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A116157
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a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) + a(n-5).
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1
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1, 1, 3, 3, 7, 8, 17, 22, 43, 60, 110, 161, 283, 428, 732, 1132, 1901, 2984, 4950, 7848, 12912, 20609, 33721, 54065, 88137, 141737, 230490, 371411, 602982, 972961, 1577840, 2548288, 4129457, 6673335, 10808634, 17474230, 28293116, 45753765, 74064872, 119794804
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OFFSET
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0,3
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COMMENTS
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A Fibonacci-Padovan sequence.
The summation over some naturally chosen planes in the pyramid composed of MacWilliams transform matrices yields this sequence, which is the convolution of the Fibonacci numbers and the (alternating) Padovan numbers. Namely, the formula F(n) = Sum_{i+k=n, i>0, k>0} binomial(k,i) = Sum_{i+k=n, i>0, k>0} Krawtchouk[{k,i},0] where Krawtchouk[{k,i},x] is the i-th Krawtchouk polynomial of order k has a natural generalization as G(n) = Sum_{i+j+k=n, i>0,j>0, k>0} Krawtchouk[{k,i},j].
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LINKS
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FORMULA
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G.f.: 1/((1-x-x^2)*(1-x^2+x^3)).
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MATHEMATICA
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a[0]=1; a[1]=1; a[2]=3; a[3]=3; a[4]=7; a[n_]:=a[n]=a[n-1]+2a[n-2]-2a[n-3]+a[n-5]; Table[a[n], {n, 0, 50}]
LinearRecurrence[{1, 2, -2, 0, 1}, {1, 1, 3, 3, 7}, 50] (* Harvey P. Dale, Mar 07 2015 *)
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PROG
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(PARI) my(x='x+O('x^50)); Vec(1/((1-x-x^2)*(1-x^2+x^3))) \\ G. C. Greubel, May 10 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/((1-x-x^2)*(1-x^2+x^3)) )); // G. C. Greubel, May 10 2019
(Sage) (1/((1-x-x^2)*(1-x^2+x^3))).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, May 10 2019
(GAP) a:=[1, 1, 3, 3, 7];; for n in [6..50] do a[n]:=a[n-1]+2*a[n-2]- 2*a[n-3]+a[n-5]; od; a; # G. C. Greubel, May 10 2019
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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Nikita Gogin & Aleksandr Myllari (alemio(AT)utu.fi), Apr 15 2007
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STATUS
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approved
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