A161700 -id:A161700 - cp's OEIS frontend

A027750 Triangle read by rows in which row n lists the divisors of n.

1, 1, 2, 1, 3, 1, 2, 4, 1, 5, 1, 2, 3, 6, 1, 7, 1, 2, 4, 8, 1, 3, 9, 1, 2, 5, 10, 1, 11, 1, 2, 3, 4, 6, 12, 1, 13, 1, 2, 7, 14, 1, 3, 5, 15, 1, 2, 4, 8, 16, 1, 17, 1, 2, 3, 6, 9, 18, 1, 19, 1, 2, 4, 5, 10, 20, 1, 3, 7, 21, 1, 2, 11, 22, 1, 23, 1, 2, 3, 4, 6, 8, 12, 24, 1, 5, 25, 1, 2, 13, 26, 1, 3, 9, 27, 1, 2, 4, 7, 14, 28, 1, 29

Offset: 1

N. J. A. Sloane

Or, in the list of natural numbers (A000027), replace n with its divisors.
This gives the first elements of the ordered pairs (a,b) a >= 1, b >= 1 ordered by their product ab.
Also, row n lists the largest parts of the partitions of n whose parts are not distinct. - Omar E. Pol, Sep 17 2008
Concatenation of n-th row gives A037278(n). - Reinhard Zumkeller, Aug 07 2011
{A210208(n,k): k=1..A073093(n)} subset of {T(n,k): k=1..A000005(n)} for all n. - Reinhard Zumkeller, Mar 18 2012
Row sums give A000203. Right border gives A000027. - Omar E. Pol, Jul 29 2012
Indices of records are in A006218. - Irina Gerasimova, Feb 27 2013
The number of primes in the n-th row is omega(n) = A001221(n). - Michel Marcus, Oct 21 2015
The row polynomials P(n,x) = Sum_{k=1..A000005(n)} T(n,k)*x^k with composite n which are irreducible over the integers are given in A292226. - Wolfdieter Lang, Nov 09 2017
T(n,k) is also the number of parts in the k-th partition of n into equal parts (see example). - Omar E. Pol, Nov 20 2019
Let there be an infinite number of tiles, each labeled with a positive integer m, initially placed on square m of an infinite 1D board. At step n, the leftmost unblocked tile (i.e., the top tile of the leftmost nonempty stack) moves forward exactly m squares, where m is its label. Tiles that land on the same square form a stack, and only the top tile of any stack may move. This sequence records the label m of the tile that moves at step n. - Ali Sada, May 23 2025
All divisors of a positive integer n form a finite set. Extending divisibility to n = 0 by using the definition (k|n <=> exists m such that m*k = n) makes the set of divisors infinite, suggesting the definition was not intended for zero, as arithmetic functions typically apply to n >= 1. So to preserve a core property when generalizing (cardinality), one can define divisors of n >= 0 as the fixed points of the greatest common divisor on the set [n] = {0, 1, 2, ..., n}. By this definition, the divisors of 0 are {0}, since 0|0 and gcd(0, 0) = 0. This definition is not circular because the gcd can be effectively calculated using the Euclidean algorithm. (Cf. links.) - Peter Luschny, Jun 02 2025

			Triangle begins:
  1;
  1, 2;
  1, 3;
  1, 2, 4;
  1, 5;
  1, 2, 3, 6;
  1, 7;
  1, 2, 4, 8;
  1, 3, 9;
  1, 2, 5, 10;
  1, 11;
  1, 2, 3, 4, 6, 12;
  ...
For n = 6 the partitions of 6 into equal parts are [6], [3,3], [2,2,2], [1,1,1,1,1,1], so the number of parts are [1, 2, 3, 6] respectively, the same as the divisors of 6. - _Omar E. Pol_, Nov 20 2019

Franklin T. Adams-Watters, Rows 1..1000, flattened
Franklin T. Adams-Watters, Rows 1..10000
Peter Luschny, A Python program using gcd for n >= 0.
Omar E. Pol, Illustration of initial terms, (2009).
Eric Weisstein's World of Mathematics, Divisor
Wikipedia, Table of divisors
Index entries for sequences related to divisors of numbers

Cf. A000005 (row length), A001221, A027749, A027751, A056534, A056538, A127093, A135010, A161700, A163280, A240698 (partial sums of rows), A240694 (partial products of rows), A247795 (parities), A292226, A244051.

a027750 n k = a027750_row n !! (k-1)
a027750_row n = filter ((== 0) . (mod n)) [1..n]
a027750_tabf = map a027750_row [1..]
-- Reinhard Zumkeller, Jan 15 2011, Oct 21 2010

Magma
```
[Divisors(n) : n in [1..20]];
```

seq(op(numtheory:-divisors(a)), a = 1 .. 20) # Matt C. Anderson, May 15 2017

Flatten[ Table[ Flatten [ Divisors[ n ] ], {n, 1, 30} ] ]

v=List();for(n=1,20,fordiv(n,d,listput(v,d)));Vec(v) \\ Charles R Greathouse IV, Apr 28 2011

from sympy import divisors
for n in range(1, 16):
    print(divisors(n)) # Indranil Ghosh, Mar 30 2017

a(A006218(n-1) + k) = k-divisor of n, 1 <= k <= A000005(n). - Reinhard Zumkeller, May 10 2006

T(n,k) = n / A056538(n,k) = A056538(n,n-k+1), 1 <= k <= A000005(n). - Reinhard Zumkeller, Sep 28 2014

More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu)

A002522 a(n) = n^2 + 1.

Original entry on oeis.org

1, 2, 5, 10, 17, 26, 37, 50, 65, 82, 101, 122, 145, 170, 197, 226, 257, 290, 325, 362, 401, 442, 485, 530, 577, 626, 677, 730, 785, 842, 901, 962, 1025, 1090, 1157, 1226, 1297, 1370, 1445, 1522, 1601, 1682, 1765, 1850, 1937, 2026, 2117, 2210, 2305, 2402, 2501

Offset: 0

N. J. A. Sloane

An n X n nonnegative matrix A is primitive (see A070322) iff every element of A^k is > 0 for some power k. If A is primitive then the power which should have all positive entries is <= n^2 - 2n + 2 (Wielandt).
a(n) = Phi_4(n), where Phi_k is the k-th cyclotomic polynomial.
As the positive solution to x=2n+1/x is x=n+sqrt(a(n)), the continued fraction expansion of sqrt(a(n)) is {n; 2n, 2n, 2n, 2n, ...}. - Benoit Cloitre, Dec 07 2001
a(n) is one less than the arithmetic mean of its neighbors: a(n) = (a(n-1) + a(n+1))/2 - 1. E.g., 2 = (1+5)/2 - 1, 5 = (2+10)/2 - 1. - Amarnath Murthy, Jul 29 2003
Equivalently, the continued fraction expansion of sqrt(a(n)) is (n;2n,2n,2n,...). - Franz Vrabec, Jan 23 2006
Number of {12,1*2*,21}-avoiding signed permutations in the hyperoctahedral group.
The number of squares of side 1 which can be drawn without lifting the pencil, starting at one corner of an n X n grid and never visiting an edge twice is n^2-2n+2. - Sébastien Dumortier, Jun 16 2005
Also, numbers m such that m^3 - m^2 is a square, (n*(1 + n^2))^2. - Zak Seidov
1 + 2/2 + 2/5 + 2/10 + ... = Pi*coth Pi [Jolley], see A113319. - Gary W. Adamson, Dec 21 2006
For n >= 1, a(n-1) is the minimal number of choices from an n-set such that at least one particular element has been chosen at least n times or each of the n elements has been chosen at least once. Some games define "matches" this way; e.g., in the classic Parker Brothers, now Hasbro, board game Risk, a(2)=5 is the number of cards of three available types (suits) required to guarantee at least one match of three different types or of three of the same type (ignoring any jokers or wildcards). - Rick L. Shepherd, Nov 18 2007
Positive X values of solutions to the equation X^3 + (X - 1)^2 + X - 2 = Y^2. To prove that X = n^2 + 1: Y^2 = X^3 + (X - 1)^2 + X - 2 = X^3 + X^2 - X - 1 = (X - 1)(X^2 + 2X + 1) = (X - 1)*(X + 1)^2 it means: (X - 1) must be a perfect square, so X = n^2 + 1 and Y = n(n^2 + 2). - Mohamed Bouhamida, Nov 29 2007
{a(k): 0 <= k < 4} = divisors of 10. - Reinhard Zumkeller, Jun 17 2009
Appears in A054413 and A086902 in relation to sequences related to the numerators and denominators of continued fractions convergents to sqrt((2*n)^2/4 + 1), n=1, 2, 3, ... . - Johannes W. Meijer, Jun 12 2010
For n > 0, continued fraction [n,n] = n/a(n); e.g., [5,5] = 5/26. - Gary W. Adamson, Jul 15 2010
The only real solution of the form f(x) = A*x^p with negative p which satisfies f^(m)(x) = f^[-1](x), x >= 0, m >= 1, with f^(m) the m-th derivative and f^[-1] the compositional inverse of f, is obtained for m=2*n, p=p(n)= -(sqrt(a(n))-n) and A=A(n)=(fallfac(p(n),2*n))^(-p(n)/(p(n)+1)), with fallfac(x,k):=Product_{j=0..k-1} (x-j) (falling factorials). See the T. Koshy reference, pp. 263-4 (there are also two solutions for positive p, see the corresponding comment in A087475). - Wolfdieter Lang, Oct 21 2010
n + sqrt(a(n)) = [2*n;2*n,2*n,...] with the regular continued fraction with period 1. This is the even case. For the general case see A087475 with the Schroeder reference and comments. For the odd case see A078370.
a(n-1) counts configurations of non-attacking bishops on a 2 X n strip [Chaiken et al., Ann. Combin. 14 (2010) 419]. - R. J. Mathar, Jun 16 2011
Also numbers k such that 4*k-4 is a square. Hence this sequence is the union of A053755 and A069894. - Arkadiusz Wesolowski, Aug 02 2011
a(n) is also the Moore lower bound on the order, A191595(n), of an (n,5)-cage. - Jason Kimberley, Oct 17 2011
Left edge of the triangle in A195437: a(n+1) = A195437(n,0). - Reinhard Zumkeller, Nov 23 2011
If h (5,17,37,65,101,...) is prime is relatively prime to 6, then h^2-1 is divisible by 24. - Vincenzo Librandi, Apr 14 2014
The identity (4*n^2+2)^2 - (n^2+1)*(4*n)^2 = 4 can be written as A005899(n)^2 - a(n)*A008586(n)^2 = 4. - Vincenzo Librandi, Jun 15 2014
a(n) is also the number of permutations simultaneously avoiding 213 and 321 in the classical sense which can be realized as labels on an increasing strict binary tree with 2n-1 nodes. See A245904 for more information on increasing strict binary trees. - Manda Riehl, Aug 07 2014
a(n-1) is the maximum number of stages in the Gale-Shapley algorithm for finding a stable matching between two sets of n elements given an ordering of preferences for each element (see Gura et al.). - Melvin Peralta, Feb 07 2016
Because of Fermat's little theorem, a(n) is never divisible by 3. - Altug Alkan, Apr 08 2016
For n > 0, if a(n) points are placed inside an n X n square, it will always be the case that at least two of the points will be a distance of sqrt(2) units apart or less. - Melvin Peralta, Jan 21 2017
Also the limit as q->1^- of the unimodal polynomial (1-q^(n*k+1))/(1-q) after making the simplification k=n. The unimodal polynomial is from O'Hara's proof of unimodality of q-binomials after making the restriction to partitions of size <= 1. See G_1(n,k) from arXiv:1711.11252. As the size restriction s increases, G_s->G_infinity=G: the q-binomials. Then substituting k=n and q=1 yields the central binomial coefficients: A000984. - Bryan T. Ek, Apr 11 2018
a(n) is the smallest number congruent to both 1 (mod n) and 2 (mod n+1). - David James Sycamore, Apr 04 2019
a(n) is the number of permutations of 1,2,...,n+1 with exactly one reduced decomposition. - Richard Stanley, Dec 22 2022
From Klaus Purath, Apr 03 2025: (Start)
The odd prime factors of these terms are always of the form 4*k + 1.
All a(n) = D satisfy the Pell equation (k*x)^2 - D*y^2 = -1. The values for k and the solutions x, y can be calculated using the following algorithm: k = n, x(0) = 1, x(1) = 4*D - 1, y(0) = 1, y(1) = 4*D - 3. The two recurrences are of the form (4*D - 2, -1). The solutions x, y of the Pell equations for n = {1 ... 14} are in OEIS.
It follows from the above that this sequence is a subsequence of A031396. (End)

