A046691 -id:A046691 - cp's OEIS frontend

A001477 The nonnegative integers.

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

N. J. A. Sloane

Although this is a list, and lists normally have offset 1, it seems better to make an exception in this case. - N. J. A. Sloane, Mar 13 2010
The subsequence 0,1,2,3,4 gives the known values of n such that 2^(2^n)+1 is a prime (see A019434, the Fermat primes). - N. J. A. Sloane, Jun 16 2010
Also: The identity map, defined on the set of nonnegative integers. The restriction to the positive integers yields the sequence A000027. - M. F. Hasler, Nov 20 2013
The number of partitions of 2n into exactly 2 parts. - Colin Barker, Mar 22 2015
The number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 8960 or 168.- Philippe A.J.G. Chevalier, Dec 29 2015
Partial sums give A000217. - Omar E. Pol, Jul 26 2018
First differences are A000012 (the "all 1's" sequence). - M. F. Hasler, May 30 2020
See A061579 for the transposed infinite square matrix, or triangle with rows reversed. - M. F. Hasler, Nov 09 2021
This is the unique sequence (a(n)) that satisfies the inequality a(n+1) > a(a(n)) for all n in N. This simple and surprising result comes from the 6th problem proposed by Bulgaria during the second day of the 19th IMO (1977) in Belgrade (see link and reference). - Bernard Schott, Jan 25 2023

			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.

Cf. A000027 (n>=1).
Cf. A000012 (first differences).
Partial sums of A057427. - Jeremy Gardiner, Sep 08 2002
Cf. A038608 (alternating signs), A001787 (binomial transform).
Cf. A055112.
Cf. Boustrophedon transforms: A231179, A000737.
Cf. A245422.
Number of orbits of Aut(Z^7) as function of the infinity norm A000579, A154286, A102860, A002412, A045943, A115067, A008586, A008585, A005843, A000217.
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. A000290.
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

Magma
```
[ n : n in [0..100]];
```
Maple
```
[ 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 *)

PARI
```
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

a(n) = n.
a(0) = 0, a(n) = a(n-1) + 1.
G.f.: x/(1-x)^2.
Multiplicative with a(p^e) = p^e. - David W. Wilson, Aug 01 2001
When seen as array: T(k, n) = n + (k+n)*(k+n+1)/2. Main diagonal is 2*n*(n+1) (A046092), antidiagonal sums are n*(n+1)*(n+2)/2 (A027480). - Ralf Stephan, Oct 17 2004
Dirichlet generating function: zeta(s-1). - Franklin T. Adams-Watters, Sep 11 2005
E.g.f.: x*e^x. - Franklin T. Adams-Watters, Sep 11 2005
a(0)=0, a(1)=1, a(n) = 2*a(n-1) - a(n-2). - Jaume Oliver Lafont, May 07 2008
Alternating partial sums give A001057 = A000217 - 2*(A008794). - Eric Desbiaux, Oct 28 2008
a(n) = 2*A080425(n) + 3*A008611(n-3), n>1. - Eric Desbiaux, Nov 15 2009
a(n) = A007966(n)*A007967(n). - Reinhard Zumkeller, Jun 18 2011
a(n) = Sum_{k>=0} A030308(n,k)*2^k. - Philippe Deléham, Oct 20 2011
a(n) = 2*A028242(n-1) + (-1)^n*A000034(n-1). - R. J. Mathar, Jul 20 2012
a(n+1) = det(C(i+1,j), 1 <= i, j <= n), where C(n,k) are binomial coefficients. - Mircea Merca, Apr 06 2013
a(n-1) = floor(n/e^(1/n)) for n > 0. - Richard R. Forberg, Jun 22 2013
a(n) = A000027(n) for all n>0.
a(n) = floor(cot(1/(n+1))). - Clark Kimberling, Oct 08 2014
a(0)=0, a(n>0) = 2*z(-1)^[( |z|/z + 3 )/2] + ( |z|/z - 1 )/2 for z = A130472(n>0); a 1 to 1 correspondence between integers and naturals. - Adriano Caroli, Mar 29 2015
G.f. as triangle: x*(1 + (x^2 - 5*x + 2)*y + x*(2*x - 1)*y^2)/((1 - x)^3*(1 - x*y)^3). - Stefano Spezia, Jul 22 2025

