A194046
Natural interspersion of A052905, a rectangular array, by antidiagonals.
Original entry on oeis.org
1, 5, 2, 10, 6, 3, 16, 11, 7, 4, 23, 17, 12, 8, 9, 31, 24, 18, 13, 14, 15, 40, 32, 25, 19, 20, 21, 22, 50, 41, 33, 26, 27, 28, 29, 30, 61, 51, 42, 34, 35, 36, 37, 38, 39, 73, 62, 52, 43, 44, 45, 46, 47, 48, 49, 86, 74, 63, 53, 54, 55, 56, 57, 58, 59, 60, 100, 87, 75
Offset: 1
Northwest corner:
1...5...10...16...23
2...6...11...17...24
3...7...12...18...25
4...8...13...19...26
9...14..20...27...35
-
z = 30;
c[k_] := (k^2 + 5 k - 4)/2;
c = Table[c[k], {k, 1, z}] (* A052905 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 255}] (* fractal sequence [A002260] *)
r[n_] := Flatten[Position[f, n]]
t[n_, k_] := r[n][[k]]
TableForm[Table[t[n, k], {n, 1, 7}, {k, 1, 7}]]
p = Flatten[Table[t[k, n - k + 1], {n, 1, 13}, {k, 1, n}]] (* A194046 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 70}]] (* A194047 *)
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) ];
-
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) */
-
def a(n): return n
print([a(n) for n in range(78)]) # Michael S. Branicky, Nov 13 2022
A028387
a(n) = n + (n+1)^2.
Original entry on oeis.org
1, 5, 11, 19, 29, 41, 55, 71, 89, 109, 131, 155, 181, 209, 239, 271, 305, 341, 379, 419, 461, 505, 551, 599, 649, 701, 755, 811, 869, 929, 991, 1055, 1121, 1189, 1259, 1331, 1405, 1481, 1559, 1639, 1721, 1805, 1891, 1979, 2069, 2161, 2255, 2351, 2449, 2549, 2651
Offset: 0
From _Ilya Gutkovskiy_, Apr 13 2016: (Start)
Illustration of initial terms:
o o
o o o o o o
o o o o o o o o o o o o
o o o o o o o o o o o o o o o o o o o o
o o o o o o o o o o o o o o o o o o o o o o o o o
n=0 n=1 n=2 n=3 n=4
(End)
From _Klaus Purath_, Mar 18 2019: (Start)
Examples:
a(0) = 1: 1^1-0*1 = 1, 0+1 = 1 (Fibonacci A000045).
a(1) = 5: 3^2-1*4 = 5, 1+3 = 4 (Lucas A000032).
a(2) = 11: 4^2-1*5 = 11, 1+4 = 5 (A000285); 5^2-2*7 = 11, 2+5 = 7 (A001060).
a(3) = 19: 5^2-1*6 = 19, 1+5 = 6 (A022095); 7^2-3*10 = 19, 3+7 = 10 (A022120).
a(4) = 29: 6^2-1*7 = 29, 1+6 = 7 (A022096); 9^2-4*13 = 29, 4+9 = 13 (A022130).
a(11)/5 = 31: 7^2-2*9 = 31, 2+7 = 9 (A022113); 8^2-3*11 = 31, 3+8 = 11 (A022121).
a(24)/11 = 59: 9^2-2*11 = 59, 2+9 = 11 (A022114); 12^2-5*17 = 59, 5+12 = 17 (A022137).
(End)
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Patrick De Geest, World!Of Numbers, Palindromes of the form n+(n+1)^2.
- Adalbert Kerber, A matrix of combinatorial numbers related to the symmetric groups, Discrete Math., 21 (1978), 319-321. [Annotated scanned copy]
- Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
- Nandini Nilakantan and Anurag Singh, Homotopy type of neighborhood complexes of Kneser graphs, KG_{2,k}, Proceeding-Mathematical Sciences, 128, Article number: 53(2018).
- Yanni Pei and Jiang Zeng, Counting signed derangements with right-to-left minima and excedances, arXiv:2206.11236 [math.CO], 2022.
- Popular Computing (Calabasas, CA), The CSR Function, Vol. 4 (No. 34, Jan 1976), pages PC34-10 to PC34-11. Annotated and scanned copy.