			G.f. = 1 + 2*x + 5*x^2 + 10*x^3 + 17*x^4 + 26*x^5 + 37*x^6 + 50*x^7 + 65*x^8 + ...

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 120).
E. Gura and M. Maschler, Insights into Game Theory: An Alternative Mathematical Experience, Cambridge, 2008; p. 26.
Thomas Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley and Sons, New York, 2001.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000. Format corrected by _Peter Kagey_, Jan 25 2016
Wawrzyniec Bieniawski, Piotr Masierak, Andrzej Tomski, and Szymon Łukaszyk, On the Salient Regularities of Strings of Assembly Theory, Preprints (2024). See p. 19.
Wawrzyniec Bieniawski, Piotr Masierak, Andrzej Tomski, and Szymon Łukaszyk, Assembly Theory - Formalizing Assembly Spaces and Discovering Patterns and Bounds, Preprints.org (2025). See p. 28.
R. P. Boas and N. J. A. Sloane, Correspondence, 1974
Yurii S. Bystryk, Vitalii L. Denysenko, and Volodymyr I. Ostryk, Lune and Lens Sequences, ResearchGate preprint, 2024. See pp. 44, 56.
Giulio Cerbai and Luca Ferrari, Permutation patterns in genome rearrangement problems: the reversal model, arXiv:1903.08774 [math.CO], 2019. See p. 19.
S. Chaiken et al., Nonattacking Queens in a Rectangular Strip, arXiv:1105.5087 [math.CO], 2011.
Bryan Ek, Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics, arXiv:1804.05933 [math.CO], 2018.
R. M. Green and Tianyuan Xu, 2-roots for simply laced Weyl groups, arXiv:2204.09765 [math.RT], 2022.
Guo-Niu Han, Enumeration of Standard Puzzles
Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]
Cheyne Homberger, Patterns in Permutations and Involutions: A Structural and Enumerative Approach, arXiv:1410.2657 [math.CO], 2014.
C. Homberger and V. Vatter, On the effective and automatic enumeration of polynomial permutation classes, arXiv:1308.4946 [math.CO], 2013.
L. B. W. Jolley, Summation of Series, Dover, 1961, p. 176.
S. J. Leon, Linear Algebra with Applications: the Perron-Frobenius Theorem [Cached copy at the Wayback Machine]
T. Mansour and J. West, Avoiding 2-letter signed patterns, arXiv:math/0207204 [math.CO], 2002.
Michelle Rudolph-Lilith, On the Product Representation of Number Sequences, with Application to the Fibonacci Family, arXiv:1508.07894 [math.NT], 2015.
Eric Weisstein's World of Mathematics, Number Picking
Eric Weisstein's World of Mathematics, Near-Square Prime
Helmut Wielandt, Unzerlegbare, nicht negative Matrizen, Math. Z. 52 (1950), 642-648. volume 52
Reinhard Zumkeller, Enumerations of Divisors
Index to values of cyclotomic polynomials of integer argument
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Left edge of A055096.
Cf. A059100, A117950, A087475, A117951, A114949, A117619 (sequences of form n^2 + K).
a(n+1) = A101220(n, n+1, 3).
Cf. A059592, A124808, A132411, A132414, A028872, A005408, A000124, A016813, A086514, A000125, A058331, A080856, A000127, A161701-A161704, A161706, A161707, A161708, A161710-A161713, A161715, A006261.
Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306 (k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), this sequence (g=5), A051890 (g=6), A188377 (g=7). - Jason Kimberley, Oct 30 2011
Cf. A002496 (primes).
Cf. A254858.
Cf. A302612, A302644, A302645, A302646.
Subsequence of A031396.

a002522 = (+ 1) . (^ 2)
a002522_list = scanl (+) 1 [1,3..]
-- Reinhard Zumkeller, Apr 06 2012

[n^2 + 1: n in [0..50]]; // Vincenzo Librandi, May 01 2011

A002522 := proc(n)
        numtheory[cyclotomic](4,n) ;
end proc:
seq(A002522(n),n=0..20) ; # R. J. Mathar, Feb 07 2014

Table[n^2 + 1, {n, 0, 50}]; (* Vladimir Joseph Stephan Orlovsky, Dec 15 2008 *)

A002522(n):=n^2+1$ makelist(A002522(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

a(n)=n^2+1 \\ Charles R Greathouse IV, Jun 10 2011

O.g.f.: (1-x+2*x^2)/((1-x)^3). - Eric Werley, Jun 27 2011
Sequences of the form a(n) = n^2 + K with offset 0 have o.g.f. (K - 2*K*x + K*x^2 + x + x^2)/(1-x)^3 and recurrence a(n) = 3*a(n-1) - 3*a(n-2) + a*(n-3). - R. J. Mathar, Apr 28 2008
For n > 0: a(n-1) = A143053(A000290(n)) - 1. - Reinhard Zumkeller, Jul 20 2008
A143053(a(n)) = A000290(n+1). - Reinhard Zumkeller, Jul 20 2008
a(n)*a(n-2) = (n-1)^4 + 4. - Reinhard Zumkeller, Feb 12 2009
a(n) = A156798(n)/A087475(n). - Reinhard Zumkeller, Feb 16 2009
From Reinhard Zumkeller, Mar 08 2010: (Start)
a(n) = A170949(A002061(n+1));
A170949(a(n)) = A132411(n+1);
A170950(a(n)) = A002061(n+1). (End)
For n > 1, a(n)^2 + (a(n) + 1)^2 + ... + (a(n) + n - 2)^2 + (a(n) + n - 1 + a(n) + n)^2 = (n+1) *(6*n^4 + 18*n^3 + 26*n^2 + 19*n + 6) / 6 = (a(n) + n)^2 + ... + (a(n) + 2*n)^2. - Charlie Marion, Jan 10 2011
From Eric Werley, Jun 27 2011: (Start)
a(n) = 2*a(n-1) - a(n-2) + 2.
a(n) = a(n-1) + 2*n - 1. (End)
a(n) = (n-1)^2 + 2(n-1) + 2 = 122 read in base n-1 (for n > 3). - Jason Kimberley, Oct 20 2011
a(n)*a(n+1) = a(n*(n+1) + 1) so a(1)*a(2) = a(3). More generally, a(n)*a(n+k) = a(n*(n+k) + 1) + k^2 - 1. - Jon Perry, Aug 01 2012
a(n) = (n!)^2* [x^n] BesselI(0, 2*sqrt(x))*(1+x). - Peter Luschny, Aug 25 2012
a(n) = A070216(n,1) for n > 0. - Reinhard Zumkeller, Nov 11 2012
E.g.f.: exp(x)*(1 + x + x^2). - Geoffrey Critzer, Aug 30 2013
a(n) = A254858(n-2,3) for n > 2. - Reinhard Zumkeller, Feb 09 2015
Sum_{n>=0} (-1)^n / a(n) = (1+Pi/sinh(Pi))/2 = 0.636014527491... = A367976 . - Vaclav Kotesovec, Feb 14 2015
Sum_{n>=0} 1/a(n) = (1 + Pi*coth(Pi))/2 = 2.076674... = A113319. - Vaclav Kotesovec, Apr 10 2016
4*a(n) = A001105(n-1) + A001105(n+1). - Bruno Berselli, Jul 03 2017
From Amiram Eldar, Jan 20 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi)*sinh(sqrt(2)*Pi).
Product_{n>=1} (1 - 1/a(n)) = Pi*csch(Pi). (End)
Sum_{n>=0} a(n)/n! = 3*e. - Davide Rotondo, Feb 16 2025

Partially edited by Joerg Arndt, Mar 11 2010

A006218 a(n) = Sum_{k=1..n} floor(n/k); also Sum_{k=1..n} d(k), where d = number of divisors (A000005); also number of solutions to x*y = z with 1 <= x,y,z <= n.

