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1, 5, -28, -513, -2800, -7849, 0, 162287, 1417472, 9546255, 59466176, 361025495, 2173796352, 13055867207, 78356634560, 470173593951, 2821093130240, 16926635307167, 101559922656192, 609359692964615, 3656158376062976, 21936950554611735, 131621703728887232, 789730222905566927
COMMENTS
6^n in the formula can be removed (for example) with the following Maple code: "with(gfun): rec1:={u1(0)=1,u1(n+1)=6*u1(n)}: rec2:={u2(n)=n^6}: poltorec(u1(n)-u2(n),[rec1,rec2],u1(n),u2(n)],a(n));". This yields a polynomial recurrence: {a(n+1)-5*n^6+6*n^5+15*n^4+20*n^3+15*n^2-6*a(n)+6*n+1, a(0) = 1} that can further be transformed into a linear recurrence with constant coefficients. - Georg Fischer, Feb 23 2021
FORMULA
a(n) = 13*a(n-1) - 63*a(n-2) + 161*a(n-3) - 245*a(n-4) + 231*a(n-5) - 133*a(n-6) + 43*a(n-7) - 6*a(n-8) for n > 7.
G.f.: (5*x^7 + 348*x^6 + 1734*x^5 + 1545*x^4 + 5*x^3 - 30*x^2 - 8*x + 1)/((x - 1)^7*(6*x - 1)). (End)
1, 6, -79, -1844, -13983, -61318, -162287, 0, 3667649, 35570638, 272475249, 1957839572, 13805455393, 96826261890, 678117659345, 4747390650568, 33232662134145, 232630103648534, 1628412985690417, 11398894291501404, 79792265017612001, 558545862282195466
COMMENTS
a(20)=79792265017612001 and a(24)=191581231375979942977 are primes, thus terms of A123206. - M. F. Hasler, Aug 20 2014
FORMULA
G.f.: (1-9*x-85*x^2-407*x^3+5991*x^4+15665*x^5+8245*x^6+831*x^7+8*x^8)/((1-7*x)*(1-x)^8). - Bruno Berselli, May 16 2011
1, 7, -192, -6049, -61440, -357857, -1417472, -3667649, 0, 91171007, 973741824, 8375575711, 68289495040, 548940083167, 4396570722048, 35181809198207, 281470681743360, 2251792837927807, 18014387489521408, 144115171092292831
LINKS
Index entries for linear recurrences with constant coefficients, signature (17, -108, 372, -798, 1134, -1092, 708, -297, 73, -8).
FORMULA
G.f.: (1 - 10*x - 203*x^2 - 2401*x^3 + 18851*x^4 + 109207*x^5 + 120743*x^6 + 34061*x^7 + 1984*x^8 + 7*x^9) / ((1-8*x)*(1-x)^9). - Bruno Berselli, May 16 2011 a(0)=1, a(1)=7, a(2)=-192, a(3)=-6049, a(4)=-61440, a(5)=-357857, a(6)=-1417472, a(7)=-3667649, a(8)=0, a(9)=91171007; for n>9, a(n) = 17*a(n-1) - 108*a(n-2) + 372*a(n-3) - 798*a(n-4) + 1134*a(n-5) - 1092*a(n-6) + 708*a(n-7) - 297*a(n-8) + 73*a(n-9) - 8*a(n-10). - Harvey P. Dale, Oct 10 2013
MATHEMATICA
Table[8^n - n^8, {n, 0, 20}] (* or *) LinearRecurrence[ {17, -108, 372, -798, 1134, -1092, 708, -297, 73, -8}, {1, 7, -192, -6049, -61440, -357857, -1417472, -3667649, 0, 91171007}, 20] (* Harvey P. Dale, Oct 10 2013 *)
1, 8, -431, -18954, -255583, -1894076, -9546255, -35570638, -91171007, 0, 2486784401, 29023111918, 277269756129, 2531261328956, 22856131408177, 205852688735274, 1852951469375105, 16677063111790072, 150094436937708753
FORMULA
a(n) = 19*a(n-1) - 135*a(n-2) + 525*a(n-3) - 1290*a(n-4) + 2142*a(n-5) - 2478*a(n-6) + 2010*a(n-7) - 1125*a(n-8) + 415*a(n-9) - 91*a(n-10) + 9*a(n-11) for n > 10.
G.f.: (-10*x^10 - 4507*x^9 - 131015*x^8 - 779378*x^7 - 1317686*x^6 - 637664*x^5 - 43448*x^4 + 10210*x^3 + 448*x^2 + 11*x - 1)/((x - 1)^10*(9*x - 1)). (End)
1, 9, -924, -58049, -1038576, -9665625, -59466176, -272475249, -973741824, -2486784401, 0, 74062575399, 938082635776, 9862141508151, 99710745345024, 999423349609375, 9998900488372224, 99997984006099551, 999996429532773376
FORMULA
a(n) = 21*a(n-1) - 165*a(n-2) + 715*a(n-3) - 1980*a(n-4) + 3762*a(n-5) - 5082*a(n-6) + 4950*a(n-7) - 3465*a(n-8) + 1705*a(n-9) - 561*a(n-10) + 111*a(n-11) - 10*a(n-12) for n > 11.