- Zdzislaw Skupień and Andrzej Żak, Pair-sums packing and rainbow cliques, in Topics In Graph Theory, A tribute to A. A. and T. E. Zykovs on the occasion of A. A. Zykov's 90th birthday, ed. R. Tyshkevich, Univ. Illinois, 2013, pages 131-144, (in English and Russian).
- Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
-
a028387 n = n + (n + 1) ^ 2 -- Reinhard Zumkeller, Jul 17 2014
-
[n + (n+1)^2: n in [0..60]]; // Vincenzo Librandi, Apr 26 2011
-
FoldList[## + 2 &, 1, 2 Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *)
Table[n + (n + 1)^2, {n, 0, 100}] (* Vincenzo Librandi, Oct 17 2012 *)
Table[ FrobeniusNumber[{n, n + 1}], {n, 2, 30}] (* Zak Seidov, Jan 14 2015 *)
-
a(n)=n^2+3*n+1 \\ Charles R Greathouse IV, Jun 10 2011
-
def a(n): return (n**2+3*n+1) # Torlach Rush, May 07 2024
-
[n+(n+1)^2 for n in range(0,48)] # Zerinvary Lajos, Jul 03 2008
Minor edits by
N. J. A. Sloane, Jul 04 2010, following suggestions from the Sequence Fans Mailing List
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 *)
A059605
a(n) = (1/3!)*(n^3 + 24*n^2 + 107*n + 90), compare A059604.
Original entry on oeis.org
15, 37, 68, 109, 161, 225, 302, 393, 499, 621, 760, 917, 1093, 1289, 1506, 1745, 2007, 2293, 2604, 2941, 3305, 3697, 4118, 4569, 5051, 5565, 6112, 6693, 7309, 7961, 8650, 9377, 10143, 10949, 11796, 12685, 13617, 14593, 15614, 16681, 17795, 18957
Offset: 0
A101882
Write three numbers, skip one, write three, skip two, write three, skip three... and so on.
Original entry on oeis.org
1, 2, 3, 5, 6, 7, 10, 11, 12, 16, 17, 18, 23, 24, 25, 31, 32, 33, 40, 41, 42, 50, 51, 52, 61, 62, 63, 73, 74, 75, 86, 87, 88, 100, 101, 102, 115, 116, 117, 131, 132, 133, 148, 149, 150, 166, 167, 168, 185, 186, 187, 205, 206, 207, 226, 227, 228, 248, 249, 250, 271
Offset: 1
Candace Mills (scorpiocand(AT)yahoo.com), Dec 19 2004
-
Flatten@Table[(n^2 + 5 n - 4)/2 + {0, 1, 2}, {n, 20}] (* Ivan Neretin, Aug 03 2016 *)
Table[Range[#, # + 2] &[(n^2 + 7 n + 2)/2], {n, 0, 20}] // Flatten (* or *)
Rest@ CoefficientList[Series[x (1 + x + x^2 - x^4 - x^5)/((1 + x + x^2)^2 (1 - x)^3), {x, 0, 61}], x] (* Michael De Vlieger, Aug 03 2016 *)
LinearRecurrence[{1,0,2,-2,0,-1,1},{1,2,3,5,6,7,10},70] (* Harvey P. Dale, Dec 26 2019 *)
-
a(n)=my(k=n%3); if(k==2, n^2+17*n-2, k==1, n^2+19*n-2, n^2+15*n)/18 \\ Charles R Greathouse IV, Aug 03 2016
A164981
A triangle with Pell numbers in the first column.
Original entry on oeis.org
1, 2, 1, 5, 3, 1, 12, 10, 4, 1, 29, 30, 16, 5, 1, 70, 87, 56, 23, 6, 1, 169, 245, 185, 91, 31, 7, 1, 408, 676, 584, 334, 136, 40, 8, 1, 985, 1836, 1784, 1158, 546, 192, 50, 9, 1, 2378, 4925, 5312, 3850, 2052, 834, 260, 61, 10, 1, 5741, 13079, 15497, 12386, 7342, 3366, 1212, 341, 73, 11, 1
Offset: 1
Triangle begins
1
2,1
5,3,1
12,10,4,1
29,30,16,5,1
70,87,56,23,6,1
169,245,185,91,31,7,1
...