Original entry on oeis.org

0, 1, 3, 5, 8, 10, 14, 16, 20, 23, 27, 29, 35, 37, 41, 45, 50, 52, 58, 60, 66, 70, 74, 76, 84, 87, 91, 95, 101, 103, 111, 113, 119, 123, 127, 131, 140, 142, 146, 150, 158, 160, 168, 170, 176, 182, 186, 188, 198, 201, 207, 211, 217, 219, 227, 231, 239, 243, 247, 249

Offset: 0

N. J. A. Sloane

The identity Sum_{k=1..n} floor(n/k) = Sum_{k=1..n} d(k) is Equation (10), p. 58, of Apostol (1976). - N. J. A. Sloane, Dec 06 2020
The "Dirichlet divisor problem" is to find a precise asymptotic estimate for this sequence - see formula lines below, also Apostol (1976), Chap. 3.
Number of increasing arithmetic progressions where n+1 is the second or later term. - Mambetov Timur, Takenov Nurdin, Haritonova Oksana (timus(AT)post.kg; oksanka-61(AT)mail.ru), Jun 13 2002. E.g., a(3) = 5 because there are 5 such arithmetic progressions: (1, 2, 3, 4); (2, 3, 4); (1, 4); (2, 4); (3, 4).
Binomial transform of A001659.
Area covered by overlapped partitions of n, i.e., sum of maximum values of the k-th part of a partition of n into k parts. - Jon Perry, Sep 08 2005
Equals inverse Mobius transform of A116477. - Gary W. Adamson, Aug 07 2008
The Polymath project (see the Tao-Croot-Helfgott link) sketches an algorithm for computing a(n) in essentially cube root time, see section 2.1. - Charles R Greathouse IV, Oct 10 2010 [Sladkey gives another. - Charles R Greathouse IV, Oct 02 2017]
The Dirichlet inverse starts (offset 1) 1, -3, -5, 1, -10, 16, -16, 1, 2, 33, -29, -6, -37, 55, 55, -1, -52, -5, -60, ... - R. J. Mathar, Oct 17 2012
The inverse Mobius transforms yields A143356. - R. J. Mathar, Oct 17 2012
An improved approximation vs. Dirichlet is: a(n) = log(Gamma(n+1)) + 2n*gamma. Using sample ranges of {n = k^2-k to k^2 + (k-1)} the means of the new error term are < +- 0.5 up to k=150, except on two values of k. These ranges appear to give means closest to zero for such small sample sizes. It is not clear sample means remain < +- 0.5 at larger k. The standard deviations are ~(n*log(n))^(1/4)/2, with n near sample range center. - Richard R. Forberg, Jan 06 2015
The values of n for which a(n) is even are given by 4*m^2 <= n <= 4*m(m+1) for m >= 0. Example: for m=1 the values of n are 4 <= n <= 8 for which a(4) to a(8) are even. - G. C. Greubel, Sep 30 2015
For n > 0, a(n) = count(x|y), 1 <= y <= x <= n, that is, the number of pairs in the ordered list of x and y, where y divides x, up to and including n. - Torlach Rush, Jan 31 2017
a(n) is also the total number of partitions of all positive integers <= n into equal parts. - Omar E. Pol, May 29 2017
a(n) is the rank of the join of the set of elements of rank n in Young's lattice, the lattice of all integer partitions ordered by inclusion of their Ferrers diagrams. - Geoffrey Critzer, Jul 11 2018
a(n) always has the same parity as floor(sqrt(n)) = A000196(n): see A211264 (proof in Diophante link). - Bernard Schott, Feb 13 2021
From Omar E. Pol, Feb 16 2021: (Start)
Apart from initial zero this is the convolution of A341062 and A000027.
Nonzero terms convolved with A341062 gives A055507. (End)
From Bernard Schott, Apr 17 2022: (Start)
a(n-1) is the number of lattice points in the first quadrant lying under the hyperbola x*y = n, excluding the lattice points on the axes.
a(n) is the number of lattice points in the first quadrant lying on or under the hyperbola x*y = n, excluding the lattice points on the axes. (Reference Hari Kishan). (End)
Let tiles Tn (for n >= 1) be initially placed on square n on an infinite 1D board. At each step, the leftmost unblocked tile (i.e., the top tile in the leftmost stack) jumps forward exactly n squares. Tiles can stack, and only the top tile of a stack can move. This sequence gives the step number when tile n moves for the first time. - Ali Sada, May 23 2025

			a(3) = 5 because 3 + floor(3/2) + 1 = 3 + 1 + 1 = 5. Or tau(1) + tau(2) + tau(3) = 1 + 2 + 2 = 5.
a(4) = 8 because 4 + floor(4/2) + floor(4/3) + 1 = 4 + 2 + 1 + 1 = 8. Or
tau(1) + tau(2) + tau(3) + tau(4) = 1 + 2 + 2 + 3 = 8.
a(5) = 10 because 5 + floor(5/2) + floor(5/3) + floor (5/4) + 1 = 5 + 2 + 1 + 1 + 1 = 10. Or tau(1) + tau(2) + tau(3) + tau(4) + tau(5) = 1 + 2 + 2 + 3 + 2 = 10.

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976.
K. Chandrasekharan, Introduction to Analytic Number Theory. Springer-Verlag, 1968, Chap. VI.
K. Chandrasekharan, Arithmetical Functions. Springer-Verlag, 1970, Chapter VIII, pp. 194-228. Springer-Verlag, Berlin.
P. G. L. Dirichlet, Werke, Vol. ii, pp. 49-66.
M. N. Huxley, The Distribution of Prime Numbers, Oxford Univ. Press, 1972, p. 7.
M. N. Huxley, Area, Lattice Points and Exponential Sums, Oxford, 1996; p. 239.
Hari Kishan, Number Theory, Krishna, Educational Publishers, 2014, Theorem 1, p. 133.
H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 56.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Nurdin N. Takenov and Oksana Haritonova, Representation of positive integers by a special set of digits and sequences, in Dolmatov, S. L. et al. editors, Materials of Science, Practical seminar "Modern Mathematics".
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Exercise 3.6.13 on page 107.

N. J. A. Sloane, Table of n, a(n) for n = 0..20000 (first 1000 terms from T. D. Noe)
Dorin Andrica and Ovidiu Bagdasar, On some results concerning the polygonal polynomials, Carpathian Journal of Mathematics (2019) Vol. 35, No. 1, 1-11.
D. Andrica and E. J. Ionascu, On the number of polynomials with coefficients in [n], An. St. Univ. Ovidius Constanța, Vol. 22(1),2014, 13-23.
R. Bellman and H. N. Shapiro, On a problem in additive number theory, Annals Math., 49 (1948), 333-340. See Eq. 1.5.
D. Berkane, O. Bordellès, and O. Ramaré, Explicit upper bounds for the remainder term in the divisor problem, Math. Comp. 81:278 (2012), pp. 1025-1051.
Peter J. Cameron and Hamid Reza Dorbidi, Minimal cover groups, arXiv:2311.15652 [math.GR], 2023. See p. 13.
Diophante, A1712, La même parité (in French).
Xiaoxi Duan and M. W. Wong, The Dirichlet divisor problem, traces and determinants for complex powers of the twisted bi-Laplacian, J. of Math. Analysis and Applications, Volume 410, Issue 1, Feb 01 2014, Pages 151-157
L. Hoehn and J. Ridenhour, Summations involving computer-related functions, Math. Mag., 62 (1989), 191-196.
M. N. Huxley, Exponential Sums and Lattice Points III, Proc. London Math. Soc., 87 (2003), pp. 591-609.
Vaclav Kotesovec, Graph - The asymptotic ratio (1000000 terms)
Richard Sladkey, A successive approximation algorithm for computing the divisor summatory function, arXiv:1206.3369 [math.NT], 2012.
Terence Tao, Ernest Croot III, and Harald Helfgott, Deterministic methods to find primes, Math. Comp. 81 (2012), 1233-1246; also at arXiv:1009.3956 [math.NT], 2010-2012.

