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A137232
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a(n) = -a(n-1) + 7*a(n-2) + 3*a(n-3) with a(0) = a(1) = 0, a(2) = 1.
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1
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0, 0, 1, -1, 8, -12, 65, -125, 544, -1224, 4657, -11593, 40520, -107700, 356561, -988901, 3161728, -9014352, 28179745, -81795025, 252010184, -740036124, 2258722337, -6682944653, 20273892640, -60278338200, 182146752721, -543273442201, 1637465696648, -4893939533892, 14726379083825
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OFFSET
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0,5
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,7,3).
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FORMULA
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From R. J. Mathar, Mar 17 2008: (Start)
O.g.f.: x^2/((1+3*x)*(1-2*x-x^2)).
a(n) = ( (-3)^n + A135532(n) )/14. (End)
a(n) = -(1/28)*(1+sqrt(2))^n - (1/14)*sqrt(2)*(1-sqrt(2))^n + (1/14)*(-3)^n -(1/28)*(1-sqrt(2))^n + (1/14)*(1+sqrt(2))^n*sqrt(2), with n>=0 - Paolo P. Lava, Jun 09 2008
a(n) = (1/14)*( (-3)^n + 5*Pell(n) - Pell(n+1) ), where Pell(n) = A000129(n). - G. C. Greubel, Apr 19 2021
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MATHEMATICA
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Table[((-3)^n + 5*Fibonacci[n, 2] -Fibonacci[n+1, 2])/14, {n, 0, 40}] (* G. C. Greubel, Apr 19 2021 *)
LinearRecurrence[{-1, 7, 3}, {0, 0, 1}, 40] (* Harvey P. Dale, Apr 26 2022 *)
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PROG
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(Magma) I:=[0, 0, 1]; [n le 3 select I[n] else -Self(n-1) +7*Self(n-2) +3*Self(n-3): n in [1..36]]; // G. C. Greubel, Apr 19 2021
(Sage) [((-3)^n +5*lucas_number1(n, 2, -1) -lucas_number1(n+1, 2, -1))/14 for n in (0..40)] # G. C. Greubel, Apr 19 2021
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CROSSREFS
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Cf. A000129, A135532.
Sequence in context: A038290 A002288 A216711 * A147764 A226259 A162466
Adjacent sequences: A137229 A137230 A137231 * A137233 A137234 A137235
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KEYWORD
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sign,easy
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AUTHOR
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Paul Curtz, Mar 08 2008
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STATUS
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approved
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