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1, 3, 8, 18, 38, 78, 158, 318, 638, 1278, 2558, 5118, 10238, 20478, 40958, 81918, 163838, 327678, 655358, 1310718, 2621438, 5242878, 10485758, 20971518, 41943038, 83886078, 167772158, 335544318, 671088638, 1342177278, 2684354558
COMMENTS
Last digit of a(n) is 8 for n > 2. - Jon Perry, Nov 19 2012
FORMULA
Binomial transform of [1, 2, 3, 2, 3, 2, 3, ...].
a(n) = 2(n-1) + Sum_{i=1..n-1} a(i). - Jon Perry, Nov 19 2012
EXAMPLE
a(4) = 18 = sum of row 4 terms of triangle A133805: (7 + 6 + 4 + 1). a(4) = 18 = (1, 3, 3, 1), dot (1, 2, 3, 2) = (1 + 6 + 9 + 2).
PROG
(Magma) a:=[1]; for n in [2..31] do Append(~a, 2*n-2+&+[a[i]:i in [1..n-1]]); end for; a; // Marius A. Burtea, Oct 15 2019 (Magma) R<x>:=PowerSeriesRing(Integers(), 31); Coefficients(R!( (1+x^2)/((1-x)*(1-2*x)))); // Marius A. Burtea, Oct 15 2019
CROSSREFS
Essentially a duplicate of A051633.
a(n) = T(8,n), array T given by A048483.
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1, 10, 28, 64, 136, 280, 568, 1144, 2296, 4600, 9208, 18424, 36856, 73720, 147448, 294904, 589816, 1179640, 2359288, 4718584, 9437176, 18874360, 37748728, 75497464, 150994936, 301989880, 603979768, 1207959544, 2415919096
COMMENTS
n-th difference of a(n), a(n-1), ..., a(0) is (9, 9, 9, ...).
FORMULA
Equals binomial transform of [1, 9, 9, 9, ...]. - Gary W. Adamson, Apr 29 2008 a(n) = 3*a(n-1) - 2*a(n-2) with a(0)=1, a(1)=10. - Philippe Deléham, Apr 15 2013
0, 3, 10, 24, 52, 108, 220, 444, 892, 1788, 3580, 7164, 14332, 28668, 57340, 114684, 229372, 458748, 917500, 1835004, 3670012, 7340028, 14680060, 29360124, 58720252, 117440508, 234881020, 469762044, 939524092, 1879048188
FORMULA
a(0) = 0, a(1) = 3, a(n+1) = 2*a(n) + 4, for n >= 1.
a(n) = 7*2^(n-1) - 4, n >= 1.
G.f.: x*(x + 3)/((x - 1)*(2*x - 1)). - Chai Wah Wu, Jul 24 2020
MATHEMATICA
Join[{0}, Table[7*2^(n-1) -4, {n, 50}]] (* G. C. Greubel, Sep 03 2018 *)
PROG
(PARI) concat([0], vector(50, n, 7*2^(n-1) -4)) \\ G. C. Greubel, Sep 03 2018 (Magma) [0] cat [7*2^(n-1) -4: n in [1..50]]; // G. C. Greubel, Sep 03 2018
Expansion of (1+5*x+7*x^2-x^3)/((1-2*x^2)*(1-x)*(1+x)).
+10
0
1, 5, 10, 14, 28, 32, 64, 68, 136, 140, 280, 284, 568, 572, 1144, 1148, 2296, 2300, 4600, 4604, 9208, 9212, 18424, 18428, 36856, 36860, 73720, 73724, 147448, 147452, 294904, 294908, 589816, 589820, 1179640, 1179644, 2359288, 2359292, 4718584, 4718588, 9437176
FORMULA
G.f.: (1+5*x+7*x^2-x^3)/((1-x)*(1+x)*(1-2*x^2)).
a(n) = a(n-1)+4 if n odd.
a(n) = a(n-1)*2 if n even.
a(2n+1) = 9*2^n - 4 = A053209(n+1). a(n) = 3*a(n-2) - 2*a(n-4) with n>3, a(0)=1, a(1)=5, a(2)=10, a(3)=14.
MATHEMATICA
CoefficientList[Series[(1+5x+7x^2-x^3)/((1-2x^2)(1-x)(1+x)), {x, 0, 40}], x] (* or *) LinearRecurrence[{0, 3, 0, -2}, {1, 5, 10, 14}, 50] (* Harvey P. Dale, Sep 17 2016 *)
Square array T(n,k) = 9*2^k - n read by ascending antidiagonals.
+10
0
9, 8, 18, 7, 17, 36, 6, 16, 35, 72, 5, 15, 34, 71, 144, 4, 14, 33, 70, 143, 288, 3, 13, 32, 69, 142, 287, 576, 2, 12, 31, 68, 141, 286, 575, 1152, 1, 11, 30, 67, 140, 285, 574, 1151, 2304, 0, 10, 29, 66, 139, 284, 573, 1150, 2303, 4608, -1, 9, 28, 65, 138, 283, 572, 1149, 2302, 4607, 9216
FORMULA
G.f.: (9 - 9*y + x*(11*y - 10))/((1 - x)^2*(1 - y)*(1 - 2*y)). - Stefano Spezia, Mar 17 2024
EXAMPLE
Table begins:
k=0 1 2 3 4 5
n=0: 9 18 36 72 144 288 ...
n=1: 8 17 35 71 143 287 ...
n=2: 7 16 34 70 142 286 ...
n=3: 6 15 33 69 141 285 ...
n=4: 5 14 32 68 140 284 ...
n=5: 4 13 31 67 139 283 ...
Every line has the signature (3,-2). For n=1: 3*17 - 2*8 = 35.
Main diagonal's difference table:
9 17 34 69 140 283 570 1145 ... = b(n)
8 17 35 71 143 287 575 1151 ... = A052996(n+2) 9 18 36 72 144 288 576 1152 ... = A005010(n) ...
MATHEMATICA
T[n_, k_] := 9*2^k - n; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Mar 06 2024 *)
CROSSREFS
Cf. A000225, A033484, A048491, A005010, A052996, A053209, A083329, A154251, A176449, A304383, A367559, A368826.
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