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A351381
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Table read by downward antidiagonals: T(n,k) = n*(k+1)^2.
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2
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4, 9, 8, 16, 18, 12, 25, 32, 27, 16, 36, 50, 48, 36, 20, 49, 72, 75, 64, 45, 24, 64, 98, 108, 100, 80, 54, 28, 81, 128, 147, 144, 125, 96, 63, 32, 100, 162, 192, 196, 180, 150, 112, 72, 36, 121, 200, 243, 256, 245, 216, 175, 128, 81, 40, 144, 242, 300, 324, 320, 294, 252, 200, 144, 90, 44
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OFFSET
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1,1
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COMMENTS
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When m and k are both positive integers and k | m, with m/k = n, then T(n,k) = S(m,k) = (m+k) + (m-k) + (m*k) + (m/k) = S(n*k,k) = n*(k+1)^2, problem proposed by Yakov Perelman.
All terms are nonsquarefree (A013929).
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REFERENCES
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I. Perelman, L'Algèbre Récréative, Chapitre IV, Les équations de Diophante, Deux nombres et quatre opérations, Editions en langues étrangères, Moscou, 1959, pp. 101-102.
Ya. I. Perelman, Algebra Can Be Fun, Chapter IV, Diophantine Equations, Two numbers and four operations, Mir Publishers Moscow, 1979, pp. 131-132.
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LINKS
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Ya. I. Perelman, Algebra Can Be Fun, Chapter IV, Diophantine Equations, Two numbers and four operations, Mir Publishers Moscow, 1979, pp. 131-132.
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FORMULA
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T(n,k) = n*(k+1)^2.
T(n,n) = (n+1)^3 - (n+1)^2 = A045991(n+1) for n >= 1.
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EXAMPLE
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Table begins:
n \ k | 1 2 3 4 5 6 7 8 9 10
----------------------------------------------------------------------------
1 | 4 9 16 25 36 49 64 81 100 121
2 | 8 18 32 50 72 98 128 162 200 242
3 | 12 27 48 75 108 147 192 243 300 363
4 | 16 36 64 100 144 196 256 324 400 484
5 | 20 45 80 125 180 245 320 405 500 605
6 | 24 54 96 150 216 294 384 486 600 726
7 | 28 63 112 175 252 343 448 567 700 847
8 | 32 72 128 200 288 392 512 648 800 968
9 | 36 81 144 225 324 441 576 729 900 1089
10 | 40 90 160 250 360 490 640 810 1000 1210
............................................................................
T(3,4) = 75 = 3*(4+1)^2 corresponds to S(3*4,4) = S(12,4) = (12+4) + (12-4) + (12*4) + 12/4 = 75.
S(10,5) = (10+5) + (10-5) + (10*5) + (10/5) = T(10/5,5) = T(2,5) = 72.
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MATHEMATICA
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T[n_, k_] := n*(k + 1)^2; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Mar 29 2022 *)
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CROSSREFS
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Cf. A000290 \ {0,1} (row 1), A001105 \ {0,2} (row 2), A033428 \ {0,3} (row 3), A016742 \ {0,4} (row 4), A033429 \ {0,5} (row 5), A033581 \ {0,6} (row 6).
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KEYWORD
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AUTHOR
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STATUS
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approved
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