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A104621
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Heptanacci-Lucas numbers.
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10
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7, 1, 3, 7, 15, 31, 63, 127, 247, 493, 983, 1959, 3903, 7775, 15487, 30847, 61447, 122401, 243819, 485679, 967455, 1927135, 3838783, 7646719, 15231991, 30341581, 60439343, 120393007, 239818559, 477709983, 951581183, 1895515647, 3775799303, 7521257025
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| This 7th-order linear recurrence is a generalization of the Lucas sequence A000032. Mario Catalani would refer to this is a generalized heptanacci sequence, had he not stopped his series of sequences after A001644 "generalized tribonacci", A073817 "generalized tetranacci", A074048 "generalized pentanacci", A074584 "generalized hexanacci." T. D. Noe and I have noted that each of these has many more primes than the corresponding tribonacci A000073 (see A104576), tetranacci A000288 (see A104577), pentanacci, hexanacci and heptanacci (see A104414). For primes in Heptanacci-Lucas numbers, see A104622. For semiprimes in Heptanacci-Lucas numbers, see A104623.
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REFERENCES
| Mario Catalani, "Polymatrix and Generalized Polynacci Numbers", arXiv:math.CO/0210201 v1, 2002
Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.4.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..200
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FORMULA
| a(0) = 7, a(1) = 1, a(2) = 3, a(3) = 7, a(4) = 15, a(5) = 31, a(6) = 63, for n > 6: a(n) = a(n-1)+a(n-2)+a(n-3)+a(n-4)+a(n-5)+a(n-6)+a(n-7).
G.f.: (-7+6*x+5*x^2+4*x^3+3*x^4+2*x^5+x^6)/(-1+x+x^2+x^3+x^4+x^5+x^6+x^7). a(n)= 7*A066178(n)-6*A066178(n-1)-5*A066178(n-2)-...-2*A066178(n-5)-A066178(n-6) if n>=6. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 16 2007
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MATHEMATICA
| a[0] = 7; a[1] = 1; a[2] = 3; a[3] = 7; a[4] = 15; a[5] = 31; a[6] = 63; a[n_] := a[n] = a[n - 1] + a[n - 2] + a[n - 3] + a[n - 4] + a[n - 5] + a[n - 6] + a[n - 7]; Table[ a[n], {n, 0, 32}] (from Robert G. Wilson v Mar 17 2005)
LinearRecurrence[{1, 1, 1, 1, 1, 1, 1}, {7, 1, 3, 7, 15, 31, 63}, 60] (* From Vladimir Joseph Stephan Orlovsky, Feb 08 2012 *)
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CROSSREFS
| Cf. A000032, A001644, A073817, A074048, A074584, A104414, A104576, A104577.
Sequence in context: A039616 A201585 A089562 * A176439 A048834 A010504
Adjacent sequences: A104618 A104619 A104620 * A104622 A104623 A104624
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KEYWORD
| easy,nonn,changed
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AUTHOR
| Jonathan Vos Post (jvospost3(AT)gmail.com), Mar 17 2005
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