A001477
The nonnegative integers.
Original entry on oeis.org
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77
Offset: 0
Triangular view:
0
1 2
3 4 5
6 7 8 9
10 11 12 13 14
15 16 17 18 19 20
21 22 23 24 25 26 27
28 29 30 31 32 33 34 35
36 37 38 39 40 41 42 43 44
45 46 47 48 49 50 51 52 53 54
- Maurice Protat, Des Olympiades à l'Agrégation, suite vérifiant f(n+1) > f(f(n)), Problème 7, pp. 31-32, Ellipses, Paris 1997.
- N. J. A. Sloane, Table of n, a(n) for n = 0..500000
- Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
- David Corneth, Counting to 13999 visualized | showing changes per digit, YouTube video, 2019.
- Hans Havermann, Table giving n and American English name for n, for 0 <= n <= 100999, without spaces or hyphens
- Hans Havermann, American English number names to one million, without spaces or hyphens
- The IMO Compendium, Problem 6, 19th IMO 1977.
- Tanya Khovanova, Recursive Sequences
- Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
- László Németh, The trinomial transform triangle, J. Int. Seqs., Vol. 21 (2018), Article 18.7.3. Also arXiv:1807.07109 [math.NT], 2018.
- N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 12.
- Eric Weisstein's World of Mathematics, Natural Number
- Eric Weisstein's World of Mathematics, Nonnegative Integer
- Index entries for "core" sequences
- Index entries for sequences that are permutations of the natural numbers
- Index entries for linear recurrences with constant coefficients, signature (2,-1).
- Index to sequences related to Olympiads.
When written as an array, the rows/columns are
A000217,
A000124,
A152948,
A152950,
A145018,
A167499,
A166136,
A167487... and
A000096,
A034856,
A055998,
A046691,
A052905,
A055999... (with appropriate offsets); cf. analogous lists for
A000027 in
A185787.
Cf.
A061579 (transposed matrix / reversed triangle).
-
a001477 = id
a001477_list = [0..] -- Reinhard Zumkeller, May 07 2012
-
print([n for n in 0:280]) # Paul Muljadi, Apr 15 2024
-
[ n : n in [0..100]];
-
[ seq(n,n=0..100) ];
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Table[n, {n, 0, 100}] (* Stefan Steinerberger, Apr 08 2006 *)
LinearRecurrence[{2, -1}, {0, 1}, 77] (* Robert G. Wilson v, May 23 2013 *)
CoefficientList[ Series[x/(x - 1)^2, {x, 0, 76}], x] (* Robert G. Wilson v, May 23 2013 *)
Range[0,100] (* Harvey P. Dale, Dec 29 2024 *)
-
A001477(n)=n /* first term is a(0) */
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def a(n): return n
print([a(n) for n in range(78)]) # Michael S. Branicky, Nov 13 2022
A055998
a(n) = n*(n+5)/2.
Original entry on oeis.org
0, 3, 7, 12, 18, 25, 33, 42, 52, 63, 75, 88, 102, 117, 133, 150, 168, 187, 207, 228, 250, 273, 297, 322, 348, 375, 403, 432, 462, 493, 525, 558, 592, 627, 663, 700, 738, 777, 817, 858, 900, 943, 987, 1032, 1078, 1125, 1173, 1222, 1272
Offset: 0
- Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 193.
- Ivan Panchenko, Table of n, a(n) for n = 0..1000
- Karl Dilcher and Larry Ericksen, Polynomials and algebraic curves related to certain binary and b-ary overpartitions, arXiv:2405.12024 [math.CO], 2024. See p. 10.
- Milan Janjic, Two Enumerative Functions.
- Kival Ngaokrajang, Illustration from A000027 (contains errors).
- Linhui Shen, Duals of semisimple Poisson-Lie groups and cluster theory of moduli spaces of G-local systems, arXiv:2003.07901 [math.RT], 2020. See p. 8.
- Leo Tavares, Illustration: Truncated Point Triangles.
- Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
a(n) =
A095660(n+1, 2): third column of (1, 3)-Pascal triangle.
Cf. other rows, columns and diagonals of
A000027 written as a table:
A034856,
A046691,
A052905,
A055999,
A155212,
A051936,
A056000,
A183897,
A056115,
A051938;
A000124,
A022856,
A152950,
A145018,
A077169,
A166136,
A167487,
A173036;
A059993,
A090288,
A054000,
A142463,
A056220,
A001105,
A001844,
A058331,
A051890,
A097080,
A093328,
A137882.
A185787
Sum of first k numbers in column k of the natural number array A000027; by antidiagonals.
