A022528
Nexus numbers (n+1)^12-n^12.
Original entry on oeis.org
1, 4095, 527345, 16245775, 227363409, 1932641711, 11664504865, 54878189535, 213710059745, 717570463519, 2138428376721, 5777672071535, 14381984674225, 33395827252815, 73052425515329, 151728638820031, 301147260519105, 574209144196415
Offset: 0
- J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 54.
- Vincenzo Librandi, Table of n, a(n) for n = 0..10000
- Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).
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[(n+1)^12-n^12: n in [0..20]]; // Vincenzo Librandi, Nov 22 2011
-
lst={};Do[a=n^6;b=(n+1)^6;s=(a+b)*(b-a);AppendTo[lst,s],{n,0,4!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 23 2009 *)
Table[(n+1)^12-n^12,{n,0,20}] (* Vincenzo Librandi, Nov 22 2011 *)
LinearRecurrence[{12,-66,220,-495,792,-924,792,-495,220,-66,12,-1},{1,4095,527345,16245775,227363409,1932641711,11664504865,54878189535,213710059745,717570463519,2138428376721,5777672071535},20] (* Harvey P. Dale, Apr 23 2019 *)
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vector(30, n, n--; (n+1)^12-n^12) \\ Colin Barker, Nov 30 2014
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.