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

Barry E. Williams, Jun 14 2000

If X is an n-set and Y a fixed (n-3)-subset of X then a(n-3) is equal to the number of 2-subsets of X intersecting Y. - Milan Janjic, Aug 15 2007
Bisection of A165157. - Jaroslav Krizek, Sep 05 2009
a(n) is the number of (w,x,y) having all terms in {0,...,n} and w=x+y-1. - Clark Kimberling, Jun 02 2012
Numbers m >= 0 such that 8m+25 is a square. - Bruce J. Nicholson, Jul 26 2017
a(n-1) = 3*(n-1) + (n-1)*(n-2)/2 is the number of connected, loopless, non-oriented, multi-edge vertex-labeled graphs with n edges and 3 vertices. Labeled multigraph analog of A253186. There are 3*(n-1) graphs with the 3 vertices on a chain (3 ways to label the middle graph, n-1 ways to pack edges on one of connections) and binomial(n-1,2) triangular graphs (one way to label the graphs, pack 1 or 2 or ...n-2 on the 1-2 edge, ...). - R. J. Mathar, Aug 10 2017
a(n) is also the number of vertices of the quiver for PGL_{n+1} (see Shen). - Stefano Spezia, Mar 24 2020
Starting from a(2) = 7, this is the 4th column of the array: natural numbers written by antidiagonals downwards. See the illustration by Kival Ngaokrajang and the cross-references. - Andrey Zabolotskiy, Dec 21 2021

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. A000096, A000217, A001477, A002522.
Row n=2 of A255961.
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.

[n*(n+5)/2: n in [0..50]]; // G. C. Greubel, Apr 05 2019

f[n_]:=n*(n+5)/2; f[Range[0,50]] (* Vladimir Joseph Stephan Orlovsky, Feb 10 2011 *)

a(n)=n*(n+5)/2 \\ Charles R Greathouse IV, Sep 24 2015

[n*(n+5)/2 for n in (0..50)] # G. C. Greubel, Apr 05 2019

G.f.: x*(3-2*x)/(1-x)^3.
a(n) = A027379(n), n > 0.
a(n) = A126890(n,2) for n > 1. - Reinhard Zumkeller, Dec 30 2006
a(n) = A000217(n) + A005843(n). - Reinhard Zumkeller, Sep 24 2008
If we define f(n,i,m) = Sum_{k=0..n-i} binomial(n,k)*Stirling1(n-k,i)*Product_{j=0..k-1} (-m-j), then a(n) = -f(n,n-1,3), for n >= 1. - Milan Janjic, Dec 20 2008
a(n) = A167544(n+8). - Philippe Deléham, Nov 25 2009
a(n) = a(n-1) + n + 2 with a(0)=0. - Vincenzo Librandi, Aug 07 2010
a(n) = Sum_{k=1..n} (k+2). - Gary Detlefs, Aug 10 2010
a(n) = A034856(n+1) - 1 = A000217(n+2) - 3. - Jaroslav Krizek, Sep 05 2009
Sum_{n>=1} 1/a(n) = 137/150. - R. J. Mathar, Jul 14 2012
a(n) = 3*n + A000217(n-1) = 3*n - floor(n/2) + floor(n^2/2). - Wesley Ivan Hurt, Jun 15 2013
a(n) = Sum_{i=3..n+2} i. - Wesley Ivan Hurt, Jun 28 2013
a(n) = 3*A000217(n) - 2*A000217(n-1). - Bruno Berselli, Dec 17 2014
a(n) = A046691(n) + 1. Also, a(n) = A052905(n-1) + 2 = A055999(n-1) + 3 for n>0. - Andrey Zabolotskiy, May 18 2016
E.g.f.: x*(6+x)*exp(x)/2. - G. C. Greubel, Apr 05 2019
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/5 - 47/150. - Amiram Eldar, Jan 10 2021
From Amiram Eldar, Feb 12 2024: (Start)
Product_{n>=1} (1 - 1/a(n)) = -5*cos(sqrt(33)*Pi/2)/(4*Pi).
Product_{n>=1} (1 + 1/a(n)) = 15*cos(sqrt(17)*Pi/2)/(2*Pi). (End)

A014616 a(n) = solution to the postage stamp problem with 2 denominations and n stamps.

