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 A303273 Array T(n,k) = binomial(n, 2) + k*n + 1 read by antidiagonals. 1
 1, 1, 1, 1, 2, 2, 1, 3, 4, 4, 1, 4, 6, 7, 7, 1, 5, 8, 10, 11, 11, 1, 6, 10, 13, 15, 16, 16, 1, 7, 12, 16, 19, 21, 22, 22, 1, 8, 14, 19, 23, 26, 28, 29, 29, 1, 9, 16, 22, 27, 31, 34, 36, 37, 37, 1, 10, 18, 25, 31, 36, 40, 43, 45, 46, 46, 1, 11, 20, 28, 35, 41 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Columns are linear recurrence sequences with signature (3,-3,1). 8*T(n,k) + A166147(k-1) are squares. Columns k are binomial transforms of [1, k, 1, 0, 0, 0, ...]. Antidiagonals sums yield A116731. REFERENCES R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 1994. LINKS FORMULA G.f.: (3*x^2*y - 3*x*y + y - 2*x^2 + 2*x - 1)/((x - 1)^3*(y - 1)^2). E.g.f.: (1/2)*(2*x*y + x^2 + 2)*exp(y + x). T(n,k) = 3*T(n-1,k) - 3*T(n-2,k) + T(n-3,k), with T(0,k) = 1, T(1,k) = k + 1 and T(2,k) = 2*k + 2. T(n,k) = T(n-1,k) + n + k - 1. T(n,k) = T(n,k-1) + n, with T(n,0) = 1. T(n,0) = A152947(n+1). T(n,1) = A000124(n). T(n,2) = A000217(n). T(n,3) = A034856(n+1). T(n,4) = A052905(n). T(n,5) = A051936(n+4). T(n,6) = A246172(n+1). T(n,7) = A302537(n). T(n,8) = A056121(n+1) + 1. T(n,9) = A056126(n+1) + 1. T(n,10) = A051942(n+10) + 1, n > 0. T(n,11) = A101859(n) + 1. T(n,12) = A132754(n+1) + 1. T(n,13) = A132755(n+1) + 1. T(n,14) = A132756(n+1) + 1. T(n,15) = A132757(n+1) + 1. T(n,16) = A132758(n+1) + 1. T(n,17) = A212427(n+1) + 1. T(n,18) = A212428(n+1) + 1. T(n,n) = A143689(n) = A300192(n,2). T(n,n+1) = A104249(n). T(n,n+2) = T(n+1,n) = A005448(n+1). T(n,n+3) = A000326(n+1). T(n,n+4) = A095794(n+1). T(n,n+5) = A133694(n+1). T(n+2,n) = A005449(n+1). T(n+3,n) = A115067(n+2). T(n+4,n) = A133694(n+2). T(2*n,n) = A054556(n+1). T(2*n,n+1) = A054567(n+1). T(2*n,n+2) = A033951(n). T(2*n,n+3) = A001107(n+1). T(2*n,n+4) = A186353(4*n+1) (conjectured). T(2*n,n+5) = A184103(8*n+1) (conjectured). T(2*n,n+6) = A250657(n-1) = A250656(3,n-1), n > 1. T(n,2*n) = A140066(n+1). T(n+1,2*n) = A005891(n). T(n+2,2*n) = A249013(5*n+4) (conjectured). T(n+3,2*n) = A186384(5*n+3) = A186386(5*n+3) (conjectured). T(2*n,2*n) = A143689(2*n). T(2*n+1,2*n+1) = A143689(2*n+1) (= A030503(3*n+3) (conjectured)). T(2*n,2*n+1) = A104249(2*n) = A093918(2*n+2) = A131355(4*n+1) (= A030503(3*n+5) (conjectured)). T(2*n+1,2*n) = A085473(n). a(n+1,5*n+1)=A051865(n+1) + 1. a(n,2*n+1) = A116668(n). a(2*n+1,n) = A054569(n+1). T(3*n,n) = A025742(3*n-1), n > 1 (conjectured). T(n,3*n) = A140063(n+1). T(n+1,3*n) = A069099(n+1). T(n,4*n) = A276819(n). T(4*n,n) = A154106(n-1), n > 0. T(2^n,2) = A028401(n+2). T(1,n)*T(n,1) = A006000(n). T(n*(n+1),n) = A211905(n+1), n > 0 (conjectured). T(n*(n+1)+1,n) = A294259(n+1). T(n,n^2+1) = A081423(n). T(n,A000217(n)) = A158842(n), n > 0. T(n,A152947(n+1)) = A060354(n+1). floor(T(n,n/2)) = A267682(n) (conjectured). floor(T(n,n/3)) = A025742(n-1), n > 0 (conjectured). floor(T(n,n/4)) = A263807(n-1), n > 0 (conjectured). ceiling(T(n,2^n)/n) = A134522(n), n > 0 (conjectured). ceiling(T(n,n/2+n)/n) = A051755(n+1) (conjectured). floor(T(n,n)/n) = A133223(n), n > 0 (conjectured). ceiling(T(n,n)/n) = A007494(n), n > 0. ceiling(T(n,n^2)/n) = A171769(n), n > 0. ceiling(T(2*n,n^2)/n) = A046092(n), n > 0. ceiling(T(2*n,2^n)/n) = A131520(n+2), n > 0. EXAMPLE The array T(n,k) begins 1    1    1    1    1    1    1    1    1    1    1    1    1  ...  