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Search: id:A000285
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| A000285 |
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a(n) = a(n-1) + a(n-2).
(Formerly M3246 N1309)
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+0
16
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| 1, 4, 5, 9, 14, 23, 37, 60, 97, 157, 254, 411, 665, 1076, 1741, 2817, 4558, 7375, 11933, 19308, 31241, 50549, 81790, 132339, 214129, 346468, 560597, 907065, 1467662, 2374727, 3842389, 6217116, 10059505
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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a(n-1)=sum(P(4;n-1-k,k),k=0..ceiling((n-1)/2)), n>=1, with a(-1)=3. These are the sums over the SW-NE diagonals in P(4;n,k), the (4,1) Pascal triangle A093561. Observation by Paul Barry (pbarry(AT)wit.ie, Apr 29 2004. Proof via recursion relations and comparison of inputs. Also SW-NE diagonal sums in the Pascal (1,3) triangle A095660.
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REFERENCES
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S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
A. Brousseau, Seeking the lost gold mine or exploring Fibonacci factorizations, Fib. Quart., 3 (1965), 129-130.
A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 53.
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 224.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..500
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Tanya Khovanova, Recursive Sequences
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FORMULA
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G.f.: (1+3*x)/(1-x-x^2).
Row sums of A131775 starting (1, 4, 5, 9, 14, 23,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 14 2007
a(n)=2*Fibonacci(n-2)+Fibonacci(n), n>=2 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 05 2007
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MAPLE
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BB := n->if n=1 then 3; > elif n=2 then 1; > else BB(n-2)+BB(n-1); > fi: > L:=[]: for k from 2 to 34 do L:=[op(L), BB(k)]: od: L; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 19 2007
with(combinat):a:=n->2*fibonacci(n-2)+fibonacci(n): seq(a(n), n=2..34); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 05 2007
A000285:=-(1+3*z)/(-1+z+z**2); [Conjectured by S. Plouffe in his 1992 dissertation.]
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CROSSREFS
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Essentially the same as A104449.
a(n) = A101220(3, 0, n+1).
a(n) = A109754(3, n+1).
a(k) = A090888(2, k-1), for k > 0.
Cf. A131775.
Adjacent sequences: A000282 A000283 A000284 this_sequence A000286 A000287 A000288
Sequence in context: A096818 A038099 A120740 this_sequence A042031 A041493 A042765
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KEYWORD
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easy,nonn,nice
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AUTHOR
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njas
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