This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
a:=n->sum(fibonacci(i,2), i=0..n): seq(a(n), n=1..29); # Zerinvary Lajos, Mar 20 2008
Join[{a=1,b=3},Table[c=2*b+a+1;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)
CoefficientList[Series[1/(1-3x+x^2+x^3),{x,0,30}],x] (* or *) LinearRecurrence[{3,-1,-1},{1,3,8},30] (* Harvey P. Dale, Jun 13 2011 *)
a(n)=local(w=quadgen(8));-1/2+(3/4+1/2*w)*(1+w)^n+(3/4-1/2*w)*(1-w)^n
vector(100, n, n--; floor((1+sqrt(2))^(n+2)/4)) \\ Altug Alkan, Oct 07 2015
Vec(1/((1-x)*(1-2*x-x^2)) + O(x^40)) \\ Michel Marcus, May 06 2017
Northwest corner: 1...3...6....10...16 2...4...7....12...19 5...9...15...24...37 8...13..21...33...51 11..18..28...43...66
(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12; c = 40; c1 = 12;
a = 3; b = 4; m[n_] := If[Mod[n, 2] == 0, 1, 0];
f[n_] := a*m[n + 1] + b*m[n] + 3*Floor[(n - 1)/2]
Table[f[n], {n, 1, 30}] (* A032766: (3+5k,4+5k, k>=0) *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i,1,10}, {j,1,10}]] (* A191655 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191655 sequence *)
Table begins: 1 2 4 7 12 20 33 54 88 143 232 376 ... 3 5 9 15 25 41 67 109 177 287 465 753 ... 6 10 17 28 46 75 122 198 321 520 842 1363 ... 8 13 22 36 59 96 156 253 410 664 1075 1740 ... 11 18 30 49 80 130 211 342 554 897 1452 2350 ... 14 23 38 62 101 164 266 431 698 1130 1829 2960 ... 16 26 43 70 114 185 300 486 787 1274 2062 3337 ... 19 31 51 83 135 219 355 575 931 1507 2439 3947 ... 21 34 56 91 148 240 389 630 1020 1651 2672 4324 ... 24 39 64 104 169 274 444 719 1164 1884 3049 4934 ... 27 44 72 117 190 308 499 808 1308 2117 3426 5544 ...
t[n_, 0] = Floor[n*GoldenRatio/(GoldenRatio - 1) + 1];
t[n_, k_] := t[n, k] = Ceiling[GoldenRatio*t[n, k-1]];
Flatten[Table[t[k-1, n-k ], {n, 12}, {k, n}] ][[;; 67]]
(* Jean-François Alcover, Jul 13 2011 *)
Table begins: 1 2 3 5 8 12 17 25 36 51 73 104 ... 4 6 9 13 19 27 39 56 80 114 162 230 ... 7 10 15 22 32 46 66 94 133 189 268 380 ... 11 16 23 33 47 67 95 135 191 271 384 544 ... 14 20 29 42 60 85 121 172 244 346 490 693 ... 18 26 37 53 75 107 152 215 305 432 611 865 ... 21 30 43 61 87 124 176 249 353 500 708 1002 ... 24 34 49 70 99 141 200 283 401 568 804 1138 ... 28 40 57 81 115 163 231 327 463 655 927 1311 ... 31 44 63 90 128 182 258 365 517 732 1036 1466 ... 35 50 71 101 143 203 288 408 577 817 1156 1635 ...
(* program generates the dispersion array T of the complement of increasing sequence f[n] *)
r = 40; r1 = 12; (* r=# rows of T, r1=# rows to show *)
c = 40; c1 = 12; (* c=# cols of T, c1=# cols to show *)
x = Sqrt[2]; f[n_] := Floor[n*x + 1]
(* f(n) is complement of column 1 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A083050 as an array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* array as a sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011, added here Jun 03 2011 by Clark Kimberling *)
From _Peter Munn_, Jul 03 2020: (Start)
Listing the terms in a snake format (with period 27) illustrates periodic and mirror symmetries. Horizontal lines mark points of mirror symmetry for 3's. Vertical lines mark further points of mirror symmetry for 2's. 79 terms are shown. (Referred to the extrapolated common origin of periodic mirror symmetry, the initial term is at offset 1.5 and the last shown is at offset 79.5 = 3^4 - 1.5.) Observe also mirror symmetry of 4's (seen vertically).
1 2 3 1 4 2 1 5 3 1 2 6
| | 1 --
1 2 3 1 5 2 1 7 3 1 2 4
_ 4
8
1 2 3 1 6 2 1 4 3 1 2 5
| | 1 --
1 2 3 1 7 2 1 4 3 1 2 9
_ 5
4
1 2 3 1 6 2 1 10 3 1 2 4
| | 1 --
1 2 3 1 4 2 1 8 3 1 2 5
(End)
From _Peter Munn_, Aug 22 2020: (Start)
The start of the sequence is shown below in conjunction with related sequences, aligning their points of mirror symmetry. The longer, and shorter, vertical lines indicate points of mirror symmetry for terms valued less than 4, and less than 3, respectively. Note each term of A051064 is the minimum of two terms displayed nearest below it, and each term of A254046 is the minimum of the two terms displayed diagonally above it.
| | |
A051064:| 1 1 2 1 1 2 1 1 3 1 1 2 1 1 2 1 1 3 1 1 2 1 1 2 1 1 4 1 1 2
| | | | | | |
[a(n)]: | 1 2 3 1 4 2 1 5 3 1 2 6 1 4 2 1 3 7 1 2 5 1 3 2 1 4 8 1 2 3
| | | | | | |
A254046:|1 2 1 1 3 1 1 2 1 1 2 1 1 4 1 1 2 1 1 2 1 1 3 1 1 2 1 1 2 1 1
| | |
(End)
A163491[n_] := A163491[n] = If[Mod[n, 3]==1, (n+2)/3, A163491[Floor[2n/3]]]; Module[{b}, b[_] = 0; a[n_] := With[{t = A163491[n]}, b[t] = b[t] + 1]]; Array[a, 105] (* Jean-François Alcover, Jan 10 2022 *)
Table begins: 1 3 8 20 49 119 288 ... 2 5 13 32 78 189 457 ... 4 10 25 61 148 358 865 ... 6 15 37 90 218 527 1273 ... 7 17 42 102 247 597 1442 ... 9 22 54 131 317 766 1850 ... 11 27 66 160 387 935 2258 ... 12 29 71 172 416 1005 2427 ... 14 34 83 201 486 1174 2835 ... 16 39 95 230 556 1343 3243 ... 18 44 107 259 626 1512 3651 ... 19 46 112 271 655 1582 3820 ... 21 51 124 300 725 1751 4228 ... 23 56 136 329 795 1920 4636 ... 24 58 141 341 824 1990 4805 ...
z:=10; x:=1+Sqrt(2); S:=[]; for n in [0..z] do for k in [0..n] do if n-k eq 0 then Append(~S, Floor(n*x/(x-1))+1); else Append(~S, Ceiling(x*S[k+1+(n*(n-1) div 2)])); end if; end for; end for; S; // Klaus Brockhaus, Jan 04 2011
(See Comments.)
z:=70; x:=1+Sqrt(2); [ Floor(n*x/(x-1))+1: n in [0..z] ]; // Klaus Brockhaus, Jan 04 2011
f[n_] := Floor[n/Sqrt@2 + n + 1]; Array[f, 68, 0]
Comments