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A116157
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a(n)=a(n-1)+2a(n-2)-2a(n-3)+a(n-5).
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0
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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[binomial[k,i],{i+k=n, i>0, k>0}]= Sum[Krawtchouk[{k,i},0],{i+k=n, i>0, k>0}], where Krawtchouk[{k,i},x] is the i-th Krawtchouk polynomial of order k has a natural generalization as G(n)= Sum[Krawtchouk[{k,i},j],{i+j+k=n, i>0,j>0, k>0}].
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REFERENCES
| N. Gogin and A. Myllari, The Fibonacci-Padovan sequence and MacWilliams transform matrices,Programming and Computing Software, Vol. 33, Issue 2 (March 2007), pp. 74 - 79.
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FORMULA
| G.f.: 1/((1-x-x^2)(1-x^2+x^3))
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MATHEMATICA
| 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}]
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CROSSREFS
| Cf. A000045, A000931.
Sequence in context: A086543 A110618 A108046 * A056357 A144554 A177936
Adjacent sequences: A116154 A116155 A116156 * A116158 A116159 A116160
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KEYWORD
| easy,nonn
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AUTHOR
| Nikita Gogin & Aleksandr Myllari (alemio(AT)utu.fi), Apr 15 2007
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