cp's OEIS Frontend

Continued fraction expansion of sqrt(2) is 1 + 1/(2 + 1/(2 + 1/(2 + ...))).

Inverse binomial transform of Mersenne numbers A000225(n+1) = 2^(n+1) - 1. - Paul Barry, Feb 28 2003

A Chebyshev transform of 2^n: if A(x) is the g.f. of a sequence, map it to ((1-x^2)/(1+x^2))A(x/(1+x^2)). - Paul Barry, Oct 31 2004

An inverse Catalan transform of A068875 under the mapping g(x)->g(x(1-x)). A068875 can be retrieved using the mapping g(x)->g(xc(x)), where c(x) is the g.f. of A000108. A040000 and A068875 may be described as a Catalan pair. - Paul Barry, Nov 14 2004

Sequence of electron arrangement in the 1s 2s and 3s atomic subshells. Cf. A001105, A016825. - Jeremy Gardiner, Dec 19 2004

Binomial transform of A165326. - Philippe Deléham, Sep 16 2009

Let m=2. We observe that a(n) = Sum_{k=0..floor(n/2)} binomial(m,n-2*k). Then there is a link with A113311 and A115291: it is the same formula with respectively m=3 and m=4. We can generalize this result with the sequence whose g.f. is given by (1+z)^(m-1)/(1-z). - Richard Choulet, Dec 08 2009

With offset 1: number of permutations where |p(i) - p(i+1)| <= 1 for n=1,2,...,n-1. This is the identical permutation and (for n>1) its reversal.

Equals INVERT transform of bar(1, 1, -1, -1, ...).

Eventual period is (2). - Zak Seidov, Mar 05 2011

Also decimal expansion of 11/90. - Vincenzo Librandi, Sep 24 2011

a(n) = 3 - A054977(n); right edge of the triangle in A182579. - Reinhard Zumkeller, May 07 2012

With offset 1: minimum cardinality of the range of a periodic sequence with (least) period n. Of course the range's maximum cardinality for a purely periodic sequence with (least) period n is n. - Rick L. Shepherd, Dec 08 2014

With offset 1: n*a(1) + (n-1)*a(2) + ... + 2*a(n-1) + a(n) = n^2. - Warren Breslow, Dec 12 2014

With offset 1: decimal expansion of gamma(4) = 11/9 where gamma(n) = Cp(n)/Cv(n) is the n-th Poisson's constant. For the definition of Cp and Cv see A272002. - Natan Arie Consigli, Sep 11 2016

a(n) equals the number of binary sequences of length n where no two consecutive terms differ. Also equals the number of binary sequences of length n where no two consecutive terms are the same. - David Nacin, May 31 2017

a(n) is the period of the continued fractions for sqrt((n+2)/(n+1)) and sqrt((n+1)/(n+2)). - A.H.M. Smeets, Dec 05 2017

Also, number of self-avoiding walks and coordination sequence for the one-dimensional lattice Z. - Sean A. Irvine, Jul 27 2020

Decimal expansion of 1/3. - Raymond Wang, Mar 06 2010

Continued fraction expansion of (3+sqrt(13))/2. - Bruno Berselli, Mar 15 2011

T(n,m) is the number of m-faces of a regular n-simplex.

An n-simplex is the n-dimensional analog of a triangle. Specifically, a simplex is the convex hull of a set of (n + 1) affinely independent points in some Euclidean space of dimension n or higher, i.e., a set of points such that no m-plane contains more than (m + 1) of them. Such points are said to be in general position.

Reversing the rows gives A074909, which as a linear sequence is essentially the same as this.

From Tom Copeland, Dec 07 2007: (Start)

T(n,k) * (k+1)! = A068424. The comment on permuted words in A068424 shows that T is related to combinations of letters defined by connectivity of regular polytope simplexes.

