A296558
Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - n, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.
Original entry on oeis.org
1, 3, 7, 13, 24, 41, 69, 114, 187, 306, 498, 809, 1312, 2126, 3443, 5574, 9022, 14601, 23628, 38235, 61869, 100110, 161985, 262101, 424092, 686199, 1110297, 1796502, 2906805, 4703313, 7610124, 12313443, 19923573, 32237022, 52160601, 84397630, 136558238
Offset: 0
a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5
a(2) = a(0) + a(1) + b(2) - 2 = 7
Complement: (b(n)) = (2, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, ...)
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a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] - n;
j = 1; While[j < 16, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
u = Table[a[n], {n, 0, k}]; (* A296558 *)
Table[b[n], {n, 0, 20}] (* complement *)
A296846
Solution of the complementary equation a(n) = a(n-1) + a(n-2) - b(n-2), where a(0) = 3, a(1) = 5, b(0) = 1, b(1) = 2, b(2) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.
Original entry on oeis.org
3, 5, 7, 10, 13, 17, 22, 30, 41, 59, 86, 130, 200, 312, 493, 785, 1257, 2019, 3252, 5246, 8472, 13691, 22135, 35797, 57901, 93666, 151534, 245166, 396665, 641795, 1038423, 1680180, 2718564, 4398704, 7117226, 11515887, 18633069, 30148911, 48781934, 78930798
Offset: 0
a(0) = 3, a(1) = 5, b(0) = 1, b(1) = 2, b(2) = 4
a(2) = a(0) + a(1) - b(0) = 7
Complement: (b(n)) = (1, 2, 4, 6, 8, 9, 11, 12, 14, 15, 16, 18, 19, 20, 21, 23, ...)
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a[0] = 3; a[1] = 5; b[0] = 1; b[1] = 2; b[2] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] - b[n - 2];
j = 1; While[j < 16, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
u = Table[a[n], {n, 0, k}]; (* A296846 *)
Table[b[n], {n, 0, 20}] (* complement *)
A305746
Solution of the complementary equation a(n) = 2*a(n-1) - a(n-3) + b(n), where a(0) = 1, a(1) = 2, a(2) = 3, b(0)= 4, b(1) = 5, b(2) = 6; b(3) = 7. See Comments.
Original entry on oeis.org
1, 2, 3, 12, 30, 66, 130, 241, 429, 742, 1258, 2103, 3481, 5722, 9360, 15259, 24817, 40296, 65356, 105919, 171567, 277804, 449716, 727893, 1178011, 1906337, 3084813, 4991648, 8076993, 13069208, 21146804, 34216652, 55364134, 89581503, 144946394, 234528695
Offset: 0
a(0) = 1, a(1) = 2, a(2) = 3, b(0)= 4, b(1) = 5, b(2) = 6; b(3) = 7, and a(3) = 2*3 - 1 + 7 = 12.
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a[0] = 1; a[1] = 2; a[2] = 3; b[0] = 4; b[1] = 5; b[2] = 6; b[3] = 7;
a[n_] := a[n] = 2*a[n - 1] - a[n - 3] + b[n];
j = 1; While[j < 12, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
Table[a[n], {n, 0, 60}] (* A305746 *)
Comments