Right edge of A056535. Cf. A000005, A001659, A052511, A143236.
Row sums of triangle A003988, A010766 and A143724.
A061017 is an inverse.
It appears that the partial sums give A078567. - N. J. A. Sloane, Nov 24 2008
Cf. A116477, A051731, A055507, A161700, A004125, A212120, A341062.

a006218 n = sum $ map (div n) [1..n]
-- Reinhard Zumkeller, Jan 29 2011

[0] cat [&+[Floor(n/k):k in [1..n]]:n in [1..60]]; // Marius A. Burtea, Aug 25 2019

with(numtheory): A006218 := n->add(sigma[0](i), i=1..n);

Table[Sum[DivisorSigma[0, k], {k, n}], {n, 70}]
FoldList[Plus, 0, Table[DivisorSigma[0, x], {x, 61}]] //Rest (* much faster *)
Join[{0},Accumulate[DivisorSigma[0,Range[60]]]] (* Harvey P. Dale, Jan 06 2016 *)

PARI
```
a(n)=sum(k=1,n,n\k)
```

a(n)=sum(k=1,sqrtint(n),n\k)*2-sqrtint(n)^2 \\ Charles R Greathouse IV, Oct 10 2010

from sympy import integer_nthroot
def A006218(n): return 2*sum(n//k for k in range(1,integer_nthroot(n,2)[0]+1))-integer_nthroot(n,2)[0]**2 # Chai Wah Wu, Mar 29 2021

a(n) = n * ( log(n) + 2*gamma - 1 ) + O(sqrt(n)), where gamma is the Euler-Mascheroni number ~ 0.57721... (see A001620), Dirichlet, 1849. Again, a(n) = n * ( log(n) + 2*gamma - 1 ) + O(log(n)*n^(1/3)). The determination of the precise size of the error term is an unsolved problem (the so-called Dirichlet divisor problem) - see references, especially Huxley (2003).
The bounds from Chandrasekharan lead to the explicit bounds n log(n) + (2 gamma - 1) n - 4 sqrt(n) - 1 <= a(n) <= n log(n) + (2 gamma - 1) n + 4 sqrt(n). - David Applegate, Oct 14 2008
a(n) = 2*(Sum_{i=1..floor(sqrt(n))} floor(n/i)) - floor(sqrt(n))^2. - Benoit Cloitre, May 12 2002
G.f.: (1/(1-x))*Sum_{k >= 1} x^k/(1-x^k). - Benoit Cloitre, Apr 23 2003
For n > 0: A027750(a(n-1) + k) = k-divisor of n, = k <= A000005(n). - Reinhard Zumkeller, May 10 2006
a(n) = A161886(n) - n + 1 = A161886(n-1) - A049820(n) + 2 = A161886(n-1) + A000005(n) - n + 2 = A006590(n) + A000005(n) - n = A006590(n+1) - n - 1 = A006590(n) + A000005(n) - n for n >= 2. a(n) = a(n-1) + A000005(n) for n >= 1. - Jaroslav Krizek, Nov 14 2009
D(n) = Sum_{m >= 2, r >= 1} (r/m^(r+1)) * Sum_{j = 1..m - 1} * Sum_{k = 0 .. m^(r+1) - 1} exp{ 2*k*pi i(p^n + (m - j)m^r) / m^(r+1) } where p is some fixed prime number. - A. Neves, Oct 04 2010
Let E(n) = a(n) - n(log n + 2 gamma - 1). Then Berkane-Bordellès-Ramaré show that |E(n)| <= 0.961 sqrt(n), |E(n)| <= 0.397 sqrt(n) for n > 5559, and |E(n)| <= 0.764 n^(1/3) log n for x > 9994. - Charles R Greathouse IV, Jul 02 2012
a(n) = Sum_{k = 1..floor(sqrt(n))} A005408(floor((n/k) - (k-1))). - Gregory R. Bryant, Apr 20 2013
Dirichlet g.f. for s > 2: Sum_{n>=1} a(n)/n^s = Sum_{k>=1} (Zeta(s-1) - Sum_{n=1..k-1} (HurwitzZeta(s,n/k)*n/k^s))/k. - Mats Granvik, Sep 24 2017
From Ridouane Oudra, Dec 31 2022: (Start)
a(n) = n^2 - Sum_{i=1..n} Sum_{j=1..n} floor(log(i*j)/log(n+1));
a(n) = floor(sqrt(n)) + 2*Sum_{i=1..n} floor((sqrt(i^2 + 4*n) - i)/2);
a(n) = n + Sum_{i=1..n} v_2(i)*round(n/i), where v_2(i) = A007814(i). (End)

A016921 a(n) = 6*n + 1.

Original entry on oeis.org

1, 7, 13, 19, 25, 31, 37, 43, 49, 55, 61, 67, 73, 79, 85, 91, 97, 103, 109, 115, 121, 127, 133, 139, 145, 151, 157, 163, 169, 175, 181, 187, 193, 199, 205, 211, 217, 223, 229, 235, 241, 247, 253, 259, 265, 271, 277, 283, 289, 295, 301, 307, 313, 319, 325, 331

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 22 ).
Also solutions to 2^x + 3^x == 5 (mod 7). - Cino Hilliard, May 10 2003
Except for 1, exponents n > 1 such that x^n - x^2 - 1 is reducible. - N. J. A. Sloane, Jul 19 2005
Let M(n) be the n X n matrix m(i,j) = min(i,j); then the trace of M(n)^(-2) is a(n-1) = 6*n - 5. - Benoit Cloitre, Feb 09 2006
If Y is a 3-subset of an (2n+1)-set X then, for n >= 3, a(n-1) is the number of 3-subsets of X having at least two elements in common with Y. - Milan Janjic, Dec 16 2007
All composite terms belong to A269345 as shown in there. - Waldemar Puszkarz, Apr 13 2016
First differences of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 773", based on the 5-celled von Neumann neighborhood. - Robert Price, May 23 2016
For b(n) = A103221(n) one has b(a(n)-1) = b(a(n)+1) = b(a(n)+2) = b(a(n)+3) = b(a(n)+4) = n+1 but b(a(n)) = n. So-called "dips" in A103221. See the Avner and Gross remark on p. 178. - Wolfdieter Lang, Sep 16 2016
A (n+1,n) pebbling move involves removing n + 1 pebbles from a vertex in a simple graph and placing n pebbles on an adjacent vertex. A two-player impartial (n+1,n) pebbling game involves two players alternating (n+1,n) pebbling moves. The first player unable to make a move loses. The sequence a(n) is also the minimum number of pebbles such that any assignment of those pebbles on a complete graph with 3 vertices is a next-player winning game in the two player impartial (k+1,k) pebbling game. These games are represented by A347637(3,n). - Joe Miller, Oct 18 2021
Interleaving of A017533 and A017605. - Leo Tavares, Nov 16 2021

			From _Ilya Gutkovskiy_, Apr 15 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
n=0  n=1     n=2     n=3
(End)

Avner Ash and Robert Gross, Summing it up, Princeton University Press, 2016, p. 178.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Kayla Barker, Mia DeStefano, Eugene Fiorini, Michael Gohn, Joe Miller, Jacob Roeder, and Tony W. H. Wong, Generalized Impartial Two-player Pebbling Games on K_3 and C_4, J. Int. Seq. (2024) Vol. 27, Issue 5, Art. No. 24.5.8. See p. 3.
Tanya Khovanova, Recursive Sequences.
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N)).
William A. Stein, The modular forms database.
Leo Tavares, Illustration: Hexagonal Lines.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A093563 ((6, 1) Pascal, column m=1).
Cf. A000567 (partial sums), A002476 (primes), A005408, A008588, A016813, A016933, A016945, A016957, A017281, A017533, A128470, A158057, A161700, A161705, A161709, A161714.
a(n) = A007310(2*(n+1)); complement of A016969 with respect to A007310.
Cf. A287326 (second column).
Cf. A000217, A003215.
Cf. A001477, A017605.

List([0..60], n-> 6*n+1); # G. C. Greubel, Sep 18 2019

a016921 = (+ 1) . (* 6)
a016921_list = [1, 7 ..]  -- Reinhard Zumkeller, Jan 15 2013

[6*n+1: n in [0..60]]; // G. C. Greubel, Sep 18 2019

a[1]:=1:for n from 2 to 100 do a[n]:=a[n-1]+6 od: seq(a[n], n=1..56); # Zerinvary Lajos, Mar 16 2008

Range[1, 500, 6] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)

a(n)=6*n+1 \\ Charles R Greathouse IV, Mar 22 2016

for n in range(0,10**5):print(6*n+1) # Soumil Mandal, Apr 14 2016

a(n) = 6*n + 1, n >= 0 (see the name).
G.f.: (1+5*x)/(1-x)^2.
A008615(a(n)) = n. - Reinhard Zumkeller, Feb 27 2008
A157176(a(n)) = A013730(n). - Reinhard Zumkeller, Feb 24 2009
a(n) = 4*(3*n-1) - a(n-1) (with a(0)=1). - Vincenzo Librandi, Nov 20 2010
E.g.f.: (1 + 6*x)*exp(x). - G. C. Greubel, Sep 18 2019
a(n) = A003215(n) - 6*A000217(n-1). See Hexagonal Lines illustration. - Leo Tavares, Sep 10 2021
From Leo Tavares, Oct 27 2021: (Start)
a(n) = 6*A001477(n-1) + 7
a(n) = A016813(n) + 2*A001477(n)
a(n) = A017605(n-1) + A008588(n-1)
a(n) = A016933(n) - 1
a(n) = A008588(n) + 1. (End)
Sum_{n>=0} (-1)^n/a(n) = Pi/6 + sqrt(3)*arccoth(sqrt(3))/3. - Amiram Eldar, Dec 10 2021

A058331 a(n) = 2*n^2 + 1.