G.f.: (9*x^11 + 10140*x^10 + 477332*x^9 + 4504245*x^8 + 12648018*x^7 + 11793648*x^6 + 3241104*x^5 + 23538*x^4 - 37875*x^3 - 948*x^2 - 12*x + 1)/((x - 1)^11*(10*x - 1)). (End)
1, 11, -3952, -529713, -16756480, -243891793, -2173796352, -13805455393, -68289495040, -277269756129, -938082635776, -2395420006033, 0, 83695120256591, 1227224552173568, 15277275236695743, 184602783918325760
COMMENTS
Conjecture: satisfies a linear recurrence having signature (25, -234, 1222, -4147, 9867, -17160, 22308, -21879, 16159, -8866, 3510, -949, 157, -12). - Harvey P. Dale, Jan 27 2019 The conjecture above is correct. From the general formula for {a(n)} we can see that the roots for the characteristic polynomial are one 12 and thirteen 1's, so the characteristic polynomial is (x - 12)*(x - 1)^13 = x^14 - 25*x^13 + 234*x^12 - ... + 12, with corresponding recurrence coefficients 25, -234, ..., -12. - Jianing Song, Jan 28 2019
LINKS
Index entries for linear recurrences with constant coefficients, signature (25,-234,1222,-4147,9867,-17160,22308,-21879,16159,-8866,3510,-949,157,-12).
CROSSREFS
Cf. A024012, A024026, A058794, A024040, A024054, A024068, A024082, A024096, A024110, A024124, A024138. - Vladimir Joseph Stephan Orlovsky, Jan 15 2009
1, 10, -1927, -175816, -4179663, -48667074, -361025495, -1957839572, -8375575711, -29023111918, -74062575399, 0, 2395420006033, 32730551749894, 375700268413577, 4168598413556276, 45932137677527745, 505412756602986138
FORMULA
a(n) = 23*a(n-1) - 198*a(n-2) + 946*a(n-3) - 2915*a(n-4) + 6237*a(n-5) - 9636*a(n-6) + 10956*a(n-7) - 9207*a(n-8) + 5665*a(n-9) - 2486*a(n-10) + 738*a(n-11) - 133*a(n-12) + 11*a(n-13) for n > 12.
G.f.: (-12*x^12 - 22383*x^11 - 1677037*x^10 - 24085511*x^9 - 104916261*x^8 - 163227822*x^7 - 91395930*x^6 - 14499462*x^5 + 523986*x^4 + 130461*x^3 + 1959*x^2 + 13*x - 1)/((x - 1)^12*(11*x - 1)). (End)
CROSSREFS
Cf. A024012, A024026, A058794, A024040, A024054, A024068, A024082, A024096, A024110, A024124. - Vladimir Joseph Stephan Orlovsky, Jan 15 2009
1, 1, -28, -235, -1008, -3093, -7712, -16679, -32512, -58537, -98976, -159003, -244736, -363101, -521440, -726607, -983040, -1288785, -1627424, -1951811, -2151424, -1986949, -959328, 1952265, 8814592, 23788807, 55227488, 119868821, 251225088
FORMULA
G:f.: (1-7*x-9*x^2-34*x^3+121*x^4+45*x^5+3*x^6) / ((1-2*x)*(1-x)^6). - Vincenzo Librandi, Oct 07 2014 a(n) = 8*a(n-1) -27*a(n-2) +50*a(n-3) -55*a(n-4) +36*a(n-5) -13*a(n-6) +2*a(n-7) for n>6. - Vincenzo Librandi, Oct 07 2014
MATHEMATICA
Table[2^n - n^5, {n, 0, 30}] (* or *) CoefficientList[Series[(1 - 7 x - 9 x^2 - 34 x^3 + 121 x^4 + 45 x^5 + 3 x^6)/((1 - 2 x) (1 - x)^6), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 07 2014 *) LinearRecurrence[{8, -27, 50, -55, 36, -13, 2}, {1, 1, -28, -235, -1008, -3093, -7712}, 30] (* Harvey P. Dale, May 14 2016 *)
PROG
(Magma) I:=[1, 1, -28, -235, -1008, -3093, -7712]; [n le 7 select I[n] else 8*Self(n-1)-27*Self(n-2)+50*Self(n-3)-55*Self(n-4)+36*Self(n-5)-13*Self(n-6)+2*Self(n-7): n in [1..35]]; // Vincenzo Librandi, Oct 07 2014
CROSSREFS
Cf. sequences of the form k^n-n^5: this sequence (k=2), A024028 (k=3), A024041 (k=4), A024054 (k=5), A024067 (k=6), A024080 (k=7), A024093 (k=8), A024106 (k=9), A024119 (k=10), A024132 (k=11), A024145 (k=12).
