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Palindromic Sums of Cubes of Consecutive Integers
Sums of Squares Sums of Primes Sums of Powers Various Palindromic Sums

  left to right (forwards) as from the right to left (backwards)

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.

There are so far only palindromic sums of cubes of type [SOCU2] of lengths 1 and 4.

There are so far only palindromic sums of cubes of type [SOCU3] of lengths 1, 2 and 5.

There is so far only one palindromic sum of cubes of type [SOCU4] of length 6.

There is so far no palindromic sum of cubes found of type [SOCU5].

There is so far no palindromic sum of cubes found of type [SOCU6].

There is so far no palindromic sum of cubes found of type [SOCU7].

There is so far no palindromic sum of cubes found of type [SOCU8].

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

59 + 60 + 61 + 62 = 242
59³ + 60³ + 61³ + 62³ = 886688

Huen Y.K. from Singapore developed a general generating function for palindromic sums and products
of consecutive integers using concise programcode written for Macsyma 2.2.1.
Global Generating Function For Palindromic Sums and Products of Consecutive Integers.

Neil Sloane's “Integer Sequences” Encyclopedia can be consulted online : Neil Sloane's Integer Sequences The regular numbers of form n3 + (n+1)3 [sums of two cubed consecutives] : %N Centered cube numbers: n^3 + (n+1)^3. under A005898. The regular numbers of form n3 + (n+1)3 + (n+2)3 [sums of three cubed consecutives] : %N n^3 + (n+1)^3 + (n+2)^3. under A027602. The regular numbers of form n3 + (n+1)3 + (n+2)3 + (n+3)3 [sums of four cubed consecutives] : %N n^3 + (n+1)^3 + (n+2)^3 + (n+3)^3. under A027603. The regular numbers of form n3 + (n+1)3 + (n+2)3 + (n+3)3 + (n+4)3 [sums of five cubed consecutives] : %N n^3 + (n+1)^3 + (n+2)^3 + (n+3)^3 + (n+4)^3. under A027604. Click here to view some of the author's [P. De Geest] entries to the table. Click here to view some entries to the table about palindromes.

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) ~).

You might be interested in my Pari/gp program which will recreate the above list given enough running time (days, even weeks).

{
for(m=2,500000, A=m; B=(3*m^2-3*m)/2; C=(2*m^3-3*m^2+m)/2; D=((m^2-m)/2)^2;
for(x=0,3000000, n=digits(A*x^3+B*x^2+C*x+D);
if(n==Vecrev(n), write("C:/Pari_gp/suco_results.txt", "m=",m," x=",x," n=",n);
print; print1("A=",A," B=",B," C=",C," D=",D);
print("  SUCO",A," ",x," ",fromdigits(n)))); print1(Strchr(13),m));
}

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Patrick De Geest - Belgium - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com