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A345095 a(n) is the sum of the two preceding terms if n is even, or of the two preceding digits if n is odd, with a(0) = 0, a(1) = 1. 2
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, 15, 111, 2, 113, 4, 117, 8, 125, 7, 132, 5, 137, 10, 147, 11, 158, 13, 171, 8, 179, 16, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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.

The repeating part (mod 100) is exactly the same as for the sister sequence A345097, where it starts already after 9 terms.

Differs from the Fibonacci sequence A000045 from a(9) = 3 on.

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).

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.

LINKS

Table of n, a(n) for n=0..99.

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)

FORMULA

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.

a(n+40) = a(n) for odd n > 32, a(n+40) = a(n) + 200 for even n >= 32, whence:

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).

a(n) = a(n-2) + a(n-40) - a(n-42) for n >= 74.

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

EXAMPLE

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.

a(8) = 21 = 8 + 13 is the sum of the two preceding terms.

a(9) = 3 = 2 + 1 is the sum of the two preceding digits.

a(10) = 24 = 21 + 3 is the sum of the two preceding terms.

a(11) = 6 = 2 + 4 is the sum of the two preceding digits.

and so on.

MATHEMATICA

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 *)

PROG

(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)

{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

CROSSREFS

Cf. A000045, A345097 (same with rule for odd/even indexed terms exchanged).

Sequence in context: A193616 A273715 A093093 * A281408 A327451 A137290

Adjacent sequences:  A345092 A345093 A345094 * A345096 A345097 A345098

KEYWORD

nonn,base

AUTHOR

M. F. Hasler and Eric Angelini, Jun 07 2021

STATUS

approved

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Last modified January 17 22:33 EST 2022. Contains 350410 sequences. (Running on oeis4.)