A171231
a(n) = (10*2^n + 3 - (-1)^n)/6.
Original entry on oeis.org
2, 4, 7, 14, 27, 54, 107, 214, 427, 854, 1707, 3414, 6827, 13654, 27307, 54614, 109227, 218454, 436907, 873814, 1747627, 3495254, 6990507, 13981014, 27962027, 55924054, 111848107, 223696214, 447392427, 894784854, 1789569707
Offset: 0
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[( 10*2^n+3-(-1)^n )/6: n in [0..40]]; // Vincenzo Librandi, Aug 05 2011
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LinearRecurrence[{2,1,-2},{2,4,7},40] (* Harvey P. Dale, Feb 11 2015 *)
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a(n)=(10<Charles R Greathouse IV, Jul 07 2011
Definition replaced by the Lava formula of 2009. Contents converted to formulas. -
R. J. Mathar, Jul 07 2011
A321373
Array T(n,k) read by antidiagonals where the first row is (-1)^k*A140966(k) and each subsequent row is obtained by adding A001045(k) to the preceding one.
Original entry on oeis.org
2, 2, -1, 2, 0, 3, 2, 1, 4, 1, 2, 2, 5, 4, 7, 2, 3, 6, 7, 12, 9, 2, 4, 7, 10, 17, 20, 23, 2, 5, 8, 13, 22, 31, 44, 41, 2, 6, 9, 16, 27, 42, 65, 84, 87, 2, 7, 10, 19, 32, 53, 86, 127, 172, 169, 2, 8, 11, 22, 37, 64, 107, 170, 257, 340, 343
Offset: 0
Triangle a(n):
2;
2, -1;
2, 0, 3;
2, 1, 4, 1;
2, 2, 5, 4, 7;
2, 3, 6, 7, 12, 9;
2, 4, 7, 10, 17, 20, 23;
etc.
Row sums: 2, 1, 5, 8, 20, 39, 83, 166, 338, 677, 1361, 2724, ... = b(n+2).
With b(0) = 2 and b(1) = 0, b(n) = b(n-1) + 2*b(n-2) + n - 4, n > 1.
b(n) = A001045(n) - A097065(n-1).
b(n) = b(n-2) + A000225(n-2).
Cf.
A000079,
A001045,
A014113,
A014551,
A048573,
A062092,
A078008,
A084247,
A092297,
A097073,
A140360,
A140966.
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T[_, 0] = 2;
T[0, k_] := (2^k + 5(-1)^k)/3;
T[n_ /; n>0, k_ /; k>0] := T[n, k] = T[n-1, k] + (2^k + (-1)^(k+1))/3;
T[, ] = 0;
Table[T[n-k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 10 2018 *)
A366987
Triangle read by rows: T(n, k) = -(2^(n - k)*(-1)^n + 2^k + (-1)^k)/3.
Original entry on oeis.org
-1, 0, 0, -2, -1, -2, 2, 1, -1, -2, -6, -3, -3, -3, -6, 10, 5, 1, -1, -5, -10, -22, -11, -7, -5, -7, -11, -22, 42, 21, 9, 3, -3, -9, -21, -42, -86, -43, -23, -13, -11, -13, -23, -43, -86, 170, 85, 41, 19, 5, -5, -19, -41, -85, -170, -342, -171, -87, -45, -27, -21, -27, -45, -87, -171, -342
Offset: 0
Triangle T(n, k) starts:
-1
0 0
-2 -1 -2
2 1 -1 -2
-6 -3 -3 -3 -6
10 5 1 -1 -5 -10
-22 -11 -7 -5 -7 -11 -22
42 21 9 3 -3 -9 -21 -42
...
Note the symmetrical distribution of the absolute values of the terms in each row.
First column: -(-1)^n *
A078008(n).
Second column: (-1)^n *
A001045(n).
Fourth column: (-1)^n *
A155980(n+2).
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T := (n, k) -> -(2^(n-k)*(-1)^n + 2^k + (-1)^k)/3:
seq(seq(T(n, k), k = 0..n), n = 0..10); # Peter Luschny, Nov 02 2023
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A366987row[n_]:=Table[-(2^(n-k)(-1)^n+2^k+(-1)^k)/3,{k,0,n}];Array[A366987row,15,0] (* Paolo Xausa, Nov 07 2023 *)
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T(n, k) = (-2^(k+1) + 2*(-1)^(k+1) + (-1)^(n+1)*2^(1+n-k))/6 \\ Thomas Scheuerle, Nov 01 2023
A355668
Array read by upwards antidiagonals T(n,k) = J(k) + n*J(k+1) where J(n) = A001045(n) is the Jacobsthal numbers.
Original entry on oeis.org
0, 1, 1, 2, 2, 1, 3, 3, 4, 3, 4, 4, 7, 8, 5, 5, 5, 10, 13, 16, 11, 6, 6, 13, 18, 27, 32, 21, 7, 7, 16, 23, 38, 53, 64, 43, 8, 8, 19, 28, 49, 74, 107, 128, 85, 9, 9, 22, 33, 60, 95, 150, 213, 256, 171, 10, 10, 25, 38, 71, 116, 193, 298, 427, 512, 341
Offset: 0
Row n=0 is A001045(k), then for further rows we successively add A001045(k+1).
k=0 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k=10
n=0: 0 1 1 3 5 11 21 43 85 171 ... = A001045
n=1: 1 2 4 8 16 32 64 128 256 512 ... = A000079
n=2: 2 3 7 13 27 53 107 213 427 853 ... = A048573
n=3: 3 4 10 18 38 74 150 298 598 1194 ... = A171160
n=4: 4 5 13 23 49 95 193 383 769 1535 ... = abs(A140683)
...
Antidiagonal sums give
A320933(n+1).
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T[n_, k_] := (2^k - (-1)^k + n*(2^(k + 1) + (-1)^k))/3; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Jul 13 2022 *)
A360033
Table T(n,k), n >= 1 and k >= 0, read by antidiagonals, related to Jacobsthal numbers A001045.
Original entry on oeis.org
1, 2, 1, 3, 3, 3, 4, 5, 7, 5, 5, 7, 11, 13, 11, 6, 9, 15, 21, 27, 21, 7, 11, 19, 29, 43, 53, 43, 8, 13, 23, 37, 59, 85, 107, 85, 9, 15, 27, 45, 75, 117, 171, 213, 171, 10, 17, 31, 53, 91, 149, 235, 341, 427, 341, 11, 19, 35, 61, 107, 181, 299, 469
Offset: 1
The array T(n,k), for n <= 1 and k >= 0, begins:
n = 1: 1, 1, 3, 5, 11, 21, 43, ... -> A001045(k+1)
n = 2: 2, 3, 7, 13, 27, 53, 107, ... -> A048573(k)
n = 3: 3, 5, 11, 21, 43, 85, 171, ... -> A001045(k+3)
n = 4: 4, 7, 15, 29, 59, 117, 235, ... -> ?
n = 5: 5, 9, 19, 37, 75, 149, 299, ... -> A062092(k+1)
n = 6: 6, 11, 23, 45, 91, 181, 363, ... -> ?
n = 7: 7, 13, 27, 53, 107, 213, 427, ... -> A048573(k+2)
Comments