Original entry on oeis.org

2, 4, 7, 10, 14, 18, 23, 28, 34, 40, 47, 54, 62, 70, 79, 88, 98, 108, 119, 130, 142, 154, 167, 180, 194, 208, 223, 238, 254, 270, 287, 304, 322, 340, 359, 378, 398, 418, 439, 460, 482, 504, 527, 550, 574, 598, 623, 648, 674, 700, 727, 754, 782, 810, 839, 868

Offset: 1

Eric W. Weisstein

Fred Lunnon [W. F. Lunnon] defines "solution" to be the smallest value not obtainable by the best set of stamps. The solutions given are one lower than this, that is, the sequence gives the largest number obtainable without a break using the best set of stamps.
a(n-2), for n >= 3, is the number of independent entries of a bisymmetric n X n matrix B_n with B_n[1,1] and B_n[n,n] fixed. Hence a(n-2) = A002620(n+1) - 2. See the Jul 07 2015 comment on A002620. For n=1 and n=2 this matrix B_n is fixed. Bisymmetric matrices B_n, with B_n[1,1] and B_n[n,n] fixed, are, for n >= 3, determined by giving the a(n-2) entries for [1,2], ...., [1,n-1]; [2,2], ..., [2,n-1]; [3,3], ..., [3,n-2]; ..., [ceiling(n/2),n-(ceiling(n/2)-1)]. - Wolfdieter Lang, Aug 16 2015
a(n-1) is the largest possible n-th element in an additive basis of order 2. - Charles R Greathouse IV, May 05 2020

			Bisymmetric matrix B_5, with B_5[1,1] and B_5[5,5] fixed, have a(3) free entries: for rows 1 and 2: each 3, row 3:  1, altogether 3 + 3 + 1 = 7 = a(5-2). Mark the corresponding matrix entries with x, and obtain a pattern symmetric around the central vertical. - _Wolfdieter Lang_, Aug 16 2015

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section C12, pp. 185-190.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Mario Bravo, Thierry Champion, and Roberto Cominetti, Universal bounds for fixed point iterations via optimal transport metrics, arXiv:2108.00300 [math.OC], 2021.
Erich Friedman, Postage stamp problem
W. F. Lunnon, A postage stamp problem, Comput. J. 12 (1969) 377-380.
Alfred Stöhr, Gelöste und ungelöste Fragen über Basen der natürlichen Zahlenreihe. I., Journal für die reine und angewandte Mathematik (1955), Volume: 194, page 40-65. See p. 47.
Hugh Thomas and Stephanie van Willigenburg, Compact symmetric solutions to the postage stamp problem, arXiv:0706.3250 [math.NT], 2007-2008.
Amitabha Tripathi, A Note on the Postage Stamp Problem, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.3.
Eric Weisstein's World of Mathematics, Postage stamp problem.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
Index to sequences related to the Postage Stamp Problem.

A row or column of the array A196416 (possibly with 1 subtracted from it).

Cf. A002620, A004116, A024206, A028884, A046691.

a014616 n = (n * (n + 6) + 1) `div` 4 -- Reinhard Zumkeller, Apr 07 2013

[(2*n*(n+6)-(-1)^n+1)/8: n in [1..60]]; // Vincenzo Librandi, Jul 09 2011

a[n_?OddQ] := (n^2 + 6*n + 1)/4; a[n_?EvenQ] := n*(n + 6)/4; Table[a[n], {n, 1, 56}] (* Jean-François Alcover, Dec 14 2011, after first formula *)
LinearRecurrence[{2,0,-2,1},{2,4,7,10},60] (* Harvey P. Dale, Oct 04 2015 *)

a(n)=(n^2 + 6*n + 1)\4 \\ Charles R Greathouse IV, Feb 06 2017

a(n) = floor((n^2 + 6*n + 1)/4).
a(n) = A002620(n+3) - 2 = A024206(n+2) - 1 = (2*n*(n+6)-(-1)^n+1)/8.
G.f.: x*(-2 + x^2)/((1 + x)*(x - 1)^3). - R. J. Mathar, Jul 09 2011
a(n) = floor(A028884(n+1)/4). - Reinhard Zumkeller, Apr 07 2013
a(n)+a(n+1) = A046691(n+1). - R. J. Mathar, Mar 13 2021
a(n) = 2*n + A002620(n-1). - Michael Chu, Apr 28 2022
a(n) = A004116(n) + 1. - Michael Chu, May 02 2022
E.g.f.: (x*(7 + x)*cosh(x) + (1 + 7*x + x^2)*sinh(x))/4. - Stefano Spezia, Nov 09 2022
Sum_{n>=1} 1/a(n) = 67/36 - cot(sqrt(2)*Pi)*Pi/(2*sqrt(2)). - Amiram Eldar, Dec 10 2022

