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A215580 Partial sums of A215602. 2
2, 5, 17, 45, 122, 320, 842, 2205, 5777, 15125, 39602, 103680, 271442, 710645, 1860497, 4870845, 12752042, 33385280, 87403802, 228826125, 599074577, 1568397605, 4106118242, 10749957120, 28143753122, 73681302245, 192900153617, 505019158605, 1322157322202, 3461452808000, 9062201101802, 23725150497405, 62113250390417, 162614600673845 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
Dividing the terms of this sequence by Fibonacci or Lucas numbers yields symmetric sets of remainders of determinable lengths. For F(n) beginning at n=3: (a) F(2n) will have a set of remainders of length 2n in which the sum of the remainders is 3*(F(2n)-n). Example for F(2*6)=144: the set of remainders is {2,5,17,45,122,32,122,45,17,5,2,0} with 2*6=12 terms and a sum of 3*(144-6)=414. (b) For F(2n+1) there will be 2*(2n+1) terms having a sum equal to (2n+1)*(F(2n+1)-3). Example for F(2*4+1)=34: the remainders are {2,5,7,11,20,14,26,29,31,29,26,14,20,11,17,5,2,0} with 2*9 terms and a sum of 9*(34-1)=279.
Using Lucas numbers starting at n=2: (a) L(2n) has 4n remainders with sum (2n+1)*(L(2n)-6*n). Example for n=4 giving L(2*4)=47, has remainders {2,5,17,45,28,38,43,43,43,38,28,45,17,5,2,0} with a sum of (8+1)*(47)-6*4=399. (B) For L(2n+1) the length of the period is 2*(2n+1) and the sum of the remainders is 4*L(2n+1)-3*(2n+1). Example for n=3 for L(2*3+1)=29 has remainders {2,5,17,16,6,1,11,6,16,17,5,2,0} with length 2*7 and sum of terms 4*29-3*7=95.
LINKS
FORMULA
a(2n) = L(4*n)-2, a(2*n+1) = L(4*n+2)-1, where L() are the Lucas numbers A000032.
G.f. ( -2+x-2*x^2 ) / ( (x-1)*(1+x)*(x^2-3*x+1) ). - R. J. Mathar, Aug 21 2012
a(n) = A005248(n+1)-A000034(n). - R. J. Mathar, Aug 21 2012
MATHEMATICA
LinearRecurrence[{3, 0, -3, 1}, {2, 5, 17, 45}, 35] (* Paolo Xausa, Feb 22 2024 *)
CROSSREFS
Sequence in context: A056304 A063106 A096295 * A275210 A219554 A074494
KEYWORD
nonn,easy
AUTHOR
J. M. Bergot, Aug 16 2012
EXTENSIONS
Edited by N. J. A. Sloane, Aug 17 2012
STATUS
approved

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Last modified June 15 18:41 EDT 2024. Contains 373410 sequences. (Running on oeis4.)