A084990 -id:A084990 - cp's OEIS frontend

A028387 a(n) = n + (n+1)^2.

1, 5, 11, 19, 29, 41, 55, 71, 89, 109, 131, 155, 181, 209, 239, 271, 305, 341, 379, 419, 461, 505, 551, 599, 649, 701, 755, 811, 869, 929, 991, 1055, 1121, 1189, 1259, 1331, 1405, 1481, 1559, 1639, 1721, 1805, 1891, 1979, 2069, 2161, 2255, 2351, 2449, 2549, 2651

Offset: 0

Patrick De Geest

a(n+1) is the least k > a(n) + 1 such that A000217(a(n)) + A000217(k) is a square. - David Wasserman, Jun 30 2005
Values of Fibonacci polynomial n^2 - n - 1 for n = 2, 3, 4, 5, ... - Artur Jasinski, Nov 19 2006
A127701 * [1, 2, 3, ...]. - Gary W. Adamson, Jan 24 2007
Row sums of triangle A135223. - Gary W. Adamson, Nov 23 2007
Equals row sums of triangle A143596. - Gary W. Adamson, Aug 26 2008
a(n-1) gives the number of n X k rectangles on an n X n chessboard (for k = 1, 2, 3, ..., n). - Aaron Dunigan AtLee, Feb 13 2009
sqrt(a(0) + sqrt(a(1) + sqrt(a(2) + sqrt(a(3) + ...)))) = sqrt(1 + sqrt(5 + sqrt(11 + sqrt(19 + ...)))) = 2. - Miklos Kristof, Dec 24 2009
When n + 1 is prime, a(n) gives the number of irreducible representations of any nonabelian group of order (n+1)^3. - Andrew Rupinski, Mar 17 2010
a(n) = A176271(n+1, n+1). - Reinhard Zumkeller, Apr 13 2010
The product of any 4 consecutive integers plus 1 is a square (see A062938); the terms of this sequence are the square roots. - Harvey P. Dale, Oct 19 2011
Or numbers not expressed in the form m + floor(sqrt(m)) with integer m. - Vladimir Shevelev, Apr 09 2012
Left edge of the triangle in A214604: a(n) = A214604(n+1,1). - Reinhard Zumkeller, Jul 25 2012
Another expression involving phi = (1 + sqrt(5))/2 is a(n) = (n + phi)(n + 1 - phi). Therefore the numbers in this sequence, even if they are prime in Z, are not prime in Z[phi]. - Alonso del Arte, Aug 03 2013
a(n-1) = n*(n+1) - 1, n>=0, with a(-1) = -1, gives the values for a*c of indefinite binary quadratic forms [a, b, c] of discriminant D = 5 for b = 2*n+1. In general D = b^2 - 4ac > 0 and the form [a, b, c] is a*x^2 + b*x*y + c*y^2. - Wolfdieter Lang, Aug 15 2013
a(n) has prime factors given by A038872. - Richard R. Forberg, Dec 10 2014
A253607(a(n)) = -1. - Reinhard Zumkeller, Jan 05 2015
An example of a quadratic sequence for which the continued square root map (see A257574) produces the number 2. There are infinitely many sequences with this property - another example is A028387. See Popular Computing link. - N. J. A. Sloane, May 03 2015
Left edge of the triangle in A260910: a(n) = A260910(n+2,1). - Reinhard Zumkeller, Aug 04 2015
Numbers m such that 4m+5 is a square. - Bruce J. Nicholson, Jul 19 2017
The numbers represented as 131 in base n: 131_4 = 29, 131_5 = 41, ... . If 'digits' larger than the base are allowed then 131_2 = 11 and 131_1 = 5 also. - Ron Knott, Nov 14 2017
From Klaus Purath, Mar 18 2019: (Start)
Let m be a(n) or a prime factor of a(n). Then, except for 1 and 5, there are, if m is a prime, exactly two squares y^2 such that the difference y^2 - m contains exactly one pair of factors {x,z} such that the following applies: x*z = y^2 - m, x + y = z with
x < y, where {x,y,z} are relatively prime numbers. {x,y,z} are the initial values of a sequence of the Fibonacci type. Thus each a(n) > 5, if it is a prime, and each prime factor p > 5 of an a(n) can be assigned to exactly two sequences of the Fibonacci type. a(0) = 1 belongs to the original Fibonacci sequence and a(1) = 5 to the Lucas sequence.
But also the reverse assignment applies. From any sequence (f(i)) of the Fibonacci type we get from its 3 initial values by f(i)^2 - f(i-1)*f(i+1) with f(i-1) < f(i) a term a(n) or a prime factor p of a(n). This relation is also valid for any i. In this case we get the absolute value |a(n)| or |p|. (End)
a(n-1) = 2*T(n) - 1, for n>=1, with T = A000217, is a proper subsequence of A089270, and the terms are 0,-1,+1 (mod 5). - Wolfdieter Lang, Jul 05 2019
a(n+1) is the number of wedged n-dimensional spheres in the homotopy of the neighborhood complex of Kneser graph KG_{2,n}. Here, KG_{2,n} is a graph whose vertex set is the collection of subsets of cardinality 2 of set {1,2,...,n+3,n+4} and two vertices are adjacent if and only if they are disjoint. - Anurag Singh, Mar 22 2021
Also the number of squares between (n+2)^2 and (n+2)^4. - Karl-Heinz Hofmann, Dec 07 2021
(x, y, z) = (A001105(n+1), -a(n-1), -a(n)) are solutions of the Diophantine equation x^3 + 4*y^3 + 4*z^3 = 8. - XU Pingya, Apr 25 2022
The least significant digit of terms of this sequence cycles through 1, 5, 1, 9, 9. - Torlach Rush, Jun 05 2024

