A341549
a(n) = Sum_{k=1..n} (-1)^(n+k)*A087322(n,k).
Original entry on oeis.org
3, 6, 23, 50, 131, 294, 687, 1530, 3419, 7502, 16391, 35490, 76467, 163830, 349535, 742730, 1572875, 3320478, 6990519, 14680050, 30758243, 64312646, 134217743, 279620250, 581610171, 1207959534, 2505397607, 5189752130, 10737418259, 22190664342
Offset: 1
A087323
a(n) = (n+1) * 2^n - 1.
Original entry on oeis.org
0, 3, 11, 31, 79, 191, 447, 1023, 2303, 5119, 11263, 24575, 53247, 114687, 245759, 524287, 1114111, 2359295, 4980735, 10485759, 22020095, 46137343, 96468991, 201326591, 419430399, 872415231, 1811939327, 3758096383, 7784628223, 16106127359, 33285996543, 68719476735
Offset: 0
Cf.
A087322 (a triangle which includes this sequence as the leading diagonal but without the initial zero).
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[((n+1)*2^n - 1): n in [1..30]]; // Vincenzo Librandi, Sep 29 2011
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Table[(n + 1)2^n - 1, {n, 0, 29}] (* Alonso del Arte, Jan 31 2014 *)
LinearRecurrence[{5,-8,4},{0,3,11},40] (* Harvey P. Dale, Sep 15 2019 *)
Formula promoted to definition and offset adjusted to 0 by
Alonso del Arte, Jan 31 2014
A190730
Let b(n,0) = n and b(n,k) = 2*b(n,k-1) + 1 for k > 0. Then a(n) = b(n,1) + b(n,2) + ... + b(n,n).
Original entry on oeis.org
3, 16, 53, 146, 367, 876, 2025, 4582, 10211, 22496, 49117, 106458, 229335, 491476, 1048529, 2228174, 4718539, 9961416, 20971461, 44040130, 92274623, 192937916, 402653113, 838860726, 1744830387, 3623878576, 7516192685, 15569256362, 32212254631, 66571992996
Offset: 1
One way to view it is to begin with n = 5, then 5 + 6 = 11 --> 11 + 12 = 23 --> 23 + 24 = 47 --> 47 + 48 = 95 --> 95 + 96 = 191. There are n steps, in this case 5, that give the sum 11 + 23 + 47 + 95 + 191 = 367. This is the same as (2*5+1) + (4*5+3) + (8*5+7) + (16*5+15) + (32*5+31). The formula gives (5+1)*2^(5+1) - 3*5 - 2 = 6*64 - 17 = 367.
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[(n+1) * 2^(n+1) - 3*n - 2 : n in [1..30]]; // Vincenzo Librandi, Sep 29 2011
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LinearRecurrence[{6,-13,12,-4},{3,16,53,146},40] (* or *)
Array[(#+1)2^(#+1)-3#-2&,40] (* Paolo Xausa, Oct 17 2023 *)
Showing 1-3 of 3 results.
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