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A345097
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a(n) is the sum of the two preceding terms if n is odd, or of the two preceding digits if n is even, with a(0) = 0, a(1) = 1.
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2
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0, 1, 1, 2, 3, 5, 8, 13, 4, 17, 8, 25, 7, 32, 5, 37, 10, 47, 11, 58, 13, 71, 8, 79, 16, 95, 14, 109, 9, 118, 9, 127, 9, 136, 9, 145, 9, 154, 9, 163, 9, 172, 9, 181, 9, 190, 9, 199, 18, 217, 8, 225, 7, 232, 5, 237, 10, 247, 11, 258, 13, 271, 8, 279, 16, 295, 14, 309, 9, 318, 9, 327
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OFFSET
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0,4
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COMMENTS
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Considering the terms modulo 100, the sequence becomes periodic with period length 40 after the first 9 terms. The period is the same as in A345095, where it starts only after the first 32 terms. This property leads to a first order recurrence and an explicit formula for a(n), see Formula section.
Differs from the Fibonacci sequence A000045 from a(8) = 4 on.
Starting with a(9) = 13, every other term a(2k-1) has at least two digits, so the next term a(2k) is equal to the sum of the last two digits of a(2k-1).
Similarly, the graph of this sequence has two components: even indexed terms repeating the pattern [8, 7, 5, 10, 11, 13, 8, 16, 14, 9, ..., 9, 18] of length 20, and odd indexed terms evolving around the straight line y(n) = 5n - 32.25, with first differences equal to the even indexed terms.
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LINKS
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Table of n, a(n) for n=0..71.
Eric Angelini, Fibonacci alternated, math-fun discussion list on xmission.com, Jul 04 2021
Index to entries for linear recurrences with constant coefficients, order 42, signature (0, 1, 0, ..., 0, 1, 0, -1)
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FORMULA
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a(n+1) = a(n) + a(n-1) if n is even or n < 6, = a(n)%10 + floor(a(n)/10)%10 if n is odd and n > 6, where % is the binary modulo (or remainder) operator.
a(n+40) = a(n) for even n > 9, a(n+40) = a(n) + 200 for odd n >= 9, hence:
a(n) = a((n-9)%40 + 9) + [floor((n-9)/40)*200 if n odd] for n >= 9, giving any term explicitly in terms of a(0..48).
a(n) = a(n-2) + a(n-40) - a(n-42) for n >= 51.
O.g.f. x*(Sum_{k=0..49} c_k x^k)/(1 - x^2 - x^40 + x^42), where c = (1, 1, 1, 2, 3, 5, 8, -4, 4, 4, 8, -1, 7, -2, 5, 5, 10, 1, 11, 2, 13, -5, 8, 8, 16, -2, 14, -5, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 8, -1, 8, -2, 6, -5, 1, 13, 14, -14).
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EXAMPLE
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Up to a(7) = 13, we have the Fibonacci sequence A000045. Then:
a(8) = 1 + 3 = 4 is the sum of the two preceding digits: those of a(7).
a(9) = 13 + 4 = 17 is the sum of the two preceding terms, a(7) + a(8).
a(10) = 1 + 7 = 8 is the sum of the two preceding digits: those of a(9).
a(11) = 17 + 8 = 25 is the sum of the two preceding terms, a(9) + a(10),
and so on.
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MATHEMATICA
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a[0]=0; a[1]=a[2]=1; a[n_]:=a[n]=If[OddQ@n, a[n-1]+a[n-2], Total[Flatten[IntegerDigits/@Array[a, n-1]][[-2;; ]]]]; Array[a, 100, 0] (* Giorgos Kalogeropoulos, Jun 08 2021 *)
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PROG
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(PARI) A345097_vec(N=99, a=List([0, 1]))={ for(n=2, N, listput(a, if(n%2 || a[n]<10, a[n-1]+a[n], sumdigits(a[n]%100)))); Vec(a)} \\ Compute the vector a(0..N)
M345097=A345097_vec(49); A345097(n)=if(n<9, M345097[n+1], n=divrem(n-9, 40); M345097[n[2]+10]+!(n[2]%2)*n[1]*200) \\ Instantly computes any term.
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CROSSREFS
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Cf. A000045, A345095 (same with rule for odd/even indexed terms exchanged).
Sequence in context: A254056 A238948 A336716 * A050416 A079345 A106005
Adjacent sequences: A345094 A345095 A345096 * A345098 A345099 A345100
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KEYWORD
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nonn,base,easy
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AUTHOR
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M. F. Hasler and Eric Angelini, Jun 07 2021
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STATUS
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approved
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