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 A343138 Array A(k, n) read by descending antidiagonals: A(k, n) = Sum_{m=0..n} F(k, m)^2, where F are the k-generalized Fibonacci numbers A092921. 2
 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 2, 1, 0, 1, 4, 6, 2, 1, 0, 1, 5, 15, 6, 2, 1, 0, 1, 6, 40, 22, 6, 2, 1, 0, 1, 7, 104, 71, 22, 6, 2, 1, 0, 1, 8, 273, 240, 86, 22, 6, 2, 1, 0, 1, 9, 714, 816, 311, 86, 22, 6, 2, 1, 0, 1, 10, 1870, 2752, 1152, 342, 86, 22, 6, 2, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 REFERENCES Greg Dresden and Yichen Wang, Sums and convolutions of k-bonacci and k-Lucas numbers, draft 2021. LINKS Russell Jay Hendel, Sums of Squares: Methods for Proving Identity Families, arXiv:2103.16756 [math.NT], 2021 FORMULA Russell Jay Hendel gives the following representation, valid for k >= 2: A(n, k) = Sum_{m=0..n} F(k, m)^2 = (1/(2*k-2)) * (2*Sum_{j=1..k-1}(j*Sum_{m=j+1..k} (m-k+1) * F(k, n+j) * F(k, n+m)) - Sum_{j=1..k}(A343125(k, j-1) * F(k, n+j)^2) + (k - 2)). - Peter Luschny, Apr 07 2021 EXAMPLE Array starts:   n = 0  1  2  3   4   5    6     7     8     9      10 ------------------------------------------------------------ [k=0] 0, 1, 1, 1,  1,  1,   1,    1,    1,     1,     1, ...  [A057427] [k=1] 0, 1, 2, 3,  4,  5,   6,    7,    8,     9,    10, ...  [A001477] [k=2] 0, 1, 2, 6, 15, 40, 104,  273,  714,  1870,  4895, ...  [A001654] [k=3] 0, 1, 2, 6, 22, 71, 240,  816, 2752,  9313, 31514, ...  [A107239] [k=4] 0, 1, 2, 6, 22, 86, 311, 1152, 4288, 15952, 59216, ... [k=5] 0, 1, 2, 6, 22, 86, 342, 1303, 5024, 19424, 75120, ... [k=6] 0, 1, 2, 6, 22, 86, 342, 1366, 5335, 20960, 82464, ... [k=7] 0, 1, 2, 6, 22, 86, 342, 1366, 5462, 21591, 85600, ... [k=8] 0, 1, 2, 6, 22, 86, 342, 1366, 5462, 21846, 86871, ... [k=9] 0, 1, 2, 6, 22, 86, 342, 1366, 5462, 21846, 87382, ... [...] [ oo] 0, 1, 2, 6, 22, 86, 342, 1366, 5462, 21846, 87382, ...  [A047849] Note that the first parameter in A(k, n) refers to rows, and the second parameter refers to columns, as always. The usual naming convention for the indices is not adhered to because the row sequences are the sums of the squares of the k-bonacci numbers. MAPLE F := (k, n) -> (F(k, n) := `if`(n<2, n, add(F(k, n-j), j = 1..min(k, n)))): A := (k, n) -> add(F(k, m)^2, m = 0..n): seq(seq(A(k, n-k), k=0..n), n = 0..11); # The following two functions implement Russell Jay Hendel's formula for k >= 2: T := (k, n) -> (n + 3)*(k - n) - 4: H := (k, n) -> (2*add(j*add((m-k+1)*F(k, n+j)*F(k, n+m), m = j+1..k), j = 1..k-1) - add(T(k, j-1)*F(k, n+j)^2, j = 1..k) + (k - 2))/(2*k - 2): seq(lprint([k], seq(H(k, n), n = 0..11)), k=2..9); # Peter Luschny, Apr 07 2021 MATHEMATICA A343138[k_, len_] := Take[Accumulate[LinearRecurrence[PadLeft[{1}, k, 1], PadLeft[{1}, k], len + k]^2], -len - 2]; A343138[0, len_] := Table[Boole[n != 0], {n, 0, len}]; A343138[1, len_] := Table[n, {n, 0, len}]; (* Table: *) Table[A343138[k, 12], {k, 0, 9}] (* Sequence / descending antidiagonals: *) Table[Table[Take[A343138[j, 12], {k + 1 - j, k + 1 - j}], {j, 0, k}], {k, 0, 10}] // Flatten (* Georg Fischer, Apr 08 2021 *) CROSSREFS Cf. A092921, A057427, A001477, A001654, A107239, A047849, A343125. Sequence in context: A244003 A332670 A118344 * A119270 A267109 A341524 Adjacent sequences:  A343135 A343136 A343137 * A343139 A343141 A343142 KEYWORD nonn,tabl,easy AUTHOR Peter Luschny, Apr 06 2021 STATUS approved

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Last modified June 19 03:38 EDT 2021. Contains 345125 sequences. (Running on oeis4.)