Original entry on oeis.org
1, 7, 25, 62, 125, 221, 357, 540, 777, 1075, 1441, 1882, 2405, 3017, 3725, 4536, 5457, 6495, 7657, 8950, 10381, 11957, 13685, 15572, 17625, 19851, 22257, 24850, 27637, 30625, 33821, 37232, 40865, 44727, 48825, 53166, 57757, 62605, 67717, 73100, 78761, 84707, 90945, 97482, 104325, 111481, 118957, 126760, 134897, 143375
Offset: 1
-
[n*(7*n^2-6*n+5)/6: n in [1..50]]; // Vincenzo Librandi, Jul 04 2012
-
f[n_,k_]:=n+(n+k-2)(n+k-1)/2;
s[k_]:=Sum[f[n,k],{n,1,k}];
Factor[s[k]]
Table[s[k],{k,1,70}] (* A185787 *)
CoefficientList[Series[(3*x^2+3*x+1)/(1-x)^4,{x,0,50}],x] (* Vincenzo Librandi, Jul 04 2012 *)
A229445
T(n,k)=Number of nXk 0..2 arrays with horizontal differences mod 3 never 1, vertical differences mod 3 never -1, and rows and columns lexicographically nondecreasing.
Original entry on oeis.org
3, 4, 5, 5, 7, 8, 6, 10, 13, 12, 7, 14, 22, 25, 17, 8, 19, 37, 53, 47, 23, 9, 25, 60, 109, 128, 84, 30, 10, 32, 93, 212, 324, 293, 142, 38, 11, 40, 138, 387, 753, 915, 625, 228, 47, 12, 49, 197, 665, 1609, 2546, 2402, 1244, 350, 57, 13, 59, 272, 1083, 3184, 6374, 8024
Offset: 1
Some solutions for n=4 k=4
..0..2..2..2....0..2..2..2....0..0..2..2....0..0..2..2....0..2..2..2
..1..0..0..2....1..0..0..0....0..0..2..2....1..1..0..0....0..2..2..2
..2..1..1..0....2..1..1..1....1..1..0..0....1..1..1..1....0..2..2..2
..2..1..1..1....2..2..2..2....1..1..1..1....2..2..1..1....1..0..0..2
A027693
a(n) = n^2 + n + 8.
Original entry on oeis.org
8, 10, 14, 20, 28, 38, 50, 64, 80, 98, 118, 140, 164, 190, 218, 248, 280, 314, 350, 388, 428, 470, 514, 560, 608, 658, 710, 764, 820, 878, 938, 1000, 1064, 1130, 1198, 1268, 1340, 1414, 1490, 1568, 1648, 1730, 1814, 1900, 1988, 2078, 2170, 2264, 2360, 2458, 2558
Offset: 0
-
List([0..50],n->n^2+n+8); # Muniru A Asiru, Jul 15 2018
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with(combinat): seq(fibonacci(3, n)+n+7, n=0..46); # Zerinvary Lajos, Jun 07 2008
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f[n_]:=n^2+n+8;f[Range[0,100]] (* Vladimir Joseph Stephan Orlovsky, Mar 12 2011 *)
LinearRecurrence[{3,-3,1},{8,10,14},60] (* Harvey P. Dale, Dec 13 2011 *)
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a(n)=n^2+(n+8) \\ Charles R Greathouse IV, Jun 17 2017
A279967
Square array read by antidiagonals upwards in which each term is the sum of prior elements in the same row, column, diagonal, or antidiagonal that divide n; the array is seeded with an initial value a(1)=1.
Original entry on oeis.org
1, 1, 2, 2, 2, 7, 2, 9, 10, 15, 2, 10, 1, 13, 17, 8, 0, 13, 1, 14, 9, 8, 0, 13, 3, 30, 13, 10, 2, 16, 1, 23, 5, 7, 14, 15, 2, 8, 28, 32, 2, 23, 2, 9, 49, 12, 0, 48, 2, 11, 1, 20, 3, 18, 13, 28, 0, 4, 1, 56, 5, 8, 16, 35, 46, 4, 2, 6, 2, 10
Offset: 1
After 6 terms, the array looks like:
.
1 2 7
1 2
2
We have a(6) = 7 because a(1) = 1, a(3) = 2, a(4) = 2, and a(5) = 2 divide 6; 1 + 2 + 2 + 2 = 7.