Original entry on oeis.org

1, 3, 9, 19, 33, 51, 73, 99, 129, 163, 201, 243, 289, 339, 393, 451, 513, 579, 649, 723, 801, 883, 969, 1059, 1153, 1251, 1353, 1459, 1569, 1683, 1801, 1923, 2049, 2179, 2313, 2451, 2593, 2739, 2889, 3043, 3201, 3363, 3529, 3699, 3873, 4051

Offset: 0

Erich Friedman, Dec 12 2000

Maximal number of regions in the plane that can be formed with n hyperbolas.
Also the number of different 2 X 2 determinants with integer entries from 0 to n.
Number of lattice points in an n-dimensional ball of radius sqrt(2). - David W. Wilson, May 03 2001
Equals A112295(unsigned) * [1, 2, 3, ...]. - Gary W. Adamson, Oct 07 2007
Binomial transform of A166926. - Gary W. Adamson, May 03 2008
a(n) = longest side a of all integer-sided triangles with sides a <= b <= c and inradius n >= 1. Triangle has sides (2n^2 + 1, 2n^2 + 2, 4n^2 + 1).
{a(k): 0 <= k < 3} = divisors of 9. - Reinhard Zumkeller, Jun 17 2009
Number of ways to partition a 3*n X 2 grid into 3 connected equal-area regions. - R. H. Hardin, Oct 31 2009
Let A be the Hessenberg matrix of order n defined by: A[1, j] = 1, A[i, i] := 2, (i > 1), A[i, i - 1] = -1, and A[i, j] = 0 otherwise. Then, for n >= 3, a(n - 1) = coeff(charpoly(A, x), x^(n - 2)). - Milan Janjic, Jan 26 2010
Except for the first term of [A002522] and [A058331] if X = [A058331], Y = [A087113], A = [A002522], we have, for all other terms, Pell's equation: [A058331]^2 - [A002522]*[A087113]^2 = 1; (X^2 - A*Y^2 = 1); e.g., 3^2 -2*2^2 = 1; 9^2 - 5*4^2 = 1; 129^2 - 65*16^2 = 1, and so on. - Vincenzo Librandi, Aug 07 2010
Niven (1961) gives this formula as an example of a formula that does not contain all odd integers, in contrast to 2n + 1 and 2n - 1. - Alonso del Arte, Dec 05 2012
Numbers m such that 2*m-2 is a square. - Vincenzo Librandi, Apr 10 2015
Number of n-tuples from the set {1,0,-1} where at most two elements are nonzero. - Michael Somos, Oct 19 2022
a(n) gives the x-value of the integral solution (x,y) of the Pellian equation x^2 - (n^2 + 1)*y^2 = 1. The y-value is given by 2*n (see Tattersall). - Stefano Spezia, Jul 23 2025

			a(1) = 3 since (0 0 / 0 0), (1 0 / 0 1) and (0 1 / 1 0) have different determinants.
G.f. = 1 + 3*x + 9*x^2 + 19*x^3 + 33*x^4 + 51*x^5 + 73*x^6 + ... - _Michael Somos_, Oct 19 2022

Ivan Niven, Numbers: Rational and Irrational, New York: Random House for Yale University (1961): 17.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 256.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Steven Edwards and William Griffiths, On Generalized Delannoy Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.3.6.
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
Leo Tavares, Illustration: Triangular Outlines
Reinhard Zumkeller, Enumerations of Divisors.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000124.
Second row of array A099597.
See A120062 for sequences related to integer-sided triangles with integer inradius n.
Cf. A112295.
Cf. A087113, A002552.
Cf. A005408, A016813, A086514, A000125, A002522, A161701, A161702, A161703, A000127, A161704, A161706, A161707, A161708, A161710, A080856, A161711, A161712, A161713, A161715, A006261.
Cf. A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A173121.
Column 2 of array A188645.
Cf. A001105 and A247375. - Bruno Berselli, Sep 16 2014
Cf. A056106, A251599.
Cf. A000384, A000217, A166926.

a058331 = (+ 1) . a001105  -- Reinhard Zumkeller, Dec 13 2014

[2*n^2 + 1 : n in [0..100]]; // Wesley Ivan Hurt, Feb 02 2017

b[g_] := Length[Union[Map[Det, Flatten[ Table[{{i, j}, {k, l}}, {i, 0, g}, {j, 0, g}, {k, 0, g}, {l, 0, g}], 3]]]] Table[b[g], {g, 0, 20}]
2*Range[0, 49]^2 + 1 (* Alonso del Arte, Dec 05 2012 *)

a(n)=2*n^2+1 \\ Charles R Greathouse IV, Jun 16 2011

G.f.: (1 + 3x^2)/(1 - x)^3. - Paul Barry, Apr 06 2003
a(n) = M^n * [1 1 1], leftmost term, where M = the 3 X 3 matrix [1 1 1 / 0 1 4 / 0 0 1]. a(0) = 1, a(1) = 3; a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). E.g., a(4) = 33 since M^4 *[1 1 1] = [33 17 1]. - Gary W. Adamson, Nov 11 2004
a(n) = cosh(2*arccosh(n)). - Artur Jasinski, Feb 10 2010
a(n) = 4*n + a(n-1) - 2 for n > 0, a(0) = 1. - Vincenzo Librandi, Aug 07 2010
a(n) = (((n-1)^2 + n^2))/2 + (n^2 + (n+1)^2)/2. - J. M. Bergot, May 31 2012
a(n) = A251599(3*n) for n > 0. - Reinhard Zumkeller, Dec 13 2014
a(n) = sqrt(8*(A000217(n-1)^2 + A000217(n)^2) + 1). - J. M. Bergot, Sep 03 2015
E.g.f.: (2*x^2 + 2*x + 1)*exp(x). - G. C. Greubel, Jul 14 2017
a(n) = A002378(n) + A002061(n). - Bruce J. Nicholson, Aug 06 2017
From Amiram Eldar, Jul 15 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + (Pi/sqrt(2))*coth(Pi/sqrt(2)))/2.
Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/sqrt(2))*csch(Pi/sqrt(2)))/2. (End)
From Amiram Eldar, Feb 05 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/sqrt(2))*sinh(Pi).
Product_{n>=1} (1 - 1/a(n)) = (Pi/sqrt(2))*csch(Pi/sqrt(2)). (End)
From Leo Tavares, May 23 2022: (Start)
a(n) = A000384(n+1) - 3*n.
a(n) = 3*A000217(n) + A000217(n-2). (End)
a(n) = a(-n) for all n in Z and A037235(n) = Sum_{k=0..n-1} a(k). - Michael Somos, Oct 19 2022

Revised description from Noam Katz (noamkj(AT)hotmail.com), Jan 28 2001

A000125 Cake numbers: maximal number of pieces resulting from n planar cuts through a cube (or cake): C(n+1,3) + n + 1.

Original entry on oeis.org

1, 2, 4, 8, 15, 26, 42, 64, 93, 130, 176, 232, 299, 378, 470, 576, 697, 834, 988, 1160, 1351, 1562, 1794, 2048, 2325, 2626, 2952, 3304, 3683, 4090, 4526, 4992, 5489, 6018, 6580, 7176, 7807, 8474, 9178, 9920, 10701, 11522, 12384, 13288, 14235, 15226

Offset: 0

N. J. A. Sloane

Note that a(n) = a(n-1) + A000124(n-1). This has the following geometrical interpretation: Define a number of planes in space to be in general arrangement when
(1) no two planes are parallel,
(2) there are no two parallel intersection lines,
(3) there is no point common to four or more planes.
Suppose there are already n-1 planes in general arrangement, thus defining the maximal number of regions in space obtainable by n-1 planes and now one more plane is added in general arrangement. Then it will cut each of the n-1 planes and acquire intersection lines which are in general arrangement. (See the comments on A000124 for general arrangement with lines.) These lines on the new plane define the maximal number of regions in 2-space definable by n-1 straight lines, hence this is A000124(n-1). Each of this regions acts as a dividing wall, thereby creating as many new regions in addition to the a(n-1) regions already there, hence a(n) = a(n-1) + A000124(n-1). - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006
More generally, we have: A000027(n) = binomial(n,0) + binomial(n,1) (the natural numbers), A000124(n) = binomial(n,0) + binomial(n,1) + binomial(n,2) (the Lazy Caterer's sequence), a(n) = binomial(n,0) + binomial(n,1) + binomial(n,2) + binomial(n,3) (Cake Numbers). - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006
If Y is a 2-subset of an n-set X then, for n>=3, a(n-3) is the number of 3-subsets of X which do not have exactly one element in common with Y. - Milan Janjic, Dec 28 2007
a(n) is the number of compositions (ordered partitions) of n+1 into four or fewer parts or equivalently the sum of the first four terms in the n-th row of Pascal's triangle. - Geoffrey Critzer, Jan 23 2009
{a(k): 0 <= k < 4} = divisors of 8. - Reinhard Zumkeller, Jun 17 2009
a(n) is also the maximum number of different values obtained by summing n consecutive positive integers with all possible 2^n sign combinations. This maximum is first reached when summing the interval [n, 2n-1]. - Olivier Gérard, Mar 22 2010
a(n) contains only 5 perfect squares > 1: 4, 64, 576, 67600, and 75203584. The incidences of > 0 are given by A047694. - Frank M Jackson, Mar 15 2013
Given n tiles with two values - an A value and a B value - a player may pick either the A value or the B value. The particular tiles are [n, 0], [n-1, 1], ..., [2, n-2] and [1, n-1]. The sequence is the number of different final A:B counts. For example, with n=4, we can have final total [5, 3] = [4, ] + [, 1] + [, 2] + [1, ] = [, 0] + [3, ] + [2, ] + [, 3], so a(4) = 2^4 - 1 = 15. The largest and smallest final A+B counts are given by A077043 and A002620 respectively. - Jon Perry, Oct 24 2014
For n>=3, a(n) is also the number of maximal cliques in the (n+1)-triangular graph (the 4-triangular graph has a(3)=8 maximal cliques). - Andrew Howroyd, Jul 19 2017
a(n) is the number of binary words of length n matching the regular expression 1*0*1*0*. Coincidentally, A000124 counts binary words of the form 0*1*0*. See Alexandersson and Nabawanda for proof. - Per W. Alexandersson, May 15 2021
For n > 0, let the n-dimensional cube, {0,1}^n be provided with the Hamming distance, d. Given an element x in {0,1}^n, a(n) is the number of elements y in {0,1}^n such that d(x, y) <= 3. Example: n = 4. Let x = (0,0,0,0) be in {0,1}^4.
d(x,y) = 0: y in {(0,0,0,0)}.
d(x,y) = 1: y in {(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1)}.
d(x,y) = 2: y in {(1,1,0,0), (1,0,1,0), (1,0,0,1), (0,1,1,0), (0,1,0,1), (0,0,1,1)}.
d(x,y) = 3: y in {(1,1,1,0), (1,1,0,1), (1,0,1,1), (0,1,1,1)}.
All these y are at a distance <= 3 from (0,0,0,0), so a(4) = 15. (See Peter C. Heinig's formula). - Yosu Yurramendi, Dec 14 2021
For n >= 2, a(n) is the number of distinct least squares regression lines fitted to n points (j,y_j), 1 <= j <= n, where each y_j is 0 or 1. The number of distinct lines with exactly k 1's among y_1, ..., y_n is A077028(n,k). The number of distinct slopes is A123596(n). - Pontus von Brömssen, Mar 16 2024
The only powers of 2 in this sequence are a(0) = 1, a(1) = 2, a(2) = 4, a(3) = 8, and a(7) = 64. - Jianing Song, Jan 02 2025