If T is the diagonally-shifted Pascal matrix, binomial(n+m, k+m), for m=1, then T is a fundamental type of matrix that is discussed in A133314 and the following hold.

The infinitesimal matrix generator is given by A132681, so T = LM(1) of A132681 with inverse LM(-1).

With a(k) = (-x)^k / k!, T * a = [ Laguerre(n,x,1) ], a vector array with index n for the Laguerre polynomials of order 1. Other formulas for the action of T are given in A132681.

T(n,k) = (1/n!) (D_x)^n (D_t)^k Gf(x,t) evaluated at x=t=0 with Gf(x,t) = exp[ t * x/(1-x) ] / (1-x)^2.

[O.g.f. for T ] = 1 / { [ 1 - t * x/(1-x) ] * (1-x)^2 }. [ O.g.f. for row sums ] = 1 / { (1-x) * (1-2x) }, giving A000225 (without a leading zero) for the row sums. Alternating sign row sums are all 1. [Sign correction noted by Vincent J. Matsko, Jul 19 2015]

O.g.f. for row polynomials = [ (1+q)**(n+1) - 1 ] / [ (1+q) -1 ] = A(1,n+1,q) on page 15 of reference on Grassmann cells in A008292. (End)

Given matrices A and B with A(n,k) = T(n,k)*a(n-k) and B(n,k) = T(n,k)*b(n-k), then A*B = C where C(n,k) = T(n,k)*[a(.)+b(.)]^(n-k), umbrally. The e.g.f. for the row polynomials of A is {(a+t) exp[(a+t)x] - a exp(a x)}/t, umbrally. - Tom Copeland, Aug 21 2008

A007318*A097806 as infinite lower triangular matrices. - Philippe Deléham, Feb 08 2009

Riordan array (1/(1-x)^2, x/(1-x)). - Philippe Deléham, Feb 22 2012

The elements of the matrix inverse are T^(-1)(n,k)=(-1)^(n+k)*T(n,k). - R. J. Mathar, Mar 12 2013

Relation to K-theory: T acting on the column vector (-0,d,-d^2,d^3,...) generates the Euler classes for a hypersurface of degree d in CP^n. Cf. Dugger p. 168 and also A104712, A111492, and A238363. - Tom Copeland, Apr 11 2014

Number of walks of length p>0 between any two distinct vertices of the complete graph K_(n+2) is W(n+2,p)=(-1)^(p-1)*Sum_{k=0..p-1} T(p-1,k)*(-n-2)^k = ((n+1)^p - (-1)^p)/(n+2) = (-1)^(p-1)*Sum_{k=0..p-1} (-n-1)^k. This is equal to (-1)^(p-1)*Phi(p,-n-1), where Phi is the cyclotomic polynomial when p is an odd prime. For K_3, see A001045; for K_4, A015518; for K_5, A015521; for K_6, A015531; for K_7, A015540. - Tom Copeland, Apr 14 2014

Consider the transformation 1 + x + x^2 + x^3 + ... + x^n = A_0*(x-1)^0 + A_1*(x-1)^1 + A_2*(x-1)^2 + ... + A_n*(x-1)^n. This sequence gives A_0, ..., A_n as the entries in the n-th row of this triangle, starting at n = 0. - Derek Orr, Oct 14 2014

See A074909 for associations among this array, the Bernoulli polynomials and their umbral compositional inverses, and the face polynomials of permutahedra and their duals (cf. A019538). - Tom Copeland, Nov 14 2014

From Wolfdieter Lang, Dec 10 2015: (Start)

A(r, n) = T(n+r-2, r-1) = risefac(n,r)/r! = binomial(n+r-1, r), for n >= 1 and r >= 1, gives the array with the number of independent components of a symmetric tensors of rank r (number of indices) and dimension n (indices run from 1 to n). Here risefac(n, k) is the rising factorial.

