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 A180664 Golden Triangle sums: a(n) = a(n-1) + A001654(n+1) with a(0)=0. 8
 0, 2, 8, 23, 63, 167, 440, 1154, 3024, 7919, 20735, 54287, 142128, 372098, 974168, 2550407, 6677055, 17480759, 45765224, 119814914, 313679520, 821223647, 2149991423, 5628750623, 14736260448, 38580030722, 101003831720 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The a(n+1) (terms doubled) are the Kn13 sums of the Golden Triangle A180662. See A180662 for information about these knight and other chess sums. LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (3,0,-3,1). FORMULA a(n+1) = Sum_{k=0..n} A180662(2*n-k+2, k+2). a(n) = (-15 + (-1)^n + (6-2*A)*A^(-n-1) + (6-2*B)*B^(-n-1))/10 with A=(3+sqrt(5))/2 and B=(3-sqrt(5))/2. G.f.: (2*x+2*x^2-x^3)/(1-3*x-x^4+3*x^3). a(n) = Sum_{i=0..n-1} F(i+2)*F(i+3), where F(i) = A000045(i). - Rigoberto Florez, Jul 07 2020 a(n) = (1/10)*((-1)^n - 15 + 2*Lucas(2*n+4)). - G. C. Greubel, Jan 21 2022 MAPLE nmax:=26: with(combinat): for n from 0 to nmax+1 do A001654(n):=fibonacci(n) * fibonacci(n+1) od: a(0):=0: for n from 1 to nmax do a(n) := a(n-1)+A001654(n+1) od: seq(a(n), n=0..nmax); MATHEMATICA Table[Sum[Fibonacci[i+2]*Fibonacci[i+3], {i, 0, n-1}], {n, 0, 40}] (* Rigoberto Florez, Jul 07 2020 *) LinearRecurrence[{3, 0, -3, 1}, {0, 2, 8, 23}, 30] (* Harvey P. Dale, Mar 30 2023 *) PROG (Magma) [(1/10)*((-1)^n - 15 + 2*Lucas(2*n+4)): n in [0..40]]; // G. C. Greubel, Jan 21 2022 (Sage) [(1/10)*((-1)^n - 15 + 2*lucas_number2(2*n+4, 1, -1)) for n in (0..40)] # G. C. Greubel, Jan 21 2022 CROSSREFS Cf. A000032, A000045, A005248, A064831, A115730, A180662, A180664, A180665, A180666. Sequence in context: A154144 A255942 A355551 * A294959 A290926 A018042 Adjacent sequences: A180661 A180662 A180663 * A180665 A180666 A180667 KEYWORD easy,nonn AUTHOR Johannes W. Meijer, Sep 21 2010 STATUS approved

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Last modified May 31 21:32 EDT 2023. Contains 363068 sequences. (Running on oeis4.)