			a(4)=15 because there are 15 compositions of 5 into four or fewer parts. a(6)=42 because the sum of the first four terms in the 6th row of Pascal's triangle is 1+6+15+20=42. - _Geoffrey Critzer_, Jan 23 2009
For n=5, (1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 35) and their opposite are the 26 different sums obtained by summing 5,6,7,8,9 with any sign combination. - _Olivier Gérard_, Mar 22 2010
G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 15*x^4 + 26*x^5 + 42*x^6 + 64*x^7 + ... - _Michael Somos_, Jul 07 2022

V. I. Arnold (ed.), Arnold's Problems, Springer, 2004, comments on Problem 1990-11 (p. 75), pp. 503-510. Numbers N_3.
R. B. Banks, Slicing Pizzas, Racing Turtles and Further Adventures in Applied Mathematics, Princeton Univ. Press, 1999. See p. 27.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 80.
H. E. Dudeney, Amusements in Mathematics, Nelson, London, 1917, page 177.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
T. H. Stickels, Mindstretching Puzzles. Sterling, NY, 1994 p. 85.
W. A. Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 30.
A. M. Yaglom and I. M. Yaglom: Challenging Mathematical Problems with Elementary Solutions. Vol. I. Combinatorial Analysis and Probability Theory. New York: Dover Publications, Inc., 1987, p. 13, #45 (First published: San Francisco: Holden-Day, Inc., 1964)

T. D. Noe, Table of n, a(n) for n = 0..1000
P. Alexandersson and O. Nabawanda, Peaks are preserved under run-sorting, arXiv:2104.04220 [math.CO], 2021.
Mohamadou Bachabi and Alain S. Togbé, Products of Fermat or Mersenne numbers in some sequences, Math. Comm. (2024) Vol. 29, 265-285.
A. M. Baxter and L. K. Pudwell, Ascent sequences avoiding pairs of patterns, 2014.
M. L. Cornelius, Variations on a geometric progression, Mathematics in School, 4 (No. 3, May 1975), p. 32. (Annotated scanned copy)
F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.
R. K. Guy, Letter to N. J. A. Sloane
Zachary Hoelscher, Semicomplete Arithmetic Sequences, Division of Hypercubes, and the Pell Constant, arXiv:2102.07083 [math.NT], 2021. Mentions this sequence.
Marie Lejeune, On the k-binomial equivalence of finite words and k-binomial complexity of infinite words, Ph. D. Thesis, Université de Liège (Belgium, 2021).
D. A. Lind, On a class of nonlinear binomial sums, Fib. Quart., 3 (1965), 292-298.
Svante Linusson, The number of M-sequences and f-vectors, Combinatorica, vol 19 no 2 (1999) 255-266.
Toufik Mansour, Howard Skogman, and Rebecca Smith, Sorting inversion sequences, arXiv:2401.06662 [math.CO], 2024. See page 7.
R. J. Mathar, The number of binary mxm matrices with at most k 1's in each row or column, (2014) Table 3 column 1.
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Sebastian Mizera and Sabrina Pasterski, Celestial Geometry, arXiv:2204.02505 [hep-th], 2022.
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
D. J. Price, Some unusual series occurring in n-dimensional geometry, Math. Gaz., 30 (1946), 149-150.
L. Pudwell and A. Baxter, Ascent sequences avoiding pairs of patterns, 2014.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
H. P. Robinson, Letter to N. J. A. Sloane, Aug 16 1971, with attachments
Eric Weisstein's World of Mathematics, Cake Number
Eric Weisstein's World of Mathematics, Cube Division by Planes
Eric Weisstein's World of Mathematics, Cylinder Cutting
Eric Weisstein's World of Mathematics, Maximal Clique
Eric Weisstein's World of Mathematics, Space Division by Planes
Eric Weisstein's World of Mathematics, Triangular Graph
Reinhard Zumkeller, Enumerations of Divisors
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Bisections give A100503, A100504.
Row sums of A077028.
Cf. A000124, A003600, A005408, A016813, A086514, A058331, A002522, A161701-A161705, A000127, A161706-A161708, A080856, A161710-A161713, A161715, A006261, A063865, A051601, A077043, A002620, A123596.
Cf. A000027, A047694, A134396, A152947, A283551.

[(n^3+5*n+6)/6: n in [0..50]]; // Vincenzo Librandi, Nov 08 2014

Maple
```
A000125 := n->(n+1)*(n^2-n+6)/6;
```

Table[(n^3 + 5 n + 6)/6, {n, 0, 50}] (* Harvey P. Dale, Jan 19 2013 *)
LinearRecurrence[{4, -6, 4, -1}, {1, 2, 4, 8}, 50] (* Harvey P. Dale, Jan 19 2013 *)
Table[Binomial[n, 3] + n, {n, 20}] (* Eric W. Weisstein, Jul 21 2017 *)

a(n)=(n^2+5)*n/6+1 \\ Charles R Greathouse IV, Jun 15 2011

Vec((1-2*x+2*x^2)/((1-x)^4) + O(x^100)) \\ Altug Alkan, Oct 16 2015

def A000125_gen():  # generator of terms
    a, b, c = 1, 1, 1
    while True:
        yield a
        a, b, c = a+b, b+c, c+1
it = A000125_gen()
A000125_list = [next(it) for  in range(50)]  # _Cole Dykstra, Aug 03 2022

a(n) = (n+1)*(n^2-n+6)/6 = (n^3 + 5*n + 6) / 6.
G.f.: (1 - 2*x + 2x^2)/(1-x)^4. - [Simon Plouffe in his 1992 dissertation.]
E.g.f.: (1 + x + x^2/2 + x^3/6)*exp(x).
a(n) = binomial(n,3) + binomial(n,2) + binomial(n,1) + binomial(n,0). - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006
Paraphrasing the previous comment: the sequence is the binomial transform of [1,1,1,1,0,0,0,...]. - Gary W. Adamson, Oct 23 2007
From Ilya Gutkovskiy, Jul 18 2016: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = Sum_{k=0..n} A152947(k+1).
Inverse binomial transform of A134396.
Sum_{n>=0} a(n)/n! = 8*exp(1)/3. (End)
a(n) = -A283551(-n). - Michael Somos, Jul 07 2022
a(n) = A046127(n+1)/2 = A033547(n)/2 + 1. - Jianing Song, Jan 02 2025

Minor typo in comments corrected by Mauro Fiorentini, Jan 02 2018

A017281 a(n) = 10*n + 1.