Numbers that can be expressed as both Sum x^y and Sum y^x where the x^y are not equal to y^x for any (x,y) pair and all (x,y) pairs are distinct.
+10
2
432, 592, 1017, 1040, 1150, 1358, 1388, 1418, 1422, 1464, 1554, 1612, 1632, 1713, 1763, 1873, 1889, 1966, 1968, 1973, 1990, 2091, 2114, 2190, 2291, 2320, 2364, 2451, 2589, 2591, 2612, 2689, 2697, 2719, 2753, 2775, 2803, 2813, 2883, 3087, 3127, 3141, 3146
COMMENTS
Numbers m of form m = Sum_{i=1...k} b_i^e_i = Sum_{i=1...k} e_i^b_i such that b_i^e_i != e_i^b_i, b_i > 1, e_i > 1, k = |{{b_i, e_i}, i = 1, 2, ...}|, k > 1.
Terms of the sequence relate to the Diophantine equation Sum_{i=1...k} x_i = 0, k > 1, x_i != 0, where x_i = (b_i^e_i - e_i^b_i) such that b_i > 1, e_i > 1 and (i != j) => ({b_i, e_i} != {b_j, e_j}). That is, we are observing linear combinations of elements from {(r^n - n^r) : n,r > 1} \ {0}, under given conditions.
For sums with k = 20 terms, one infinite family of examples is known: "2^(2t) + t^(4) + 2^(2t+8) + (t+4)^(4) + 2^(2t+16) + (t+8)^(4) + 2^(2t+32) + (t+16)^(4) + 2^(2t+34) + (t+17)^(4) + 4^(t+1) + (2t+2)^(2) + 4^(t+2) + (2t+4)^(2) + 4^(t+10) + (2t+20)^(2) + 4^(t+14) + (2t+28)^(2) + 4^(t+18) + (2t+36)^(2)" is a term of the sequence, for every t > 4.
EXAMPLE
17 = 2^3 + 3^2 = 3^2 + 2^3 is not in the sequence because {2,3} = {3,2} are not distinct.
25 = 3^3 + 2^4 = 3^3 + 4^2 is not in the sequence because 3^3 = 3^3 and 2^4 = 4^2 are commutative.
The smallest term of the sequence is:
a(1) = 432 = 3^2 + 5^2 + 2^6 + 3^4 + 5^3 + 2^7
= 2^3 + 2^5 + 6^2 + 4^3 + 3^5 + 7^2.
The smallest term that has more than one representation is:
a(11) = 1554 = 3^2 + 7^2 + 6^3 + 2^8 + 4^5
= 2^3 + 2^7 + 3^6 + 8^2 + 5^4,
a(11) = 1554 = 3^2 + 5^2 + 2^6 + 10^2 + 2^7 + 3^5 + 2^8 + 3^6
= 2^3 + 2^5 + 6^2 + 2^10 + 7^2 + 5^3 + 8^2 + 6^3.
Smallest terms with k = 5, 6, 7, 8, 9, 10 summands are:
a(9) = 1422 = 5^2 + 7^2 + 9^2 + 3^5 + 4^5
= 2^5 + 2^7 + 2^9 + 5^3 + 5^4,
a(1) = 432 = 3^2 + 5^2 + 2^6 + 3^4 + 5^3 + 2^7
= 2^3 + 2^5 + 6^2 + 4^3 + 3^5 + 7^2,
a(2) = 592 = 3^2 + 5^2 + 7^2 + 4^3 + 2^6 + 5^3 + 2^8
= 2^3 + 2^5 + 2^7 + 3^4 + 6^2 + 3^5 + 8^2,
a(11) = 1554 = 3^2 + 5^2 + 2^6 + 10^2 + 2^7 + 3^5 + 2^8 + 3^6
= 2^3 + 2^5 + 6^2 + 2^10 + 7^2 + 5^3 + 8^2 + 6^3,
a(14) = 1713 = 3^2 + 2^5 + 6^2 + 8^2 + 4^3 + 2^7 + 3^5 + 2^9 + 5^4
= 2^3 + 5^2 + 2^6 + 2^8 + 3^4 + 7^2 + 5^3 + 9^2 + 4^5,
a(28) = 2451 = 3^2 + 5^2 + 6^2 + 8^2 + 3^4 + 2^7 + 6^3 + 3^5 + 5^4 + 2^10
= 2^3 + 2^5 + 2^6 + 2^8 + 4^3 + 7^2 + 3^6 + 5^3 + 4^5 + 10^2.
CROSSREFS
Cf. A337671 (subsequence for k <= 5). Cf. A005188 (perfect digital invariants). Cf. Nonnegative numbers of the form (r^n - n^r), for n,r > 1: A045575. Cf. Numbers of the form (r^n - n^r): A024012 (r = 2), A024026 (r = 3), A024040 (r = 4), A024054 (r = 5), A024068 (r = 6), A024082 (r = 7), A024096 (r = 8), A024110 (r = 9), A024124 (r = 10), A024138 (r = 11), A024152 (r = 12).
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