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%I #45 Mar 30 2024 23:08:03
%S 0,1,1,2,3,5,8,13,21,3,24,6,30,3,33,6,39,12,51,6,57,12,69,15,84,12,96,
%T 15,111,2,113,4,117,8,125,7,132,5,137,10,147,11,158,13,171,8,179,16,
%U 195,14,209,9,218,9,227,9,236,9,245,9,254,9,263,9,272,9,281,9,290,9,299,18,317,8,325,7,332,5,337,10,347,11,358,13,371,8,379,16,395,14,409,9,418,9,427,9,436,9,445,9
%N a(n) is the sum of the two preceding terms if n is even or n < 8, and the sum of the two rightmost digits of a(n-1) if n is odd, with a(0) = 0, a(1) = 1.
%C Considering the terms modulo 100, the sequence becomes periodic with period 40 after the first 32 terms. [Observation by _Hans Havermann_.] See more precise formula: a(n+40) = a(n) [+ 200 if n even], n >= 32, where 200 is the sum of every other term [i.e., the odd-indexed terms] of the repeating part.
%C The repeating part (mod 100) is exactly the same as for the sister sequence A345097, where it starts already after 9 terms.
%C Differs from the Fibonacci sequence A000045 from a(9) = 3 on.
%C After a(6) = 8, every other term a(2k), computed as the sum of the two preceding terms, has at least two digits, so the subsequent term a(2k+1) is always equal to the sum of the last two digits of the preceding term a(2k).
%C In the same way, the graph of this sequence has two components: odd-indexed terms repeating the pattern [8, 7, 5, 10, 11, 13, 8, 16, 14, 9, ..., 9, 18] of length 20, and even-indexed terms evolving around the straight line y(n) = 5n - 47.25 with first differences equal to the odd-indexed terms.
%H Eric Angelini, <a href="https://mailman.xmission.com/hyperkitty/list/math-fun@mailman.xmission.com/thread/ERBI7PWVRAJJ5ZTGSKX2C7F4PTAVTEOR/">Fibonacci alternated</a>, math-fun discussion list on xmission.com, Jul 04 2021
%H <a href="/index/Rec#order_42">Index entries for linear recurrences with constant coefficients</a>, signature (0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1).
%F a(n+1) = a(n) + a(n-1) if n is odd, = a(n)%10 + floor(a(n)/10)%10 if n is even, where % is the binary modulo (or remainder) operator.
%F a(n+40) = a(n) for odd n > 32, a(n+40) = a(n) + 200 for even n >= 32, whence:
%F a(n) = a((n-32)%40 + 32) + [floor((n-32)/40)*200 if n even], n >= 32, giving any a(n) explicitly in terms of a(0..71).
%F a(n) = a(n-2) + a(n-40) - a(n-42) for n >= 74.
%F O.g.f.: x*(1 + x - x^2)*(Sum_{k=0..35} c_k x^2k)/(1 - x^2 - x^40 + x^42), where c = (1, 2, 5, 13, 3, 6, 3, 6, 12, 6, 12, 15, 12, 15, 2, 4, 8, 7, 5, 10, 10, 11, 3, 3, 11, 3, 6, 3, -3, 3, -3, -6, -3, -6, 7, 14). - _M. F. Hasler_, Jun 10 2021
%e Up to a(6) = 8, the terms have only one digit and therefore the sequence coincides with the Fibonacci sequence A000045 up to a(7) = 13.
%e a(8) = 21 = 8 + 13 is the sum of the two preceding terms.
%e a(9) = 3 = 2 + 1 is the sum of the two preceding digits.
%e a(10) = 24 = 21 + 3 is the sum of the two preceding terms.
%e a(11) = 6 = 2 + 4 is the sum of the two preceding digits.
%e and so on.
%t a[0]=0;a[1]=1;a[n_]:=a[n]=If[EvenQ@n,a[n-1]+a[n-2],Total[Flatten[IntegerDigits/@Array[a,n-1]][[-2;;]]]];Array[a,100,0] (* _Giorgos Kalogeropoulos_, Jun 08 2021 *)
%o (PARI) A345095_vec(N=99,a=List([0,1]))={ for(n=2,N, listput(a, if(n%2==0 || a[n]<10, a[n-1]+a[n], sumdigits(a[n]%100))));Vec(a)} \\ Compute the vector a(0..N)
%o {M345095=A345095_vec(72); A345095(n)=if(n<32, M345095[n+1], n=divrem(n-32, 40); M345095[n[2]+33]+!(n[2]%2)*n[1]*200)} \\ _M. F. Hasler_, Jun 10 2021
%Y Cf. A000045, A345097 (same with rule for odd-/even-indexed terms exchanged).
%K nonn,base
%O 0,4
%A _M. F. Hasler_ and _Eric Angelini_, Jun 07 2021
%E Definition amended by _Georg Fischer_, Aug 08 2023
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