As(r, n) = T(n+1, r+1) = fallfac(n, r)/r! = binomial(n, r), r >= 1 and n >= 1 (with the triangle entries T(n, k) = 0 for n < k) gives the array with the number of independent components of an antisymmetric tensor of rank r and dimension n. Here fallfac is the falling factorial. (End)

The h-vectors associated to these f-vectors are given by A000012 regarded as a lower triangular matrix. Read as bivariate polynomials, the h-polynomials are the complete homogeneous symmetric polynomials in two variables, found in the compositional inverse of an e.g.f. for A008292, the h-vectors of the permutahedra. - Tom Copeland, Jan 10 2017

For a correlation between the states of a quantum system and the combinatorics of the n-simplex, see Boya and Dixit. - Tom Copeland, Jul 24 2017

Any number of the form b^k+1 is composite for b>2 and k odd since b+1 algebraically divides b^k+1. - Robert G. Wilson v, Aug 25 2002

The first three terms (indices 0, 1 and 2) are the only known primes. Moreover, the terms not of the form a(2^k) are all composite, except for a(0). Indeed, for all n >= 0, a(2n+1) is divisible by 11, a(4n+2) is divisible by 101, a(8n+4) is divisible by 73, a(16n+8) is divisible by 17, a(32n+16) is divisible by 353, a(64n+32) is divisible by 19841, etc. - M. F. Hasler, Nov 03 2018 [Edited based on the comment by Jeppe Stig Nielsen, Oct 17 2019]

This sequence also results when each term is generated by converting the previous term into a Roman numeral, then replacing each letter with its corresponding decimal value, provided that the vinculum is used and numerals are written in a specific way for integers greater than 3999, e.g., IV with a vinculum over the I and V for 4000. - Jamie Robert Creasey, Apr 14 2021

By Mihăilescu's theorem, a(n) can never be a perfect power (see "Catalan's conjecture" in Links). - Marco Ripà, Mar 10 2025

The conjecture is that the leading term is always 1.

Odlyzko has checked it for primes up to pi(10^13) = 3*10^11.

From M. F. Hasler, Jun 02 2012: (Start)

The second column, omitting the initial 3, is given in A089582. The number of "0"s preceding the first term > 1 in the n-th row is given in A213014. The first term > 1 in any row must equal 2, else the conjecture is violated: Obviously all terms except for the first one are even. Thus, if the 2nd term in some row is > 2, it is >= 4, and the first term of the subsequent row is >= 3. If there is a positive number of zeros preceding a first term > 2 (thus >= 4), this "jump" will remain constant and "propagate" (in subsequent rows) to the beginning of the row, and the previously discussed case applies.

The previous statement can also be formulated as: Gilbreath's conjecture is equivalent to: A036277(n) > A213014(n)+2 for all n.

CAVEAT: While table A036261 starts with the first absolute differences of the primes in its first row, the present sequence has the primes themselves in its uppermost row, which is sometimes referred to as "row 0". Thus, "first row" of this table A036262 may either refer to row 1 (1,2,2,...), or to row 0 (2,3,5,7,...), while the latter might, however, as well be referred to "row 1 of A036262" in other sequences or papers.

(End)

From Clark Kimberling, Nov 27 2022: (Start)

Suppose that S = (s(k)), for k >= 1, is a sequence of real numbers. For n >= 1, let g(1,n) = |s(n+1)-s(n)| and g(k,n) = |g(k-1,n+1) - g(k-1,n)| for k >= 2.

Call (g(k,n)) the Gilbreath array of S. Call the first column of this array the Gilbreath transform of S. Denote this transform by G(S), so that G(S) is the sequence (g(n,1)). If S is the sequence of primes, then the Gilbreath conjecture holds that G(S) consists exclusively of 1's. More generally, it appears that there are many S such that G(S) is eventually periodic. See A358691 for conjectured examples. (End)

T(n,k) is the number of ways to distribute k identical objects in n distinct containers; containers may be left empty.