A000012 1    2    3    4    5    6    7    8    9   10   11   12   13  ...  A000027 2    4    6    8   10   12   14   16   18   20   22   24   26  ...  A005843 4    7   10   13   16   19   22   25   28   31   34   37   40  ...  A016777 7   11   15   19   23   27   31   35   39   43   47   51   55  ...  A004767 11  16   21   26   31   36   41   46   51   56   61   66   71  ...  A016861 16  22   28   34   40   46   52   58   64   70   76   82   88  ...  A016957 22  29   36   43   50   57   64   71   78   85   92   99  106  ...  A016993 29  37   45   53   61   69   77   85   93  101  109  117  125  ...  A004770 37  46   55   64   73   82   91  100  109  118  127  136  145  ...  A017173 46  56   66   76   86   96  106  116  126  136  146  156  166  ...  A017341 56  67   78   89  100  111  122  133  144  155  166  177  188  ...  A017401 67  79   91  103  115  127  139  151  163  175  187  199  211  ...  A017605 79  92  105  118  131  144  157  170  183  196  209  222  235  ...  A190991 ... The inverse binomial transforms of the columns are 1    1    1    1    1    1    1    1    1    1    1    1    1  ... 0    1    2    3    4    5    6    7    8    9   10   11   12  ... 1    1    1    1    1    1    1    1    1    1    1    1    1  ... 0    0    0    0    0    0    0    0    0    0    0    0    0  ... 0    0    0    0    0    0    0    0    0    0    0    0    0  ... 0    0    0    0    0    0    0    0    0    0    0    0    0  ... ... T(k,n-k) = A087401(n,k) + 1 as triangle 1 1   1 1   2   2 1   3   4   4 1   4   6   7   7 1   5   8  10  11  11 1   6  10  13  15  16  16 1   7  12  16  19  21  22  22 1   8  14  19  23  26  28  29  29 1   9  16  22  27  31  34  36  37  37 1  10  18  25  31  36  40  43  45  46  46 ... MAPLE T := (n, k) -> binomial(n, 2) + k*n + 1; for n from 0 to 20 do seq(T(n, k), k = 0 .. 20) od; MATHEMATICA Table[With[{n = m - k}, Binomial[n, 2] + k n + 1], {m, 0, 11}, {k, m, 0, -1}] // Flatten (* Michael De Vlieger, Apr 21 2018 *) PROG (Maxima) T(n, k) := binomial(n, 2)+ k*n + 1\$ for n:0 thru 20 do     print(makelist(T(n, k), k, 0, 20)); (PARI) T(n, k) = binomial(n, 2) + k*n + 1; tabl(nn) = for (n=0, nn, for (k=0, nn, print1(T(n, k), ", ")); print); \\ Michel Marcus, May 17 2018 CROSSREFS Cf. A085475, A086270, A086271, A086272, A086273, A130154, A159798, A162609, A162610, A300401. Sequence in context: A118032 A089692 A066201 * A193820 A216368 A123956 Adjacent sequences:  A303270 A303271 A303272 * A303274 A303275 A303276 KEYWORD nonn,tabl AUTHOR Franck Maminirina Ramaharo, Apr 20 2018 STATUS approved

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Last modified December 7 05:14 EST 2019. Contains 329839 sequences. (Running on oeis4.)