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 A141419 Triangle read by rows: T(n, k) = A000217(n) - A000217(n - k) with 1 <= k <= n. 16
 1, 2, 3, 3, 5, 6, 4, 7, 9, 10, 5, 9, 12, 14, 15, 6, 11, 15, 18, 20, 21, 7, 13, 18, 22, 25, 27, 28, 8, 15, 21, 26, 30, 33, 35, 36, 9, 17, 24, 30, 35, 39, 42, 44, 45, 10, 19, 27, 34, 40, 45, 49, 52, 54, 55 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS As a rectangle, the accumulation array of A051340. From Clark Kimberling, Feb 05 2011: (Start) Here all the weights are divided by two where they aren't in Cahn. As a rectangle, A141419 is in the accumulation chain ... < A051340 < A141419 < A185874 < A185875 < A185876 < ... (See A144112 for the definition of accumulation array.) row 1: A000027 col 1: A000217 diag (1,5,...): A000326 (pentagonal numbers) diag (2,7,...): A005449 (second pentagonal numbers) diag (3,9,...): A045943 (triangular matchstick numbers) diag (4,11,...): A115067 diag (5,13,...): A140090 diag (6,15,...): A140091 diag (7,17,...): A059845 diag (8,19,...): A140672 (End) Let N=2*n+1 and k=1,2,...,n. Let A_{N,n-1} = [0,...,0,1; 0,...,0,1,1; ...; 0,1,...,1; 1,...,1], an n X n unit-primitive matrix (see [Jeffery]). Let M_n=[A_{N,n-1}]^4. Then t(n,k)=[M_n]_(1,k), that is, the n-th row of the triangle is given by the first row of M_n. - L. Edson Jeffery, Nov 20 2011 Conjecture. Let N=2*n+1 and k=1,...,n. Let A_{N,0}, A_{N,1}, ..., A_{N,n-1} be the n X n unit-primitive matrices (again see [Jeffery]) associated with N, and define the Chebyshev polynomials of the second kind by the recurrence U_0(x) = 1, U_1(x) = 2*x and U_r(x) = 2*x*U_(r-1)(x) - U_(r-2)(x)  (r>1). Define the column vectors V_(k-1) = (U_(k-1)(cos(Pi/N)), U_(k-1)(cos(3*Pi/N)), ..., U_(k-1)(cos((2*n-1)*Pi/N)))^T, where T denotes matrix transpose. Let S_N = [V_0, V_1, ..., V_(n-1)] be the n X n matrix formed by taking V_(k-1) as column k-1. Let X_N = [S_N]^T*S_N, and let [X_N]_(i,j) denote the entry in row i and column j of X_N, i,j in {0,...,n-1}. Then t(n,k) = [X_N]_(k-1,k-1), and row n of the triangle is given by the main diagonal entries of X_N. Remarks: Hence t(n,k) is the sum of squares t(n,k) = sum[m=1,...,n (U_(k-1)(cos((2*m-1)*Pi/N)))^2]. Finally, this sequence is related to A057059, since X_N = [sum_{m=1,...,n} A057059(n,m)*A_{N,m-1}] is also an integral linear combination of unit-primitive matrices from the N-th set. - L. Edson Jeffery, Jan 20 2012 Row sums: n*(n+1)*(2*n+1)/6. - L. Edson Jeffery, Jan 25 2013 n-th row = partial sums of n-th row of A004736. - Reinhard Zumkeller, Aug 04 2014 T(n,k) is the number of distinct sums made by at most k elements in {1, 2, ... n}, for 1 <= k <= n, e.g., T(6,2) = the number of distinct sums made by at most 2 elements in {1,2,3,4,5,6}. The sums range from 1, to 5+6=11. So there are 11 distinct sums. - Derek Orr, Nov 26 2014 A number n occurs in this sequence A001227(n) times, the number of odd divisors of n, see A209260. - Hartmut F. W. Hoft, Apr 14 2016 Conjecture: 2*n + 1 is composite if and only if gcd(t(n,m),m) != 1, for some m. - L. Edson Jeffery, Jan 30 2018 From Peter Munn, Aug 21 2019 in respect of the sequence read as a triangle: (Start) A number m can be found in column k if and only if A286013(m, k) is nonzero, in which case m occurs in column k on row A286013(m, k). The first occurrence of m is in row A212652(m) column A109814(m), which is the rightmost column in which m occurs. This occurrence determines where m appears in A209260. The last occurrence of m is in row m column 1. Viewed as a sequence of rows, consider the subsequences (of rows) that contain every positive integer. The lexicographically latest of these subsequences consists of the rows with row numbers in A270877; this is the only one that contains its own row numbers