Entry improved by comments from John Seldon (johnseldon(AT)onetel.com), Sep 15 2004

More terms from John W. Layman, Apr 13 1999

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

Clark Kimberling, Feb 03 2011

This is one of many interesting sequences and arrays that stem from the natural number array A000027, of which a northwest corner is as follows:
  1....2.....4.....7...11...16...22...29...
  3....5.....8....12...17...23...30...38...
  6....9....13....18...24...31...39...48...
  10...14...19....25...32...40...49...59...
  15...20...26....33...41...50...60...71...
  21...27...34....42...51...61...72...84...
  28...35...43....52...62...73...85...98...
Blocking out all terms below the main diagonal leaves columns whose sums comprise A185787.  Deleting the main diagonal and then summing give A185787.  Analogous treatments to the left of the main diagonal give A100182 and A101165.  Further sequences obtained directly from this array are easily obtained using the following formula for the array: T(n,k)=n+(n+k-2)(n+k-1)/2.
Examples:
row 1:  A000124
row 2:  A022856
row 3:  A016028
row 4:  A145018
row 5:  A077169
col 1:  A000217
col 2:  A000096
col 3:  A034856
col 4:  A055998
col 5:  A046691
col 6:  A052905
col 7:  A055999
diag. (1,5,...) ...... A001844
diag. (2,8,...) ...... A001105
diag. (4,12,...)...... A046092
diag. (7,17,...)...... A056220
diag. (11,23,...) .... A132209
diag. (16,30,...) .... A054000
diag. (22,38,...) .... A090288
diag. (3,9,...) ...... A058331
diag. (6,14,...) ..... A051890
diag. (10,20,...) .... A005893
diag. (15,27,...) .... A097080
diag. (21,35,...) .... A093328
antidiagonal sums:  (1,5,15,34,...)=A006003=partial sums of A002817.
Let S(n,k) denote the n-th partial sum of column k.  Then
S(n,k)=n*(n^2+3k*n+3*k^2-6*k+5)/6.
S(n,1)=n(n+1)(n+2)/6
S(n,2)=n(n+1)(n+5)/6
S(n,3)=n(n+2)(n+7)/6
S(n,4)=n(n^2+12n+29)/6
S(n,5)=n(n+5)(n+10)/6
S(n,6)=n(n+7)(n+11)/6
S(n,7)=n(n+10)(n+11)/6
Weight array of T: A144112
Accumulation array of T: A185506
Second rectangular sum array of T: A185507
Third rectangular sum array of T: A185508
Fourth rectangular sum array of T: A185509

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000027, A185788, A100182, A101165, A079824.

[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 *)

a(n)=n*(7*n^2-6*n+5)/6.

G.f.: x*(3*x^2+3*x+1)/(1-x)^4. - Vincenzo Librandi, Jul 04 2012

Edited by Clark Kimberling, Feb 25 2023

A219883 T(n,k)=Number of nXk arrays of the minimum value of corresponding elements and their horizontal, diagonal or antidiagonal neighbors in a random, but sorted with lexicographically nondecreasing rows and nonincreasing columns, 0..2 nXk array.

Original entry on oeis.org

3, 3, 6, 6, 7, 10, 10, 11, 21, 15, 15, 17, 47, 46, 21, 21, 24, 129, 146, 87, 28, 28, 32, 292, 621, 410, 151, 36, 36, 41, 600, 2190, 2645, 1069, 247, 45, 45, 51, 1158, 6965, 14536, 10350, 2701, 386, 55, 55, 62, 2148, 21035, 74998, 89795, 40239, 6645, 581, 66, 66, 74