Original entry on oeis.org

1, 11, 21, 31, 41, 51, 61, 71, 81, 91, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 201, 211, 221, 231, 241, 251, 261, 271, 281, 291, 301, 311, 321, 331, 341, 351, 361, 371, 381, 391, 401, 411, 421, 431, 441, 451, 461, 471, 481, 491, 501, 511, 521, 531

Offset: 0

N. J. A. Sloane

Equals [1, 2, 3, ...] convolved with [1, 9, 0, 0, 0, ...]. - Gary W. Adamson, May 30 2009
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=10, (i>1), A[i,i-1] = -1, and A[i,j]=0 otherwise. Then, for n>=2, a(n-1) = -coeff(charpoly(A,x),x^(n-1)). - Milan Janjic, Feb 21 2010
Positive integers with last decimal digit = 1. - Wesley Ivan Hurt, Jun 17 2015
Also the number of (not necessarily maximal) cliques in the 2n-crossed prism graph. - Eric W. Weisstein, Nov 29 2017
From Martin Renner, May 28 2024: (Start)
Also number of squares in a grid cross with equally long arms and a width of two points (cf. A017113), e.g. for n = 2 there are nine squares of size 1 unit of area, four of size 2, two of size 5, four of size 8 and two of size 13, thus a total of 21 squares.
    · ·           · ·           · ·           · ·           * ·
    · ·           · ·           * ·           * ·           · ·
* * · · · ·   · · * · · ·   · · · · * ·   · · · · · ·   · · · · · *
* * · · · ·   · * · * · ·   · * · · · ·   * · · · * ·   * · · · · ·
    · ·           * ·           · *           · ·           · ·
    · ·           · ·           · ·           * ·           · *
The possible areas of the squares are given by ceiling(k^2/2) for 1 <= k <= 2*n+1, cf. A000982. In general, there are 4*n + 1 squares with one unit area to be found in the cross, cf. A016813, for n > 0 always four squares of even area and two squares of odd area > 1. (End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Milan Janjic, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8.
Tanya Khovanova, Recursive Sequences
Eric Weisstein's World of Mathematics, Clique
Eric Weisstein's World of Mathematics, Crossed Prism Graph
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A093645 (column 1).
Subsequence of A034709, together with A017293, A017329, A139222, A139245, A139249, A139264, A139279 and A139280.
Cf. A048161, A154428.
Cf. A005408, A016813, A016921, A017533, A161700, A128470, A158057, A161705, A161709, A161714.
Cf. A030430 (primes).
Cf. A272914, first comment. [Bruno Berselli, May 26 2016]

List([0..60], n-> 10*n+1 ); # G. C. Greubel, Sep 18 2019

a017281 = (+ 1) . (* 10)
a017281_list = [1,11..]  -- Reinhard Zumkeller, Apr 16 2012

[10*n+1 : n in [0..60]]; // Zaki Khandaker, May 16 2015

A017281:=n->10*n + 1; seq(A017281(n), n=0..80); # Wesley Ivan Hurt, Jan 29 2014

f[n_] := FromDigits[IntegerDigits[n^2, n + 1]]; Array[f, 60] (* Robert G. Wilson v, Apr 14 2009 *)
Range[1, 1000, 10] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)
(* From Eric W. Weisstein, Nov 29 2017: (Start) *)
Table[10n+1, {n, 0, 60}]
10*Range[0, 60] + 1
LinearRecurrence[{2, -1}, {11, 21}, {0, 60}]
CoefficientList[Series[(1+9x)/(1-x)^2, {x, 0, 60}], x] (* End *)

Vec((1+9*x)/(1-x)^2 + O(x^80)) \\ Michel Marcus, Jun 17 2015

[10*n+1 for n in (0..60)] # G. C. Greubel, Sep 18 2019

G.f.: (1+9*x)/(1-x)^2.
a(n) = 20*n - a(n-1) - 8, with a(0)=1. - Vincenzo Librandi, Nov 20 2010
a(n) = 2*a(n-1) - a(n-2), for n > 2. - Wesley Ivan Hurt, Jun 17 2015
E.g.f.: (1 + 10*x)*exp(x). - G. C. Greubel, Sep 18 2019

A000127 Maximal number of regions obtained by joining n points around a circle by straight lines. Also number of regions in 4-space formed by n-1 hyperplanes.

Original entry on oeis.org

1, 2, 4, 8, 16, 31, 57, 99, 163, 256, 386, 562, 794, 1093, 1471, 1941, 2517, 3214, 4048, 5036, 6196, 7547, 9109, 10903, 12951, 15276, 17902, 20854, 24158, 27841, 31931, 36457, 41449, 46938, 52956, 59536, 66712, 74519, 82993, 92171, 102091, 112792, 124314, 136698

Offset: 1

N. J. A. Sloane

a(n) is the sum of the first five terms in the n-th row of Pascal's triangle. - Geoffrey Critzer, Jan 18 2009
{a(k): 1 <= k <= 5} = divisors of 16. - Reinhard Zumkeller, Jun 17 2009
Equals binomial transform of [1, 1, 1, 1, 1, 0, 0, 0, ...]. - Gary W. Adamson, Mar 02 2010
From Bernard Schott, Apr 05 2021: (Start)
As a(n) = 2^(n-1) for n = 1..5, it is misleading to believe that a(n) = 2^(n-1) for n > 5 (see Patrick Popescu-Pampu link); other curiosities: a(6) = 2^5 - 1 and a(10) = 2^8.
The sequence of the first differences is A000125, the sequence of the second differences is A000124, the sequence of the third differences is A000027 and the sequence of the fourth differences is the all 1's sequence A000012 (see J. H. Conway and R. K. Guy reference, p. 80). (End)
a(n) is the number of binary words of length n matching the regular expression 0*1*0*1*0*. A000124 and A000125 count binary words of the form 0*1*0* and 1*0*1*0*, respectively. - Manfred Scheucher, Jun 22 2023

			a(7)=99 because the first five terms in the 7th row of Pascal's triangle are 1 + 7 + 21 + 35 + 35 = 99. - _Geoffrey Critzer_, Jan 18 2009
G.f. = x + 2*x^2 + 4*x^3 + 8*x^4 + 16*x^5 + 31*x^6 + 57*x^7 + 99*x^8 + 163*x^9 + ...

R. B. Banks, Slicing Pizzas, Racing Turtles and Further Adventures in Applied Mathematics, Princeton Univ. Press, 1999. See p. 28.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.
J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, Chap. 3.
J. H. Conway and R. K. Guy, Le Livre des Nombres, Eyrolles, 1998, p. 80.
J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique Des Nombres, Problem 33 pp. 18; 128 Ellipses Paris 2004.
A. Deledicq and D. Missenard, A La Recherche des Régions Perdues, Math. & Malices, No. 22 Summer 1995 issue pp. 22-3 ACL-Editions Paris.
M. Gardner, Mathematical Circus, pp. 177; 180-1 Alfred A. Knopf NY 1979.
M. Gardner, The Colossal Book of Mathematics, 2001, p. 561.
James Gleick, Faster, Vintage Books, NY, 2000 (see pp. 259-261).
M. de Guzman, Aventures Mathématiques, Prob. B pp. 115-120 PPUR Lausanne 1990.
Ross Honsberger; Mathematical Gems I, Chap. 9.
Ross Honsberger; Mathematical Morsels, Chap. 3.
Jeux Mathématiques et Logiques, Vol. 3 pp. 12; 51 Prob. 14 FFJM-SERMAP Paris 1988.
J. N. Kapur, Reflections of a Mathematician, Chap.36, pp. 337-343, Arya Book Depot, New Delhi 1996.
C. D. Miller, V. E. Heeren, J. Hornsby, M. L. Morrow and J. Van Newenhizen, Mathematical Ideas, Tenth Edition, Pearson, Addison-Wesley, Boston, 2003, Cptr 1, 'The Art of Problem Solving, page 6.
I. Niven, Mathematics of Choice, pp. 158; 195 Prob. 40 NML 15 MAA 1965.
C. S. Ogilvy, Tomorrow's Math, pp. 144-6 OUP 1972.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 252-255.
Alfred S. Posamentier & Ingmar Lehmann, The (Fabulous) Fibonacci Numbers, Prometheus Books, NY, 2007, page 81-87.
A. M. Robert, A Course in p-adic Analysis, Springer-Verlag, 2000; p. 213.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
Alan Calvitti, Illustration of initial terms
M. L. Cornelius, Variations on a geometric progression, Mathematics in School, 4 (No. 3, May 1975), p. 32. (Annotated scanned copy)
Colin Defant, Meeting Covered Elements in nu-Tamari Lattices, arXiv:2104.03890 [math.CO], 2021.
M. Griffiths, Remodified Bessel Functions via Coincidences and Near Coincidences, Journal of Integer Sequences, Vol. 14 (2011), Article 11.7.1.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
R. K. Guy, Letter to N. J. A. Sloane
D. A. Lind, On a class of nonlinear binomial sums, Fib. Quart., 3 (1965), 292-298.
Math Forum, Regions of a circle Cut by Chords to n points.
R. J. Mathar, The number of binary nxm matrices with at most k 1's in each row or column, (2014) Table 4 column 1.
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Leo Moser and W. Bruce Ross, Mathematical Miscellany, On the Danger of Induction, Mathematics Magazine, Vol. 23, No. 2 (Nov. - Dec., 1949), pp. 109-114.
M. Noy, A Short Solution of a Problem in Combinatorial Geometry, Mathematics Magazine, pp. 52-3 69(1) 1996 MAA.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Patrick Popescu-Pampu, Démarrage trompeur, Images des Mathématiques, CNRS, 2017, rediffusion 2021 (in French).
D. J. Price, Some unusual series occurring in n-dimensional geometry, Math. Gaz., 30 (1946), 149-150.
H. P. Robinson, Letter to N. J. A. Sloane, Mar 21 1985
Grant Sanderson, Circle division solution, video, 2015.
Grant Sanderson, Circle division solution, updated video (2023).
K. Uhland, A Blase of Glory
K. Uhland, Moser's Problem
Prasad Balakrishnan Warrier, The physiognomy of the Erdős-Szekeres conjecture (happy ending problem), Math. Student (Indian Math. Soc., 2024) Vol. 93, Nos. 3-4, 28-48.
Eric Weisstein's World of Mathematics, Circle Division by Chords
Eric Weisstein's World of Mathematics, Strong Law of Small Numbers
Reinhart Zumkeller, Enumerations of Divisors
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000012, A000027, A000124, A000125, A002522, A005408, A016813, A086514, A058331, A161701, A161702, A161703, A161704, A161706, A161707, A161708, A161710, A080856, A161711, A161712, A161713, A161715, A006261, A007318, A008859-A008863, A219531, A223718.

