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Search: id:A065874
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| A065874 |
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A second order recurrence of promic type (integer roots). |
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+0
2
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| 1, 1, 43, 85, 1891, 5461, 84883, 314245, 3879331, 17077621, 180009523, 897269605, 8457669571, 46142992981, 401365114963, 2339370820165, 19196705648611, 117450280095541, 923711917337203, 5856623681349925, 44652524209512451
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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If the number j = A002378(m) is promic ( = i(i+1)), then a(n) = a(n-1)+j*a(n-2),a(0) = a(1) = 1 has a closed form solution involving only powers of integers. The binomial coefficient sum solves the recurrence regardless of promicity (cf. GKP reference)
Hankel transform is := 1,42,0,0,0,0,0,0,0,0,0,0,... [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 02 2008]
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REFERENCES
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R. L. Graham, D. E. Knuth, O. Patashnik, "Concrete Mathematics", Addison-Wesley, 1994, p. 204.
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LINKS
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Harry J. Smith, Table of n, a(n) for n=0,...,150
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FORMULA
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a(n)=a(n-1)+42a(n-2); a(0)=a(1)=1. a(n)=[7^(n+1) - (-6)^(n+1)]/13.
G.f.: -1/(6*x+1)/(7*x-1). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 16 2007
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MAPLE
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n->sum(binomial(n-k, k)*(42)^k, k=0..n)
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PROGRAM
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(PARI) { for (n=0, 150, if (n>1, a=a1 + 42*a2; a2=a1; a1=a, a=a1=a2=1); write("b065874.txt", n, " ", a) ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Nov 02 2009]
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CROSSREFS
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Cf. A001045 (j=2), A015441 (j=6), A053404 (j=12), A053428 (j=20), A053430 (j=30).
Sequence in context: A139982 A119487 A063351 this_sequence A062060 A037986 A039526
Adjacent sequences: A065871 A065872 A065873 this_sequence A065875 A065876 A065877
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KEYWORD
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nonn,new
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AUTHOR
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Len Smiley (smiley(AT)math.uaa.alaska.edu), Dec 07 2001
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