Offset: 1

R. H. Hardin Nov 30 2012

Table starts
..3....3......6......10........15........21.........28.........36........45
..6....7.....11......17........24........32.........41.........51........62
.10...21.....47.....129.......292.......600.......1158.......2148......3863
.15...46....146.....621......2190......6965......21035......60891....171009
.21...87....410....2645.....14536.....74998.....371384....1780796...8327951
.28..151...1069...10350.....89795....764027....6337533...51044999.398508546
.36..247...2701...40239....565824...8017398..113313853.1558416287
.45..386...6645..155199...3605746..85092182.2047979807
.55..581..15787..581728..22430565.882578607
.66..847..36047.2085519.132945658
.78.1201..79071.7121374
.91.1662.166909

			Some solutions for n=3 k=4
..1..1..0..0....2..2..0..0....1..0..0..0....1..0..0..0....0..0..0..0
..1..1..0..1....2..2..0..0....0..0..0..0....1..0..0..0....0..0..0..0
..1..1..0..0....2..2..0..0....0..0..0..2....2..0..2..0....0..2..0..2

R. H. Hardin, Table of n, a(n) for n = 1..111

Column 1 is A000217(n+1)
Column 2 is A219589
Row 1 is A000217 for n>1
Row 2 is A046691 for n>2

A337484 Number of ordered triples of positive integers summing to n that are neither strictly increasing nor strictly decreasing.

Original entry on oeis.org

0, 0, 0, 1, 3, 6, 8, 13, 17, 22, 28, 35, 41, 50, 58, 67, 77, 88, 98, 111, 123, 136, 150, 165, 179, 196, 212, 229, 247, 266, 284, 305, 325, 346, 368, 391, 413, 438, 462, 487, 513, 540, 566, 595, 623, 652, 682, 713, 743, 776, 808, 841, 875, 910, 944, 981, 1017

Offset: 0

Gus Wiseman, Sep 11 2020

nonn

			The a(3) = 1 through a(7) = 13 triples:
  (1,1,1)  (1,1,2)  (1,1,3)  (1,1,4)  (1,1,5)
           (1,2,1)  (1,2,2)  (1,3,2)  (1,3,3)
           (2,1,1)  (1,3,1)  (1,4,1)  (1,4,2)
                    (2,1,2)  (2,1,3)  (1,5,1)
                    (2,2,1)  (2,2,2)  (2,1,4)
                    (3,1,1)  (2,3,1)  (2,2,3)
                             (3,1,2)  (2,3,2)
                             (4,1,1)  (2,4,1)
                                      (3,1,3)
                                      (3,2,2)
                                      (3,3,1)
                                      (4,1,2)
                                      (5,1,1)

A140106 is the unordered case.
A242771 allows strictly increasing but not strictly decreasing triples.
A337481 counts these compositions of any length.
A001399(n - 6) counts unordered strict triples.
A001523 counts unimodal compositions, with complement A115981.
A007318 and A097805 count compositions by length.
A069905 counts unordered triples.
A218004 counts strictly increasing or weakly decreasing compositions.
A332745 counts partitions with weakly increasing or weakly decreasing run-lengths.
A332835 counts compositions with weakly increasing or weakly decreasing run-lengths.
A337483 counts triples either weakly increasing or weakly decreasing.
Cf. A000212, A000217, A001840, A014311, A046691, A128422, A156040, A332834, A337461, A337482, A337561, A337603, A337604.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],!Less@@#&&!Greater@@#&]],{n,0,15}]

a(n) = 2*A242771(n - 1) - A000217(n - 1), n > 0.
2*A001399(n - 6) = 2*A069905(n - 3) = 2*A211540(n - 1) is the complement.
4*A001399(n - 6) = 4*A069905(n - 3) = 4*A211540(n - 1) is the strict case.
Conjectures from Colin Barker, Sep 13 2020: (Start)
G.f.: x^3*(1 + 2*x + 2*x^2 - x^3) / ((1 - x)^3*(1 + x)*(1 + x + x^2)).
a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6) for n>6.
(End)

A027379 Expansion of (1+x^2-x^3)/(1-x)^3.

Original entry on oeis.org

1, 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

Offset: 0

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Essentially the triangular numbers (A000217) minus 3.
Also essentially the same as A055998.
Cf. A000217, A034856, A046691.