T(n,k) is the number of nondecreasing functions f from {1,...,k} to {1,...,n}. - Dennis P. Walsh, Apr 07 2011

Coefficients of Faber polynomials for function x^2/(x-1). - Michael Somos, Sep 09 2003

Consider k-fold Cartesian products CP(n,k) of the vector A(n)=[1,2,3,...,n].

An element of CP(n,k) is a n-tuple T_t of the form T_t=[i_1,i_2,i_3,...,i_k] with t=1,...,n^k.

We count members T of CP(n,k) which satisfy some condition delta(T_t), so delta(.) is an indicator function which attains values of 1 or 0 depending on whether T_t is to be counted or not; the summation sum_{CP(n,k)} delta(T_t) over all elements T_t of CP produces the count.

For the triangle here we have delta(T_t) = 0 if for any two i_j, i_(j+1) in T_t one has i_j > i_(j+1), T(n,k) = Sum_{CP(n,k)} delta(T_t) = Sum_{CP(n,k)} delta(i_j > i_(j+1)).

The indicator function which tests on i_j = i_(j+1) generates A158497, which contains further examples of this type of counting.

Triangle of the numbers of combinations of k elements with repetitions from n elements {1,2,...,n} (when every element i, i=1,...,n, appears in a k-combination either 0, or 1, or 2, ..., or k times). - Vladimir Shevelev, Jun 19 2012

G.f. for Faber polynomials is -log(-t*x-(1-sqrt(1-4*t))/2+1)=sum(n>0, T(n,k)*t^k/n). - Vladimir Kruchinin, Jul 04 2013

Values of complete homogeneous symmetric polynomials with all arguments equal to 1, or, equivalently, the number of monomials of degree k in n variables. - Tom Copeland, Apr 07 2014

Row k >= 0 of the infinite square array A[k,n] = C(n,k), n >= 0, would start with k zeros in front of the first nonzero element C(k,k) = 1; this here is the triangle obtained by taking the first k+1 nonzero terms C(k .. 2k, k) of rows k = 0, 1, 2, ... of that array. - M. F. Hasler, Mar 05 2017

Each set partition is written as a sequence of blocks, ordered by the smallest elements in the blocks.

The EG2[2*m,n] matrix coefficients were introduced in A008955. We discovered that EG2[2m,n] = Sum_{k = 1..n} (-1)^(k+n)*t1(n-1,k-1)*2*eta(2*m-2*n+2*k)/((n-1)!)^2 with t1(n,m) the central factorial numbers A008955 and eta(m) = (1-2^(1-m))*zeta(m) with eta(m) the Dirichlet eta function and zeta(m) the Riemann zeta function.

A different way to define these matrix coefficients is EG2[2*m,n] = (1/m)*Sum_{k = 0..m-1} ZETA(2*m-2*k, n-1)*EG2[2*k, n] with ZETA(2*m, n-1) = zeta(2*m) - Sum_{k = 1..n-1} (k)^(-2*m) and EG2[0, n] = 1, for m = 0, 1, 2, ..., and n = 1, 2, 3, ... .

We define the row sums of the EG2 matrix rs(2*m,p) = Sum_{n >= 1} (n^p)*EG2(2*m,n) for p = -2, -1, 0, 1, ... and m >= p+2. We discovered that rs(2*m,p=-2) = 2*eta(2*m+2) = (1 - 2^(1-(2*m+2)))*zeta(2*m+2). This formula is quite unlike the other rs(2*m,p) formulas, see the examples.

The series expansions of the row generators RGEG2(z,2*m) about z = 0 lead to the EG2[2*m,n] coefficients while the series expansions about z = 1 lead to the ZG1[2*m-1,n] coefficients, see the formulas.

The first Maple program gives the triangle coefficients. Adding the second program to the first one gives information about the row sums rs(2*m,p).

The a(n) formulas of the right hand columns are related to sequence A036283, see also A161740 and A161741.