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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) ~).

Index	Parameters A, B, C, D of the cubic equation	SUCO(#terms=A)	Startvalue	Palindrome	Length	Record & Date
1	A=2 B=3 C=3 D=1	SUCO2	0	1	1
2	A=2 B=3 C=3 D=1	SUCO2	1	9	1
3	A=2 B=3 C=3 D=1	SUCO2	16	9009	4
4	A=3 B=9 C=15 D=9	SUCO3	0	9	1
5	A=3 B=9 C=15 D=9	SUCO3	2	99	2
6	A=3 B=9 C=15 D=9	SUCO3	16	14841	5
7	A=4 B=18 C=42 D=36	SUCO4	59	886688	6
8	A=9 B=108 C=612 D=1296	SUCO9	1722	46277277264	11
9	A=11 B=165 C=1155 D=3025	SUCO11	16	108801	6
10	A=11 B=165 C=1155 D=3025	SUCO11	37	828828	6
11	A=11 B=165 C=1155 D=3025	SUCO11	226	135666531	9
12	A=16 B=360 C=3720 D=14400	SUCO16	43	2112112	7
13	A=19 B=513 C=6327 D=29241	SUCO19	13	239932	6
14	A=21 B=630 C=8610 D=44100	SUCO21	3	76167	5
15	A=23 B=759 C=11385 D=64009	SUCO23	47	4663664	7
16	A=24 B=828 C=12972 D=76176	SUCO24	290	658808856	9
17	A=46 B=3105 C=94185 D=1071225	SUCO46	28	7152517	7
18	A=62 B=5673 C=232593 D=3575881	SUCO62	30	17333371	8
19	A=77 B=8778 C=447678 D=8561476	SUCO77	15	17511571	8
20	A=83 B=10209 C=561495 D=11580409	SUCO83	43	61200216	8
21	A=88 B=11484 C=669900 D=14653584	SUCO88	27	42844824	8
22	A=91 B=12285 C=741195 D=16769025	SUCO91	3073	2759066609572	13
23	A=261 B=101790 C=17677530 D=1151244900	SUCO261	328	27110501172	11
24	A=333 B=165834 C=36759870 D=3055657284	SUCO333	76	6953443596	10
25	A=393 B=231084 C=60466980 D=5933312784	SUCO393	16	6961551696	10
26	A=402 B=241803 C=64722603 D=6496521201	SUCO402	1483	1945416145491	13
27	A=481 B=346320 C=110937840 D=13326393600	SUCO481	684	405162261504	12
28	A=630 B=594405 C=249451965 D=39257478225	SUCO630	629	588114411885	12
29	A=909 B=1238058 C=749850462 D=170309734596	SUCO909	9214	823251525152328	15
30	A=1111 B=1849815 C=1369479705 D=380201726025	SUCO1111	69	483867768384	12
31	A=1195 B=2140245 C=1704348435 D=508960962225	SUCO1195	8	522733337225	12
32	A=1750 B=4591125 C=5354782125 D=2342047640625	SUCO1750	388	5213088803125	13
33	A=2063 B=6380859 C=8773681125 D=4523929064209	SUCO2063	40208	144775552255577441	18
34	A=3108 B=14484834 C=30007747770 D=23312268445284	SUCO3108	3367	407192737291704	15
35	A=4329 B=28103868 C=81098395092 D=87758599617936	SUCO4329	104341	5232110564650112325	19
36	A=5656 B=47977020 C=180889357740 D=255754938675600	SUCO5656	1376	610232727232016	15
37	A=6890 B=71197815 C=327011564295 D=563236540086025	SUCO6890	5372	5442734224372445	16
38	A=7124 B=76116378 C=361476679122 D=643744777759876	SUCO7124	10493	21047802520874012	17
39	A=15477 B=359283078 C=3706963037778 D=14342703348572676	SUCO15477	1163	19164202420246191	17
40	A=24934 B=932519133 C=15500643868437 D=96621325934563521	SUCO24934	2485	141281564465182141	18
41	A=25297 B=959869368 C=16187556978408 D=102372133736079936	SUCO25297	1715	133084588885480331	18
42	A=53025 B=4217396400 C=149083556941200 D=1976270266081440000	SUCO53025	16147	5706337582857336075	19
43	A=54367 B=4433574483 C=160691951420013 D=2184064744034301921	SUCO54367	179381	487479173565371974784	21
44	A=62763 B=5908697109 C=247229734869075 D=3879189058433884209	SUCO62763	4534	5127444581854447215	19
45	A=66146 B=6562840755 C=289401588773235 D=4785653197276552225	SUCO66146	11784	9215533523253355129	19
46	A=78172 B=9166175118 C=477689105491158 D=9335418477093590436	SUCO78172	356231	4876531221111221356784	22
47	A=85187 B=10885109673 C=618176263439343 D=13165068065915351881	SUCO85187	129679	462153418202814351264	21
48	A=122174 B=22389546153 C=1823606144615697 D=55699086326368566601	SUCO122174	1977100	1035381727498947271835301	25	Record Largest [ September 2, 2023 ]
49	A=131455 B=25920428355 C=2271571299461565 D=74652067345187556225	SUCO131455	281638	5707061327337231607075	22
50	A=139204 B=29066421618 C=2697431747800842 D=93872985075037526436	SUCO139204	61742	403984947020749489304	21
51	A=203085 B=61864971210 C=8375877830464830 D=425252740312680984900	SUCO203085	11385	528930630414036039825	21
52	A=237454 B=84576246993 C=13388617243568217 D=794793506157882393561	SUCO237454	296985	18450539336563393505481	23
53	A=257848 B=99727999884 C=17143076966726460 D=1105074884540344890384	SUCO257848	482743	61629063499899436092616	23
54	A=351192 B=185003204508 C=43314368597314188 D=3802909519803207946896	SUCO351192	322431	48774189806260898147784	23

{
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