[n eq 0 select 1 else n*(n+5)/2: n in [0..50]]; // G. C. Greubel, Jul 31 2022

a(n)=if(n,n*(n+5)/2,1) \\ Charles R Greathouse IV, Jun 11 2015

[n*(n+5)/2 + bool(n==0) for n in (0..50)] # G. C. Greubel, Jul 31 2022

a(n) = n*(n+5)/2 = A000217(n+2) - 3 for n>=1. - Emeric Deutsch, Mar 01 2004
a(n) = n + a(n-1) + 2, n>1. - Vincenzo Librandi, Dec 06 2009
E.g.f.: 1 + x*(x + 6)*exp(x)/2. - G. C. Greubel, May 14 2017

A209268 Inverse permutation A054582.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 10, 7, 15, 9, 21, 8, 28, 14, 36, 11, 45, 20, 55, 13, 66, 27, 78, 12, 91, 35, 105, 19, 120, 44, 136, 16, 153, 54, 171, 26, 190, 65, 210, 18, 231, 77, 253, 34, 276, 90, 300, 17, 325, 104, 351, 43, 378, 119, 406, 25, 435, 135, 465, 53, 496, 152

Offset: 1

Boris Putievskiy, Jan 15 2013

nonn

Permutation of the natural numbers.

a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.

			The start of the sequence for n = 1..32 as table, distributed by exponent of highest power of 2 dividing n:
   |   Exponent of highest power of 2 dividing n
n  |--------------------------------------------------
   |    0      1      2       3      4         5    ...
------------------------------------------------------
1  |....1
2  |...........2
3  |....3
4  |..................4
5  |....6
6  |...........5
7  |...10
8  |..........................7
9  |...15
10 |...........9
11 |...21
12 |..................8
13 |...28
14 |..........14
15 |...36
16 |................................11
17 |...45
18 |..........20
19 |...55
20 |.................13
21 |...66
22 |..........27
23 |...78
24 |................................12
25 |...91
26 |..........35
27 |..105
28 |.................19
29 |..120
30 |..........44
31 |..136
32 |.........................................16
. . .
Let r_c be number row inside the column number c.
r_c = (n+2^c)/2^(c+1).
The column number 0 contains numbers r_0*(r_0+1)/2,     A000217,
The column number 1 contains numbers r_1*(r_1+3)/2,     A000096,
The column number 2 contains numbers r_2*(r_2+5)/2 + 1, A034856,
The column number 3 contains numbers r_3*(r_3+7)/2 + 3, A055998,
The column number 4 contains numbers r_4*(r_4+9)/2 + 6, A046691.

Boris Putievskiy, Table of n, a(n) for n = 1..10000
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.
R. J. Mathar, oeisPy
Eric W. Weisstein, MathWorld: Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A054582, A003602, A007814, A000217, A000096, A034856, A055998, A046691, A014132.

a[n_] := (v = IntegerExponent[n, 2]; (1/2)*(((1/2)*(n/2^v + 1) + v)^2 + (1/2)*(n/2^v + 1) - v)); Table[a[n], {n, 1, 55}] (* Jean-François Alcover, Jan 15 2013, from 1st formula *)

f = open("result.csv", "w")
def A007814(n):
### author        Richard J. Mathar 2010-09-06 (Start)
### http://oeis.org/wiki/User:R._J._Mathar/oeisPy/oeisPy/oeis_bulk.py
        a = 0
        nshft = n
        while (nshft %2 == 0):
                a += 1
                nshft >>= 1
        return a
###(End)
for  n in range(1,10001):
     x = A007814(n)
     y = (n+2**x)/2**(x+1)
     m = ((x+y)**2-x+y)/2
     f.write('%d;%d;%d;%d;\n' % (n, x, y, m))
f.close()

a(n) = (((A003602)+A007814(n))^2 - A007814(n) + A003602(n))/2.

a(n) = ((x+y)^2-x+y)/2, where x = max {k: 2^k | n}, y = (n+2^x)/2^(x+1).