From _Philippe Deléham_, Oct 10 2013: (Start)
Triangle (0, 2, 1/2, -1/2, 0, 0, ...) DELTA (1, 0, -1/2, 1/2, 0, 0, ...):
1
0, 1
0, 2, 1
0, 5, 3, 1
0, 12, 10, 4, 1
0, 29, 30, 16, 5, 1
0, 70, 87, 56, 23, 6, 1
0, 169, 245, 185, 91, 31, 7, 1
... (End)
-
A164981 := proc(n,k) option remember; if n <1 or k<1 or k>n then 0; elif n = 1 then 1; else 2*procname(n-1,k)+procname(n-1,k-1)+procname(n-2,k)-procname(n-2,k-1) ; end if; end proc:
-
T[n_, k_] := T[n, k] = Which[n < 1 || k < 1 || k > n, 0, n == 1, 1, True, 2*T[n-1, k] + T[n-1, k-1] + T[n-2, k] - T[n-2, k-1]];
Table[T[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 06 2023 *)
-
T(n,k) = if ((n==1) && (k==1), return(1)); if ((n<=0) || (k<=0) || (nMichel Marcus, Feb 01 2023
A210552
Triangle of coefficients of polynomials u(n,x) jointly generated with A210553; see the Formula section.
Original entry on oeis.org
1, 1, 2, 1, 3, 3, 1, 4, 5, 5, 1, 5, 7, 10, 8, 1, 6, 9, 16, 18, 13, 1, 7, 11, 23, 31, 33, 21, 1, 8, 13, 31, 47, 62, 59, 34, 1, 9, 15, 40, 66, 101, 119, 105, 55, 1, 10, 17, 50, 88, 151, 205, 227, 185, 89, 1, 11, 19, 61, 113, 213, 321, 414, 426, 324, 144, 1, 12, 21, 73
Offset: 1
First five rows:
1
1...2
1...3...3
1...4...5...5
1...5...7...10...8
First three polynomials u(n,x): 1, 1 + 2x, 1 + 3x + 3x^2.
-
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
v[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%] (* A210552 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A210553 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* A000225 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A094024 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A052551 *)
A302537
a(n) = (n^2 + 13*n + 2)/2.
Original entry on oeis.org
1, 8, 16, 25, 35, 46, 58, 71, 85, 100, 116, 133, 151, 170, 190, 211, 233, 256, 280, 305, 331, 358, 386, 415, 445, 476, 508, 541, 575, 610, 646, 683, 721, 760, 800, 841, 883, 926, 970, 1015, 1061, 1108, 1156, 1205, 1255, 1306, 1358, 1411, 1465, 1520, 1576
Offset: 0
Illustration of initial terms (by the formula a(n) = A052905(n) + 3*n):
. o
. o o
. o o o o
. o o o o o o
. o o o o o o o o o
. o o o o o o o o o o o o
. o o o o o o o o o o . . . . . o
. o o o o o o o . . . . o o . . . . . o
. o o o o o . . . o o . . . . o o . . . . . o
. o o o . . o o . . . o o . . . . o o . . . . . o
. o o . o o . . o o . . . o o . . . . o o . . . . . o
. o o o . o o . . o o . . . o o . . . . o o . . . . . o
. o o o o o o o o o o o o o o o o o o o o o o o o o o o o
. o o o o o o o o o o o o o o o o o o o o o
. o o o o o o o o o o o o o o o o o o o o o
. o o o o o o o o o o o o o o o o o o o o o
----------------------------------------------------------------------
. 1 8 16 25 35 46 58
- R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 1994.
Sequences whose n-th terms are of the form binomial(n, 2) + n*k + 1:
-
A302537:= func< n | ((n+1)^2 +12*n +1)/2 >;
[A302537(n): n in [0..50]]; // G. C. Greubel, Jan 21 2025
-
a := n -> (n^2 + 13*n + 2)/2;
seq(a(n), n = 0 .. 100);
-
Table[(n^2 + 13 n + 2)/2, {n, 0, 100}]
CoefficientList[ Series[(5x^2 - 5x - 1)/(x - 1)^3, {x, 0, 50}], x] (* or *)
LinearRecurrence[{3, -3, 1}, {1, 8, 16}, 51] (* Robert G. Wilson v, May 19 2018 *)
-
makelist((n^2 + 13*n + 2)/2, n, 0, 100);
-
a(n) = (n^2 + 13*n + 2)/2; \\ Altug Alkan, Apr 12 2018
-
def A302537(n): return (n**2 + 13*n + 2)//2
print([A302537(n) for n in range(51)]) # G. C. Greubel, Jan 21 2025
Showing 1-10 of 19 results.
Comments