a000127 = sum . take 5 . a007318_row  -- Reinhard Zumkeller, Nov 24 2012

[(n^4-6*n^3+23*n^2-18*n+24)/24: n in [1..50]]; // Vincenzo Librandi, Feb 16 2015

A000127 := n->(n^4 - 6*n^3 + 23*n^2 - 18*n + 24)/24;
with (combstruct):ZL:=[S, {S=Sequence(U, card=1)}, unlabeled]: seq(count(subs(r=6, ZL), size=m), m=0..41); # Zerinvary Lajos, Mar 08 2008

f[n_] := Sum[Binomial[n, i], {i, 0, 4}]; Table[f@n, {n, 0, 40}] (* Robert G. Wilson v, Jun 29 2007 *)
Total/@Table[Binomial[n-1,k],{n,50},{k,0,4}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{1,2,4,8,16},50] (* Harvey P. Dale, Aug 24 2011 *)
Table[(n^4 - 6 n^3 + 23 n^2 - 18 n + 24) / 24, {n, 100}] (* Vincenzo Librandi, Feb 16 2015 *)
a[ n_] := Binomial[n, 4] + Binomial[n, 2] + 1; (* Michael Somos, Dec 23 2017 *)

a(n)=(n^4-6*n^3+23*n^2-18*n+24)/24 \\ Charles R Greathouse IV, Mar 22 2016

{a(n) = binomial(n, 4) + binomial(n, 2) + 1}; /* Michael Somos, Dec 23 2017 */

def A000127(n): return n*(n*(n*(n - 6) + 23) - 18)//24 + 1 # Chai Wah Wu, Sep 18 2021

a(n) = C(n-1, 4) + C(n-1, 3) + ... + C(n-1, 0) = A055795(n) + 1 = C(n, 4) + C(n-1, 2) + n.
a(n) = Sum_{k=0..2} C(n, 2k). - Joel Sanderi (sanderi(AT)itstud.chalmers.se), Sep 08 2004
a(n) = (n^4 - 6*n^3 + 23*n^2 - 18*n + 24)/24.
G.f.: (1 - 3*x + 4*x^2 - 2*x^3 + x^4)/(1-x)^5. (for offset 0) - Simon Plouffe in his 1992 dissertation
E.g.f.: (1 + x + x^2/2 + x^3/6 + x^4/24)*exp(x) (for offset 0). [Typos corrected by Juan M. Marquez, Jan 24 2011]
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), n > 4. - Harvey P. Dale, Aug 24 2011
a(n) = A000124(A000217(n-1)) - n*A000217(n-2) - A034827(n), n > 1. - Melvin Peralta, Feb 15 2016
a(n) = A223718(-n). - Michael Somos, Dec 23 2017
For n > 2, a(n) = n + 1 + sum_{i=2..(n-2)}sum_{j=1..(n-i)}(1+(i-1)(j-1)). - Alec Jones, Nov 17 2019

Formula corrected and additional references from torsten.sillke(AT)lhsystems.com

Additional correction from Jonas Paulson (jonasso(AT)sdf.lonestar.org), Oct 30 2003

A017533 a(n) = 12*n + 1.

Original entry on oeis.org

1, 13, 25, 37, 49, 61, 73, 85, 97, 109, 121, 133, 145, 157, 169, 181, 193, 205, 217, 229, 241, 253, 265, 277, 289, 301, 313, 325, 337, 349, 361, 373, 385, 397, 409, 421, 433, 445, 457, 469, 481, 493, 505, 517, 529, 541, 553, 565, 577, 589, 601, 613, 625, 637

Offset: 0

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A161700, A005408, A016813, A016921, A017281, A158057, A161705, A161709, A161714, A128470, A016945, A287326 (third column).

#include
int main(){
int n;
for(n=0; n<=100; n++)
printf("%d, ", 12*n + 1);
return 0;
} /* Indranil Ghosh, Apr 05 2017 */

List([0..60], n-> 12*n + 1); # G. C. Greubel, Sep 18 2019

[12*n+1: n in [0..60]]; // G. C. Greubel, Sep 18 2019

seq(12*n+1, n=0..60); # G. C. Greubel, Sep 18 2019

Array[12*#+1&,60,0] (* Vladimir Joseph Stephan Orlovsky, Dec 14 2009 *)
CoefficientList[Series[(1+11x)/(1-x)^2, {x, 0,60}], x] (* Michael De Vlieger, Apr 05 2017 *)

a(n)=12*n+1 \\ Charles R Greathouse IV, Jul 10 2016

def a(n): return 12*n + 1 # Indranil Ghosh, Apr 05 2017

[12*n + 1 for n in (0..60)] # G. C. Greubel, Sep 18 2019

a(n) = 12*n + 1, n >= 0.
a(n) = 24*n - 10 - a(n-1), (with a(0)=1). - Vincenzo Librandi, Dec 24 2010
G.f.: (1 + 11*x)/(1-x)^2. - Indranil Ghosh, Apr 05 2017
E.g.f.: (1 + 12*x)*exp(x). - G. C. Greubel, Sep 18 2019

A006261 a(n) = Sum_{k=0..5} binomial(n,k).

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 63, 120, 219, 382, 638, 1024, 1586, 2380, 3473, 4944, 6885, 9402, 12616, 16664, 21700, 27896, 35443, 44552, 55455, 68406, 83682, 101584, 122438, 146596, 174437, 206368, 242825, 284274, 331212, 384168, 443704, 510416, 584935, 667928, 760099, 862190

Offset: 0

N. J. A. Sloane, based on a suggestion from S. C. Chan, Jun 10 1975

a(n) is the sum of the first six terms of the n-th row in Pascal's triangle. - Geoffrey Critzer, Jan 19 2009
Also the interpolating polynomial for the divisors of 32: {a(k): 0 <= k < 6} = {1,2,4,8,16,32}. - Reinhard Zumkeller, Jun 17 2009
a(n) is the maximal number of regions in 5-space formed by n-1 4-dimensional hypercubes. - Carl Schildkraut, May 26 2015
a(n) is the number of binary words of length n matching the regular expression 1*0*1*0*1*0*. A000124, A000125, A000127 count binary words of the form 0*1*0*, 1*0*1*0*, and 0*1*0*1*0*, respectively. - Manfred Scheucher, Jun 22 2023

			a(7) = 120 because the first six terms in the 7th row of Pascal's triangle 1 + 7 + 21 + 35 + 35 + 21 = 120. - _Geoffrey Critzer_, Jan 19 2009

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
S. C. Chan, Letter to N. J. A. Sloane, Oct. 1975
M. L. Cornelius, Variations on a geometric progression, Mathematics in School, 4 (No. 3, May 1975), p. 32.
M. L. Cornelius, Variations on a geometric progression, Mathematics in School, 4 (No. 3, May 1975), p. 32. (Annotated scanned copy)
R. K. Guy, Letter to N. J. A. Sloane
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Reinhard Zumkeller, Enumerations of Divisors
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A057703.

Cf. A005408, A000124, A016813, A086514, A000125, A058331, A002522, A161701, A161702, A161703, A000127, A161704, A161706, A161707, A161708, A161710, A080856, A161711, A161712, A161713, A161715, A007318, A008859, A008860, A008861, A008862, A008863, A219531.

a006261 = sum . take 6 . a007318_row  -- Reinhard Zumkeller, Nov 24 2012

[(n^5 - 5*n^4 + 25*n^3 + 5*n^2 + 94*n + 120)/120: n in [0..40]]; // Vincenzo Librandi, Jul 17 2011

A006261:=(z**2-z+1)*(3*z**2-3*z+1)/(z-1)**6; # Simon Plouffe in his 1992 dissertation

CoefficientList[
  Series[(1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120) Exp[x], {x, 0,
    52}], x]*Table[n!, {n, 0, 52}]

a(n)=sum(k=0,5,binomial(n,k)) \\ Charles R Greathouse IV, Apr 08 2016

A006261_list, m = [], [1, -3, 4, -2, 1, 1]
for _ in range(10**2):
    A006261_list.append(m[-1])
    for i in range(5):
        m[i+1] += m[i] # Chai Wah Wu, Jan 24 2016

[binomial(n,1)+binomial(n,3)+binomial(n,5) for n in range(1, 38)] # Zerinvary Lajos, May 17 2009

a(n) = A057703(n) + 1.
a(n) = binomial(n+1, 5) + binomial(n+1, 3) + binomial(n+1, 1). - Len Smiley, Oct 20 2001
G.f.: (1 - 4*x + 7*x^2 - 6*x^3 + 3*x^4)/(1-x)^6. - Geoffrey Critzer, Jan 19 2009
E.g.f.: (1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120)*exp(x).
a(n) = (n^5 - 5*n^4 + 25*n^3 + 5*n^2 + 94*n + 120)/120. - Reinhard Zumkeller, Jun 17 2009
a(n) = a(n-1) + A000127(n-1). - Christian Schroeder, Jan 04 2016

cp's OEIS Frontend

A027750 Triangle read by rows in which row n lists the divisors of n.

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A002522 a(n) = n^2 + 1.

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A006218 a(n) = Sum_{k=1..n} floor(n/k); also Sum_{k=1..n} d(k), where d = number of divisors (A000005); also number of solutions to x*y = z with 1 <= x,y,z <= n.

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A016921 a(n) = 6*n + 1.

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A058331 a(n) = 2*n^2 + 1.

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