A022527
Nexus numbers: a(n) = (n+1)^11 - n^11.
Original entry on oeis.org
1, 2047, 175099, 4017157, 44633821, 313968931, 1614529687, 6612607849, 22791125017, 68618940391, 185311670611, 457696700077, 1049152023349, 2257404775627, 4600190689711, 8942430185041, 16679710263217, 29996513771599, 52221848818987, 88309741101781
Offset: 0
- J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 54.
- T. D. Noe, Table of n, a(n) for n = 0..1000
- H. D. Nguyen, D. Taggart, Mining the OEIS: Ten Experimental Conjectures, 2013. Mentions this sequence. - From _N. J. A. Sloane_, Mar 16 2014
- Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
-
[(n+1)^11-n^11: n in [0..40]]; // Vincenzo Librandi, Jan 26 2011
-
b:=11: a:=n->(n+1)^b-n^b: seq(a(n),n=0..18); # Muniru A Asiru, Feb 28 2018
-
q=11; Table[(n+1)^q-n^q, {n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Jan 23 2009 *)
Differences[Range[0, 20]^11] (* Harvey P. Dale, Jan 25 2011 *)
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for(n=0,20, print1((n+1)^11 - n^11, ", ")) \\ G. C. Greubel, Feb 27 2018
A341050
Cube array read by upward antidiagonals ignoring zero and empty terms: T(n, k, r) is the number of n-ary strings of length k, containing r consecutive 0's.
Original entry on oeis.org
1, 1, 1, 3, 1, 1, 3, 1, 5, 8, 1, 1, 3, 1, 5, 8, 1, 7, 21, 19, 1, 1, 3, 1, 5, 8, 1, 7, 21, 20, 1, 9, 40, 81, 43, 1, 1, 3, 1, 5, 8, 1, 7, 21, 20, 1, 9, 40, 81, 47, 1, 11, 65, 208, 295, 94, 1, 1, 3, 1, 5, 8, 1, 7, 21, 20, 1, 9, 40, 81, 48, 1, 11, 65, 208, 297, 107, 1, 13, 96, 425, 1024, 1037, 201
Offset: 2
For n = 5, k = 6 and r = 4, there are 65 strings: {000000, 000001, 000002, 000003, 000004, 000010, 000011, 000012, 000013, 000014, 000020, 000021, 000022, 000023, 000024, 000030, 000031, 000032, 000033, 000034, 000040, 000041, 000042, 000043, 000044, 010000, 020000, 030000, 040000, 100000, 100001, 100002, 100003, 100004, 110000, 120000, 130000, 140000, 200000, 200001, 200002, 200003, 200004, 210000, 220000, 230000, 240000, 300000, 300001, 300002, 300003, 300004, 310000, 320000, 330000, 340000, 400000, 400001, 400002, 400003, 400004, 410000, 420000, 430000, 440000}
The first seven slices of the tetrahedron (or pyramid) are:
-----------------Slice 1-----------------
1
-----------------Slice 2-----------------
1
1 3
-----------------Slice 3-----------------
1
1 3
1 5 8
-----------------Slice 4-----------------
1
1 3
1 5 8
1 7 21 19
-----------------Slice 5-----------------
1
1 3
1 5 8
1 7 21 20
1 9 40 81 43
-----------------Slice 6-----------------
1
1 3
1 5 8
1 7 21 20
1 9 40 81 47
1 11 65 208 295 94
-----------------Slice 7-----------------
1
1 3
1 5 8
1 7 21 20
1 9 40 81 48
1 11 65 208 297 107
1 13 96 425 1024 1037 201
Cf.
A005408,
A003215,
A005917,
A022521,
A022522,
A022523,
A022524,
A022525,
A022526,
A022527,
A022528,
A022529,
A022530,
A022531,
A022532,
A022533,
A022534,
A022535,
A022536,
A022537,
A022538,
A022539,
A022540 (k=x, r=1, where x is the x-th Nexus Number).
Cf.
A000567 [(k=4, r=2),(k=5, r=3),(k=6, r=4),...,(k=x, r=x-2)].
Cf.
A103532 [(k=6, r=3),(k=7, r=4),(k=8, r=5),...,(k=x, r=x-3)].
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m[r_, n_] := Normal[With[{p = 1/n}, SparseArray[{Band[{1, 2}] -> p, {i_, 1} /; i <= r -> 1 - p, {r + 1, r + 1} -> 1}]]]; T[n_, k_, r_] := MatrixPower[m[r, n], k][[1, r + 1]]*n^k; DeleteCases[Transpose[PadLeft[Reverse[Table[T[n, k, r], {k, 2, 8}, {r, 2, k}, {n, 2, r}], 2]], 2 <-> 3], 0, 3] // Flatten
Showing 1-2 of 2 results.