only once. (End) REFERENCES R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 139. LINKS Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p. 24. Carlton Gamer, David W. Roeder, and John J. Watkins, Trapezoidal Numbers, Mathematics Magazine 58:2 (1985), pp. 108-110. L. E. Jeffery, Unit-primitive matrices M. A. Nyblom, On the representation of the integers as a difference of nonconsecutive triangular numbers, Fibonacci Quarterly 39:3 (2001), pp. 256-263. FORMULA t(n,m) = m*(2*n - m + 1)/2. t(n,m) = A000217(n) - A000217(n-m). - L. Edson Jeffery, Jan 16 2013 Let v = d*h with h odd be an integer factorization, then v = t(d+(h-1)/2, h) if h+1 <= 2*d, and v = t(d+(h-1)/2, 2*d) if h+1 > 2*d; see A209260. - Hartmut F. W. Hoft, Apr 14 2016 G.f.: y*(-x + y)/((-1 + x)^2*(-1 + y)^3). - Stefano Spezia, Oct 14 2018 T(n, 2) = A060747(n) for n > 1. T(n, 3) = A008585(n - 1) for n > 2. T(n, 4) = A016825(n - 2) for n > 3. T(n, 5) = A008587(n - 2) for n > 4. T(n, 6) = A016945(n - 3) for n > 5. T(n, 7) = A008589(n - 3) for n > 6. T(n, 8) = A017113(n - 4) for n > 7.r n > 5. T(n, 7) = A008589(n - 3) for n > 6. T(n, 8) = A017113(n - 4) for n > 7. T(n, 9) = A008591(n - 4) for n > 8. T(n, 10) = A017329(n - 5) for n > 9. T(n, 11) = A008593(n - 5) for n > 10.  T(n, 12) = A017593(n - 6) for n > 11. T(n, 13) = A008595(n - 6) for n > 12. T(n, 14) = A147587(n - 7) for n > 13. T(n, 15) = A008597(n - 7) for n > 14. T(n, 16) = A051062(n - 8) for n > 15. T(n, 17) = A008599(n - 8) for n > 16. - Stefano Spezia, Oct 14 2018 T(2*n-k, k) = A070543(n, k). - Peter Munn, Aug 21 2019 EXAMPLE As a triangle:    1,    2,  3,    3,  5,  6,    4,  7,  9, 10,    5,  9, 12, 14, 15,    6, 11, 15, 18, 20, 21,    7, 13, 18, 22, 25, 27, 28,    8, 15, 21, 26, 30, 33, 35, 36,    9, 17, 24, 30, 35, 39, 42, 44, 45,   10, 19, 27, 34, 40, 45, 49, 52, 54, 55; As a rectangle:    1   2   3   4   5   6   7   8   9  10    3   5   7   9  11  13  15  17  19  21    6   9  12  15  18  21  24  27  30  33   10  14  18  22  26  30  34  38  42  46   15  20  25  30  35  40  45  50  55  60   21  27  33  39  45  51  57  63  69  75   28  35  42  49  56  63  70  77  84  91   36  44  52  60  68  76  84  92 100 108   45  54  63  72  81  90  99 108 117 126   55  65  75  85  95 105 115 125 135 145 Since the odd divisors of 15 are 1, 3, 5 and 15, number 15 appears four times in the triangle at t(3+(5-1)/2, 5) in column 5 since 5+1 <= 2*3, t(5+(3-1)/2, 3), t(1+(15-1)/2, 2*1) in column 2 since 15+1 > 2*1, and t(15+(1-1)/2, 1). - Hartmut F. W. Hoft, Apr 14 2016 MAPLE a:=(n, k)->k*n-binomial(k, 2): seq(seq(a(n, k), k=1..n), n=1..12); # Muniru A Asiru, Oct 14 2018 MATHEMATICA T[n_, m_] = m*(2*n - m + 1)/2; a = Table[Table[T[n, m], {m, 1, n}], {n, 1, 10}]; Flatten[a] PROG (Haskell) a141419 n k =  k * (2 * n - k + 1) `div` 2 a141419_row n = a141419_tabl !! (n-1) a141419_tabl = map (scanl1 (+)) a004736_tabl -- Reinhard Zumkeller, Aug 04 2014 CROSSREFS Cf. A000330 (row sums), A004736, A057059, A070543. A144112, A051340, A141419, A185874, A185875, A185876 are accumulation chain related. A141418 is a variant. Cf. A001227, A209260. - Hartmut F. W. Hoft, Apr 14 2016 A109814, A212652, A270877, A286013 relate to where each natural number appears in this sequence. A000027, A000217, A000326, A005449, A045943, A059845, A115067, A140090, A140091, A140672 are rows, columns or diagonals - refer to comments. Sequence in context: A216066 A234094 A301853 * A072451 A349669 A023156 Adjacent sequences:  A141416 A141417 A141418 * A141420 A141421 A141422 KEYWORD nonn,tabl,easy AUTHOR Roger L. Bagula, Aug 05 2008 EXTENSIONS Simpler name by Stefano Spezia, Oct 14 2018 STATUS approved

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Last modified October 3 22:17 EDT 2022. Contains 357237 sequences. (Running on oeis4.)