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A188646 Array of a(n)=a(n-1)*k-((k-1)/(k^n)) where a(0)=1 and k=(sqrt(x^2-1)+x)^2 for integers x>=1. 10
1, 1, 1, 1, 13, 1, 1, 181, 33, 1, 1, 2521, 1121, 61, 1, 1, 35113, 38081, 3781, 97, 1, 1, 489061, 1293633, 234361, 9505, 141, 1, 1, 6811741, 43945441, 14526601, 931393, 20021, 193, 1, 1, 94875313, 1492851361, 900414901, 91267009, 2842841, 37441, 253, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Conjecture: Given function f(x, y)=(sqrt(x^2+y)+x)^2; constant k=f(x, y); and initial term a(0)=1; then for all integers x>=1 and y=[+-]1, k may be irrational, but sequence a(n)=a(n-1)*k-((k-1)/(k^n)) always produces integer sequences; y=-1 results shown here; y=1 results are A188647.

Also square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is (1/n) * T_{2*k+1}(n), with the Chebyshev polynomials of the first kind (type T). - Seiichi Manyama, Jan 01 2019

LINKS

Table of n, a(n) for n=0..44.

Wikipedia, Chebyshev polynomials.

Index entries for sequences related to Chebyshev polynomials.

FORMULA

A(n,k) = 2 * A188644(n,k) - A(n,k-1).

A(n,k) = Sum_{j=0..k} binomial(2*k+1,2*j+1)*(n^2-1)^(k-j)*n^(2*j). - Seiichi Manyama, Jan 01 2019

EXAMPLE

Square array begins:

     | 0    1       2          3             4

-----+---------------------------------------------

   1 | 1,   1,      1,         1,            1, ...

   2 | 1,  13,    181,      2521,        35113, ...

   3 | 1,  33,   1121,     38081,      1293633, ...

   4 | 1,  61,   3781,    234361,     14526601, ...

   5 | 1,  97,   9505,    931393,     91267009, ...

   6 | 1, 141,  20021,   2842841,    403663401, ...

   7 | 1, 193,  37441,   7263361,   1409054593, ...

   8 | 1, 253,  64261,  16322041,   4145734153, ...

   9 | 1, 321, 103361,  33281921,  10716675201, ...

  10 | 1, 397, 158005,  62885593,  25028308009, ...

  11 | 1, 481, 231841, 111746881,  53861764801, ...

  12 | 1, 573, 328901, 188788601, 108364328073, ...

  13 | 1, 673, 453601, 305726401, 206059140673, ...

  14 | 1, 781, 610741, 477598681, 373481557801, ...

  15 | 1, 897, 805505, 723342593, 649560843009, ...

  ...

MATHEMATICA

A[n_, k_] := 1/n ChebyshevT[2k+1, n];

Table[A[n-k, k], {n, 1, 9}, {k, n-1, 0, -1}] // Flatten (* Jean-Fran├žois Alcover, Jan 02 2019, after Seiichi Manyama *)

CROSSREFS

Row 1-8 give A000012, A001570, A077420, A302329, A302330, A302331, A302332, A253880.

Column 1 is A082109(n-1).

Cf. A188644, A188647 (f(x, y) as above with y=1).

Diagonal gives A322904.

Sequence in context: A156539 A172300 A022176 * A174791 A015132 A066036

Adjacent sequences:  A188643 A188644 A188645 * A188647 A188648 A188649

KEYWORD

nonn,tabl

AUTHOR

Charles L. Hohn, Apr 06 2011

EXTENSIONS

Edited and extended by Seiichi Manyama, Jan 01 2019

STATUS

approved

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Last modified June 1 11:51 EDT 2020. Contains 334762 sequences. (Running on oeis4.)