Search: a093178 -id:a093178
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1, 2, 3, 6, 7, 12, 13, 20, 21, 30, 31, 42, 43, 56, 57, 72, 73, 90, 91, 110, 111, 132, 133, 156, 157, 182, 183, 210, 211, 240, 241, 272, 273, 306, 307, 342, 343, 380, 381, 420, 421, 462, 463, 506, 507, 552, 553, 600, 601, 650, 651, 702, 703, 756, 757, 812, 813
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1, 1, 1, 1, 2, 3, 1, 3, 9, 1, 1, 4, 18, 4, 5, 1, 5, 30, 10, 25, 1, 1, 6, 45, 20, 75, 6, 7, 1, 7, 63, 35, 175, 21, 49, 1, 1, 8, 84, 56, 350, 56, 196, 8, 9, 1, 9, 108, 84, 630, 126, 588, 36, 81, 1
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FORMULA
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Binomial transform of an infinite lower triangular matrix with A093178: (1, 1, 3, 1, 5, 1, 7, ...) in the main diagonal and the rest zeros.
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CROSSREFS
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A133080
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Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.
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+10
29
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1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1
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COMMENTS
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A133080^(-1) * [1,2,3,...] = A093178: (1, 1, 3, 1, 5, 1, 7, 1, 9, ...).
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CROSSREFS
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A049471
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Decimal expansion of tan(1).
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+10
14
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1, 5, 5, 7, 4, 0, 7, 7, 2, 4, 6, 5, 4, 9, 0, 2, 2, 3, 0, 5, 0, 6, 9, 7, 4, 8, 0, 7, 4, 5, 8, 3, 6, 0, 1, 7, 3, 0, 8, 7, 2, 5, 0, 7, 7, 2, 3, 8, 1, 5, 2, 0, 0, 3, 8, 3, 8, 3, 9, 4, 6, 6, 0, 5, 6, 9, 8, 8, 6, 1, 3, 9, 7, 1, 5, 1, 7, 2, 7, 2, 8, 9, 5, 5, 5, 0, 9, 9, 9, 6, 5, 2, 0, 2, 2, 4, 2, 9, 8
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A243366
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Number T(n,k) of Dyck paths of semilength n having exactly k (possibly overlapping) occurrences of the consecutive steps UDUUDU (with U=(1,1), D=(1,-1)); triangle T(n,k), n>=0, 0<=k<=max(0,floor(n/2)-1), read by rows.
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+10
13
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1, 1, 2, 5, 13, 1, 37, 5, 112, 19, 1, 352, 70, 7, 1136, 259, 34, 1, 3742, 962, 149, 9, 12529, 3585, 627, 54, 1, 42513, 13399, 2584, 279, 11, 145868, 50201, 10529, 1334, 79, 1, 505234, 188481, 42606, 6092, 474, 13, 1764157, 709001, 171563, 27048, 2561, 109, 1
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CROSSREFS
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T(n,floor(n/2)-1) gives A093178(n) for n>3.
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A124625
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Even numbers sandwiched between 1's.
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+10
11
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1, 0, 1, 2, 1, 4, 1, 6, 1, 8, 1, 10, 1, 12, 1, 14, 1, 16, 1, 18, 1, 20, 1, 22, 1, 24, 1, 26, 1, 28, 1, 30, 1, 32, 1, 34, 1, 36, 1, 38, 1, 40, 1, 42, 1, 44, 1, 46, 1, 48, 1, 50, 1, 52, 1, 54, 1, 56, 1, 58, 1, 60, 1, 62, 1, 64, 1, 66, 1, 68, 1, 70, 1, 72, 1, 74, 1, 76, 1, 78, 1, 80, 1, 82, 1, 84
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A241269
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Denominator of c(n) = (n^2+n+2)/((n+1)*(n+2)*(n+3)).
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+10
8
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3, 6, 15, 60, 105, 21, 126, 360, 495, 330, 429, 1092, 1365, 420, 1020, 2448, 2907, 1710, 1995, 4620, 5313, 759, 3450, 7800, 8775, 4914, 5481, 12180, 13485, 3720, 8184, 17952, 19635, 10710, 11655, 25308, 27417, 3705, 15990, 34440, 37023, 19866, 21285, 45540
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COMMENTS
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The first column is 1, 0, 1/3, 0, 1/5, 0, 1/7, 0, ... . Numerators: A093178(n+1). This incites, considering tan(1), to introduce before the first row
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A009001
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Expansion of e.g.f: (1+x)*cos(x).
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+10
6
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1, 1, -1, -3, 1, 5, -1, -7, 1, 9, -1, -11, 1, 13, -1, -15, 1, 17, -1, -19, 1, 21, -1, -23, 1, 25, -1, -27, 1, 29, -1, -31, 1, 33, -1, -35, 1, 37, -1, -39, 1, 41, -1, -43, 1, 45, -1, -47, 1, 49, -1, -51, 1, 53, -1, -55, 1, 57, -1, -59, 1, 61, -1, -63, 1, 65, -1, -67, 1, 69, -1, -71, 1, 73, -1, -75, 1, 77, -1, -79, 1, 81, -1, -83, 1, 85
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COMMENTS
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If signs are ignored, continued fraction for tan(1) (cf. A093178).
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A224344
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Number T(n,k) of compositions of n using exactly k primes (counted with multiplicity); triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.
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+10
6
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1, 1, 1, 1, 1, 3, 2, 5, 1, 3, 8, 5, 5, 13, 13, 1, 7, 23, 27, 7, 11, 39, 52, 25, 1, 17, 65, 99, 66, 9, 27, 106, 186, 151, 41, 1, 40, 177, 340, 323, 133, 11, 61, 293, 608, 666, 358, 61, 1, 92, 482, 1076, 1330, 867, 236, 13, 142, 781, 1894, 2581, 1971, 737, 85, 1
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A019426
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Continued fraction for tan(1/3).
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+10
4
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0, 2, 1, 7, 1, 13, 1, 19, 1, 25, 1, 31, 1, 37, 1, 43, 1, 49, 1, 55, 1, 61, 1, 67, 1, 73, 1, 79, 1, 85, 1, 91, 1, 97, 1, 103, 1, 109, 1, 115, 1, 121, 1, 127, 1, 133, 1, 139, 1, 145, 1, 151, 1, 157, 1, 163, 1, 169, 1, 175, 1, 181, 1, 187, 1, 193, 1, 199, 1, 205, 1, 211, 1, 217, 1, 223, 1
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