The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A180664

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A180664

Golden Triangle sums: a(n) = a(n-1) + A001654(n+1) with a(0)=0.
(history; published version)

#27 by Harvey P. Dale at Thu Mar 30 16:55:14 EDT 2023

STATUS

editing

approved

#26 by Harvey P. Dale at Thu Mar 30 16:55:12 EDT 2023

MATHEMATICA

LinearRecurrence[{3, 0, -3, 1}, {0, 2, 8, 23}, 30] (* Harvey P. Dale, Mar 30 2023 *)

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approved

editing

#25 by Michael De Vlieger at Sat Jan 22 08:44:32 EST 2022

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reviewed

approved

#24 by Michel Marcus at Sat Jan 22 01:25:42 EST 2022

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proposed

reviewed

#23 by G. C. Greubel at Fri Jan 21 14:01:57 EST 2022

STATUS

editing

proposed

#22 by G. C. Greubel at Fri Jan 21 14:01:49 EST 2022

FORMULA

From _a(n) = (1/10)*((-1)^n - 15 + 2*Lucas(2*n+4)). - _G. C. Greubel_, Jan 21 2022: (Start)

a(n) = (1/10)*((-1)^n - 15 + 2*Lucas(2*n+4)).

a(n) = (1/10)*((-1)^n - 15 + 14*ChebyshevU(n, 3/2) - 6*ChebyshevU(n-1, 3/2)). (End)

CROSSREFS

Cf. A000032, A000045, A005248, A064831, A115730, A180662, A180664, A180665, A180666.

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proposed

editing

#21 by G. C. Greubel at Fri Jan 21 02:06:01 EST 2022

STATUS

editing

proposed

Discussion

Fri Jan 21

02:15

Michel Marcus: I wonder if the ChebyshevU formula should rather be in A005248 ?

#20 by G. C. Greubel at Fri Jan 21 02:05:58 EST 2022

FORMULA

a(n) = (1/10)*((-1)^n - 15 + 2*LucasLLucas(2*n+4)).

STATUS

proposed

editing

#19 by G. C. Greubel at Fri Jan 21 01:59:55 EST 2022

STATUS

editing

proposed

#18 by G. C. Greubel at Fri Jan 21 01:59:43 EST 2022

LINKS

G. C. Greubel, <a href="/A180664/b180664.txt">Table of n, a(n) for n = 0..1000</a>

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(9-45))/2 and B=(3-sqrt(9-45))/2.

From G. C. Greubel, Jan 21 2022: (Start)

a(n) = (1/10)*((-1)^n - 15 + 2*LucasL(2*n+4)).

a(n) = (1/10)*((-1)^n - 15 + 14*ChebyshevU(n, 3/2) - 6*ChebyshevU(n-1, 3/2)). (End)

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, A064831, A115730, A180662, A180664, A180665, A115730, A180666.

STATUS

approved

editing