From _Hartmut F. W. Hoft_, Jan 23 2017: (Start)
1 2 7 15 17 9 10 15 49 13 4 31 22
1 2 10 13 14 13 14 9 18 46 12 66
2 9 1 1 30 7 2 3 35 12 3
2 10 13 3 5 23 20 16 14 17
2 0 13 23 2 1 8 11 2
8 0 1 32 11 5 3 6
8 16 28 2 56 42 8
2 8 48 1 2 104
2 0 4 10 1
12 0 2 10
28 6 2
2 42
2
.
Expanded the triangle to the first 13 antidiagonals of the array, i.e. a(1) ... a(91), to show the start of the 2- and 0-value patterns in columns 1 and 2. The first 0 beyond column 2 is a(677) in row 27, column 11 of the triangle.
A188382(n)=2*n^2+n+1 for n>=0 are the alternate sequence indices for column 1 starting in row 1, 2*n^2+n+2 for n>=1 are the alternate sequence indices for column 2 starting in row 2, and 2*n^2+n+11 for n>=5 are the alternate sequence indices for column 11 starting in row 1.
The sequence indices in the triangle for row positions k>=1 in columns 1,..., 5 are given in sequences A000124(k), A152948(k+3), A152950(k+3), A145018(k+4) and A167499(k+4).
(End)
Cf.
A279966 for the related sequence which counts prior terms.
Cf.
A269347 for a one-dimensional version of this sequence.
-
(* printing of the triangle is commented out of function a279967[] *)
pCol[{i_, j_}] := Map[{#, j}&, Range[1, i-1]]
pDiag[{i_, j_}] := If[j>=i, Map[{#, j-i+#}&, Range[1, i-1]], Map[{i-j+#, #}&, Range[1, j-1]]]
pRow[{i_, j_}] := Map[{i, #}&, Range[1, j-1]]
pAdiag[{i_, j_}] := Map[{i+j-#, #}&, Range[1, j-1]]
priorPos[{i_, j_}] := Join[pCol[{i, j}], pDiag[{i, j}], pRow[{i, j}], pAdiag[{i, j}]]
seqPos[{i_, j_}] := (i+j-2)(i+j-1)/2+j
antiDiag[k_] := Map[{k+1-#, #}&, Range[1, k]]
upperTriangle[k_] := Flatten[Map[antiDiag, Range[1, k]], 1]
a279967[k_] := Module[{ut=upperTriangle[k], ms=Table[" ", {i, 1, k}, {j, 1, k}], h, pos, val, seqL={1}}, ms[[1, 1]]=1; For[h=2, h<=Length[ut], h++, pos=ut[[h]]; val=Apply[Plus, Select[Map[ms[[Apply[Sequence, #]]]&, priorPos[pos]], #!=0 && Mod[seqPos[pos], #]==0&]]; AppendTo[seqL, val]; ms[[Apply[Sequence, pos]]]=val]; (* Print[TableForm[ms]]; *) seqL]
a279967[13] (* values in first 13 antidiagonals *)
(* Hartmut F. W. Hoft, Jan 23 2017 *)
A166136
a(n) = n*(n+3)/2 + 7.
Original entry on oeis.org
9, 12, 16, 21, 27, 34, 42, 51, 61, 72, 84, 97, 111, 126, 142, 159, 177, 196, 216, 237, 259, 282, 306, 331, 357, 384, 412, 441, 471, 502, 534, 567, 601, 636, 672, 709, 747, 786, 826, 867, 909, 952, 996, 1041, 1087, 1134, 1182, 1231, 1281, 1332, 1384, 1437
Offset: 1
-
I:=[9, 12, 16]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Mar 15 2012
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Table[n*(n+3)/2+7, {n, 1, 40}] (* or *) LinearRecurrence[{3,-3,1}, {9, 12, 16}, 40] (* Vincenzo Librandi, Mar 15 2012 *)
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for(n=1, 40, print1(n*(n+3)/2+7, ", ")); \\ Vincenzo Librandi, Mar 15 2012
Definition replaced by polynomial from
R. J. Mathar, Oct 12 2009
A245300
Triangle T(n,k) = (n+k)*(n+k+1)/2 + k, 0 <= k <= n, read by rows.