A046726 Triangle of numbers of semi-meanders of order n with k components.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 6, 4, 1, 4, 11, 16, 10, 1, 5, 17, 37, 48, 24, 1, 6, 24, 66, 126, 140, 66, 1, 7, 32, 104, 254, 430, 428, 174, 1, 8, 41, 152, 438, 956, 1454, 1308, 504, 1, 9, 51, 211, 690, 1796, 3584, 4976, 4072, 1406, 1, 10, 62, 282, 1023, 3028, 7238, 13256, 16880, 12796, 4210

Offset: 1

N. J. A. Sloane

Rows are in order of decreasing number of components. Diagonals give number of semi-meanders with k components. - Andrew Howroyd, Nov 27 2015

			Triangle starts:
  1;
  1, 1;
  1, 2,  2;
  1, 3,  6,   4;
  1, 4, 11,  16,  10;
  1, 5, 17,  37,  48,  24;
  1, 6, 24,  66, 126, 140,   66;
  1, 7, 32, 104, 254, 430,  428,  174;
  1, 8, 41, 152, 438, 956, 1454, 1308, 504;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..820
P. Di Francesco, O. Golinelli, and E. Guitter, Meander, folding and arch statistics, arXiv:hep-th/9506030, 1995.
P. Di Francesco, O. Golinelli, and E. Guitter, Meander, folding and arch statistics, Mathematical and Computer Modelling 26 (1997), 97-147.

Diagonals include A000682, A046721, A046722, A046723, A046724, A046725. Columns include A000027, A046691. Row sums are in A000108 (Catalan numbers).

More terms from Larry Reeves (larryr(AT)acm.org), Apr 05 2000

T(12,k)-T(40,k) from Andrew Howroyd, Dec 07 2015

A371912 Maximum Zagreb index of maximal 3-degenerate graphs with n vertices.

Original entry on oeis.org

12, 36, 66, 102, 144, 192, 246, 306, 372, 444, 522, 606, 696, 792, 894, 1002, 1116, 1236, 1362, 1494, 1632, 1776, 1926, 2082, 2244, 2412, 2586, 2766, 2952, 3144, 3342, 3546, 3756, 3972, 4194, 4422, 4656, 4896, 5142, 5394, 5652, 5916, 6186, 6462, 6744, 7032, 7326, 7626, 7932

Offset: 3

Allan Bickle, Apr 11 2024

The Zagreb index of a graph is the sum of the squares of the degrees over all vertices of the graph.

A maximal 3-degenerate graph can be constructed from a 3-clique by iteratively adding a new 3-leaf (vertex of degree 3) adjacent to three existing vertices. The extremal graphs are 3-stars, so the bound also applies to 3-trees.

			The graph K_3 has 3 degree 2 vertices, so a(3) = 3*4 = 12.

Paolo Xausa, Table of n, a(n) for n = 3..10000
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
Allan Bickle, Zagreb Indices of Maximal k-degenerate Graphs, Australas. J. Combin. 89 1 (2024) 167-178.
J. Estes and B. Wei, Sharp bounds of the Zagreb indices of k-trees, J Comb Optim 27 (2014), 271-291.
I. Gutman and K. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50 (2004), 83-92.
A. Hou, S. Li, L. Song, and B. Wei, Sharp bounds for Zagreb indices of maximal outerplanar graphs, J Comb Optim 22 (2011), 252-269.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A002378, A152811, A371912 (Zagreb indices of maximal k-degenerate graphs).

Array[3*(#^2 + # - 8) &, 50, 3] (* Paolo Xausa, Jun 09 2024 *)

a(n) = 3*(n-1)^2 + 9*(n-3).
a(n) = 6*A046691(n-2) for n>2.
a(n) = 6*A060577(n-1) for n>3.
G.f.: 6*x^3*(2 - x^2)/(1 - x)^3. - Stefano Spezia, Apr 12 2024
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 5. - Chai Wah Wu, Apr 16 2024
Sum_{n>=3} 1/a(n) = 19/72 + Pi*tan(Pi*sqrt(33)/2)*sqrt(33)/99 = 0.1865497.... - R. J. Mathar, Apr 22 2024

cp's OEIS Frontend

A001477 The nonnegative integers.

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A055998 a(n) = n*(n+5)/2.

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A014616 a(n) = solution to the postage stamp problem with 2 denominations and n stamps.

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A185787 Sum of first k numbers in column k of the natural number array A000027; by antidiagonals.

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A219883 T(n,k)=Number of nXk arrays of the minimum value of corresponding elements and their horizontal, diagonal or antidiagonal neighbors in a random, but sorted with lexicographically nondecreasing rows and nonincreasing columns, 0..2 nXk array.

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A337484 Number of ordered triples of positive integers summing to n that are neither strictly increasing nor strictly decreasing.

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A027379 Expansion of (1+x^2-x^3)/(1-x)^3.

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