 97, 98 or 99
There is so far no palindromic sum of cubes found of type [SOCU6].
Sums of SEVEN cubed consecutives of the form 7*x^3 + 63*x^2 + 273*x + 441 can only start or end with a 0, 1, 4, 5, 6 or 9.
Alas, my palindromes may not have leading 0's! So the zero option must not be investigated.
1 can be followed by any number : 10, 11, 12, 13, 14, 15, 16, 17, 18 or 19
4 can only be followed by an even number : 40, 42, 44, 46 or 48
5 can be followed by any number : 50, 51, 52, 53, 54, 55, 56, 57, 58 or 59
6 can only be followed by an odd number : 61, 63, 65, 67 or 69
9 can be followed by any number : 90, 91, 92, 93, 94, 95, 96, 97, 98 or 99
There is so far no palindromic sum of cubes found of type [SOCU7].
Sums of EIGHT cubed consecutives of the form 8*x^3 + 84*x^2 + 420*x + 784 can only start or end with a 0, 4 or 6.
Alas, my palindromes may not have leading 0's! So the zero option must not be investigated.
4 can only be followed by an even number : 40, 42, 44, 46 or 48
6 can only be followed by an odd number : 61, 63, 65, 67 or 69
There is so far no palindromic sum of cubes found of type [SOCU8].
Sums of NINE cubed consecutives of the form 9*x^3 + 108*x^2 + 612*x + 1296 can start or end with any digit 0, 1, 2, 3, 4, 5, 6, 7, 8 or 9.
Alas, my palindromes may not have leading 0's! So the zero option must not be investigated.
1 can be followed by any number : 10, 11, 12, 13, 14, 15, 16, 17, 18 or 19
2 can only be followed by an odd number : 21, 23, 25, 27 or 29
3 can be followed by any number : 30, 31, 32, 33, 34, 35, 36, 37, 38 or 39
4 can only be followed by an even number : 40, 42, 44, 46 or 48
5 can only be followed by 2 or 7 : 52 or 57
6 can only be followed by an odd number : 61, 63, 65, 67 or 69
7 can be followed by any number : 70, 71, 72, 73, 74, 75, 76, 77, 78 or 79
8 can only be followed by an even number : 80, 82, 84, 86 or 88
9 can be followed by any number : 90, 91, 92, 93, 94, 95, 96, 97, 98 or 99
There is so far only one palindromic sum of cubes found of type [SOCU9] of length 11.
A nice coincidence occurred with TWO Cubed Consecutive Integers
16 + 17 = 33
162 + 172 = 545
163 + 173 = 9009
A nice coincidence occurred with FOUR Cubed Consecutive Integers
Searching palindromes equal to sums of cubes of consecutive integers seems very difficult.
Here is an overview of all the palindromes I could find so far with terms m from 2 up to 500000 and
with starting values of the consecutives x from 0 up to 3000000.
With this limit we can arrive at around 25-digit palindromes (~ PL = digitlength(3000000^3 * m) ~).