Original entry on oeis.org
0, 1, 4, 3, 7, 12, 6, 11, 17, 24, 10, 16, 23, 31, 40, 15, 22, 30, 39, 49, 60, 21, 29, 38, 48, 59, 71, 84, 28, 37, 47, 58, 70, 83, 97, 112, 36, 46, 57, 69, 82, 96, 111, 127, 144, 45, 56, 68, 81, 95, 110, 126, 143, 161, 180, 55, 67, 80, 94, 109, 125, 142, 160, 179, 199, 220
Offset: 0
First rows and their row sums (A245301):
0 0;
1, 4 5;
3, 7, 12 22;
6, 11, 17, 24 58;
10, 16, 23, 31, 40 120;
15, 22, 30, 39, 49, 60 215;
21, 29, 38, 48, 59, 71, 84 350;
28, 37, 47, 58, 70, 83, 97, 112 532;
36, 46, 57, 69, 82, 96, 111, 127, 144 768;
45, 56, 68, 81, 95, 110, 126, 143, 161, 180 1065;
55, 67, 80, 94, 109, 125, 142, 160, 179, 199, 220 1430;
66, 79, 93, 108, 124, 141, 159, 178, 198, 219, 241, 264 1870;
78, 92, 107, 123, 140, 158, 177, 197, 218, 240, 263, 287, 312 2392.
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a245300 n k = (n + k) * (n + k + 1) `div` 2 + k
a245300_row n = map (a245300 n) [0..n]
a245300_tabl = map a245300_row [0..]
a245300_list = concat a245300_tabl
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[k + Binomial(n+k+1,2): k in [0..n], n in [0..15]]; // G. C. Greubel, Apr 01 2021
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Table[k + Binomial[n+k+1,2], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 01 2021 *)
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flatten([[k + binomial(n+k+1,2) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Apr 01 2021
Original entry on oeis.org
17, 23, 47, 57, 93, 107, 173, 233, 353, 437, 467, 563, 677, 743, 817, 829, 851, 863, 955, 1037, 1187, 1213, 1277, 1387, 1433, 1487, 1549, 2089, 2147, 2213, 2287, 2293, 2417, 2473, 2689, 2777, 2911, 3083, 3323, 3391, 6691, 9337, 22969, 38557, 47347, 75391, 104999, 130927, 146719
Offset: 1
Number 817 = 19*43, equivalent to array position (4, 37), is in the sequence since none of the numbers in the prior column, diagonal, row and antidiagonal contain the counts of 1, 19, 43 and 817. - _Hartmut F. W. Hoft_, Jan 23 2017
-
(* support functions are in A279967 *)
a278436[k_] := Module[{ut=upperTriangle[k], ms=Table[" ", {i, 1, k}, {j, 1, k}], h, pos, val, seqL={}}, ms[[1, 1]]=1; For[h=2, h<=Length[ut], h++, pos=ut[[h]]; val=Length[Select[Map[ms[[Apply[Sequence, #]]]&, priorPos[pos]], #!=0 && Mod[seqPos[pos], #]==0&]]; If[val==0, AppendTo[seqL, h]]; ms[[Apply[Sequence, pos]]]=val]; seqL]
a278436[100] (* data through 3391. - Hartmut F. W. Hoft, Jan 23 2017 *)
A328000
a(n) = Sum_{k=0..n}(k!*(n - k)!)/(floor(k/2)!*floor((n - k)/2)!)^2.
Original entry on oeis.org
1, 2, 5, 16, 28, 96, 160, 512, 896, 2560, 4864, 12288, 25600, 57344, 131072, 262144, 655360, 1179648, 3211264, 5242880, 15466496, 23068672, 73400320, 100663296, 343932928, 436207616, 1593835520, 1879048192, 7314866176, 8053063680, 33285996544, 34359738368
Offset: 0
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[IsOdd(n) select 2^(n - 1)*(n + 1) else 2^(n - 5)*(n*(n + 2) + 32):n in [0..30]]; // Marius A. Burtea, Feb 05 2020
-
swing := n -> n!/iquo(n,2)!^2: a := n -> add(swing(k)*swing(n-k), k=0..n):
seq(`if`(irem(n, 2) = 0, 2 + n*(n + 2)/16, n + 1)*2^(n - 1), n=0..31);
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A328000List[len_] := CoefficientList[Series[(4 x^2 - x - 1)^2 / (1 - 4 x^2)^3 , {x, 0, len}], x]; A328000List[31]
LinearRecurrence[{0,12,0,-48,0,64},{1,2,5,16,28,96},40] (* Harvey P. Dale, Jun 19 2022 *)
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x='x + O('x^32);
Vec(serlaplace(((3*x + 8)*sinh(2*x) + (2*x^2 + 16*(x + 1))*cosh(2*x))/16))
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Vec((1 + x - 4*x^2)^2 / ((1 - 2*x)^3*(1 + 2*x)^3) + O(x^30)) \\ Colin Barker, Feb 05 2020
Showing 1-10 of 11 results.
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