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.

A nice coincidence occurred with TWO Cubed Consecutive Integers

16 + 17 = 33

16^{2}+ 17^{2}= 545

16^{3}+ 17^{3}= 9009

A nice coincidence occurred with FOUR Cubed Consecutive Integers

59 + 60 + 61 + 62 = 242

59^{3}+ 60^{3}+ 61^{3}+ 62^{3}= 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 :The regular numbers of form n ^{3} + (n+1)^{3} [sums of two cubed consecutives] :The regular numbers of form n ^{3} + (n+1)^{3} + (n+2)^{3} [sums of three cubed consecutives] :The regular numbers of form n ^{3} + (n+1)^{3} + (n+2)^{3} + (n+3)^{3} [sums of four cubed consecutives] :The regular numbers of form n ^{3} + (n+1)^{3} + (n+2)^{3} + (n+3)^{3} + (n+4)^{3} [sums of five cubed consecutives] :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 for the first values m up to 50000.

With this limit we can arrive at around 25-digit palindromes (~ PL = digitlength(3000000^3 * m) ~).

From index 49 (m > 50000) on the startvalue x was searched only from 0 and 500000.

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 |

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

{ for(m=2,50000, 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