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A225209
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a(n) = (392*16^n -1620*8^n +1890*4^n -767)/105.
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1
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1, 249, 8537, 186073, 3427545, 58664153, 970097881, 15776875737, 254486643929, 4088295982297, 65545039643865, 1049779971687641, 16804957869966553, 268947166998693081, 4303697458594972889, 68863501862374868185
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OFFSET
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1,2
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COMMENTS
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Starting at n=1, a cube has an edge=2^(n+1)-3. The beginning cube has a value of 1 and is surrounded by 2^n layers of cubes each valued at 2^n. The sum of all cubes with values of 2^n is a(n).
Indices of primes in this sequence: 3, 10, 12, 21, 37, 70, 102, 201, 961, 1854, ....
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 1..825
Index entries for linear recurrences with constant coefficients, signature (29,-252,736,-512).
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FORMULA
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a(n) = 29*a(n-1) - 252*a(n-2) + 736*a(n-3) - 512*a(n-4).
a(n) = a(n-1) + 7*2^(4*n-1) - 27*2^(3*n-1) + 27*2^(2*n-1), for n>0.
G.f. x*(1 +220*x +1568*x^2 +512*x^3)/( (1-x)*(1-4*x)*(1-8*x)*(1-16*x) ). - R. J. Mathar, May 09 2013
a(n) = a(n-1) +2^(n-1)*(A036563(n+1)^3 -A036563(n)^3). - R. J. Mathar, May 18 2013
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EXAMPLE
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The first cubes has value 1=a(1). The second cube has 2 layers of cubes each valued at 2 surrounding the cube of value 1 to give (5^3-1)*2+1=249=a(2). Next surround by 2^2 layers of cubes each valued at 2^2: (13^3-5^3)*4+249=8537=a(3). Finally, surround by 2^3 layers of cubes each of value 2^3 to get (29^3-13^3)*8 + 8537 = 186073 = a(4).
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MAPLE
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seq( (392*2^(4*n) -1620*2^(3*n) +1890*2^(2*n) -767)/105, n=1..20); # G. C. Greubel, Dec 31 2019
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MATHEMATICA
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LinearRecurrence[{29, -252, 736, -512}, {1, 249, 8537, 186073}, 20] (* Harvey P. Dale, Apr 22 2018 *)
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PROG
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(PARI) vector(20, n, (392*2^(4*n) -1620*2^(3*n) +1890*2^(2*n) -767)/105) \\ G. C. Greubel, Dec 31 2019
(Magma) [(392*2^(4*n) -1620*2^(3*n) +1890*2^(2*n) -767)/105: n in [1..20]]; // G. C. Greubel, Dec 31 2019
(Sage) [(392*2^(4*n) -1620*2^(3*n) +1890*2^(2*n) -767)/105 for n in (1..20)] # G. C. Greubel, Dec 31 2019
(GAP) List([1..20], n-> (392*2^(4*n) -1620*2^(3*n) +1890*2^(2*n) -767)/105); # G. C. Greubel, Dec 31 2019
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CROSSREFS
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Sequence in context: A045254 A197349 A197400 * A197363 A069154 A045169
Adjacent sequences: A225206 A225207 A225208 * A225210 A225211 A225212
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KEYWORD
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nonn,easy
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AUTHOR
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J. M. Bergot, May 01 2013
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STATUS
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approved
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