			From _Ilya Gutkovskiy_, Apr 13 2016: (Start)
Illustration of initial terms:
                                        o               o
                        o           o   o o           o o
            o       o   o o       o o   o o o       o o o
    o   o   o o   o o   o o o   o o o   o o o o   o o o o
o   o o o   o o o o o   o o o o o o o   o o o o o o o o o
n=0  n=1       n=2           n=3               n=4
(End)
From _Klaus Purath_, Mar 18 2019: (Start)
Examples:
a(0) = 1: 1^1-0*1 = 1, 0+1 = 1 (Fibonacci A000045).
a(1) = 5: 3^2-1*4 = 5, 1+3 = 4 (Lucas A000032).
a(2) = 11: 4^2-1*5 = 11, 1+4 = 5 (A000285); 5^2-2*7 = 11, 2+5 = 7 (A001060).
a(3) = 19: 5^2-1*6 = 19, 1+5 = 6 (A022095); 7^2-3*10 = 19, 3+7 = 10 (A022120).
a(4) = 29: 6^2-1*7 = 29, 1+6 = 7 (A022096); 9^2-4*13 = 29, 4+9 = 13 (A022130).
a(11)/5 = 31: 7^2-2*9 = 31, 2+7 = 9 (A022113); 8^2-3*11 = 31, 3+8 = 11 (A022121).
a(24)/11 = 59: 9^2-2*11 = 59, 2+9 = 11 (A022114); 12^2-5*17 = 59, 5+12 = 17 (A022137).
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Patrick De Geest, World!Of Numbers, Palindromes of the form n+(n+1)^2.
Adalbert Kerber, A matrix of combinatorial numbers related to the symmetric groups, Discrete Math., 21 (1978), 319-321. [Annotated scanned copy]
Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
Nandini Nilakantan and Anurag Singh, Homotopy type of neighborhood complexes of Kneser graphs, KG_{2,k}, Proceeding-Mathematical Sciences, 128, Article number: 53(2018).
Yanni Pei and Jiang Zeng, Counting signed derangements with right-to-left minima and excedances, arXiv:2206.11236 [math.CO], 2022.
Popular Computing (Calabasas, CA), The CSR Function, Vol. 4 (No. 34, Jan 1976), pages PC34-10 to PC34-11. Annotated and scanned copy.
Zdzislaw Skupień and Andrzej Żak, Pair-sums packing and rainbow cliques, in Topics In Graph Theory, A tribute to A. A. and T. E. Zykovs on the occasion of A. A. Zykov's 90th birthday, ed. R. Tyshkevich, Univ. Illinois, 2013, pages 131-144, (in English and Russian).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Complement of A028392. Third column of array A094954.
Cf. A000217, A002522, A062392, A062786, A127701, A135223, A143596, A052905, A162997, A062938 (squares of this sequence).
A110331 and A165900 are signed versions.
Cf. A002327 (primes), A094210.
Frobenius number for k successive numbers: this sequence (k=2), A079326 (k=3), A138984 (k=4), A138985 (k=5), A138986 (k=6), A138987 (k=7), A138988 (k=8).
Cf. also A135223, A176271, A214604, A105728, A005408, A002378, A084990, A253607, A260910, A089270.

a028387 n = n + (n + 1) ^ 2  -- Reinhard Zumkeller, Jul 17 2014

[n + (n+1)^2: n in [0..60]]; // Vincenzo Librandi, Apr 26 2011

FoldList[## + 2 &, 1, 2 Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *)
Table[n + (n + 1)^2, {n, 0, 100}] (* Vincenzo Librandi, Oct 17 2012 *)
Table[ FrobeniusNumber[{n, n + 1}], {n, 2, 30}] (* Zak Seidov, Jan 14 2015 *)

a(n)=n^2+3*n+1 \\ Charles R Greathouse IV, Jun 10 2011

def a(n): return (n**2+3*n+1) # Torlach Rush, May 07 2024

[n+(n+1)^2 for n in range(0,48)] # Zerinvary Lajos, Jul 03 2008

a(n) = sqrt(A062938(n)). - Floor van Lamoen, Oct 08 2001
a(0) = 1, a(1) = 5, a(n) = (n+1)*a(n-1) - (n+2)*a(n-2) for n > 1. - Gerald McGarvey, Sep 24 2004
a(n) = A105728(n+2, n+1). - Reinhard Zumkeller, Apr 18 2005
a(n) = A109128(n+2, 2). - Reinhard Zumkeller, Jun 20 2005
a(n) = 2*T(n+1) - 1, where T(n) = A000217(n). - Gary W. Adamson, Aug 15 2007
a(n) = A005408(n) + A002378(n); A084990(n+1) = Sum_{k=0..n} a(k). - Reinhard Zumkeller, Aug 20 2007
Binomial transform of [1, 4, 2, 0, 0, 0, ...] = (1, 5, 11, 19, ...). - Gary W. Adamson, Sep 20 2007
G.f.: (1+2*x-x^2)/(1-x)^3. a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - R. J. Mathar, Jul 11 2009
a(n) = (n + 2 + 1/phi) * (n + 2 - phi); where phi = 1.618033989... Example: a(3) = 19 = (5 + .6180339...) * (3.381966...). Cf. next to leftmost column in A162997 array. - Gary W. Adamson, Jul 23 2009
a(n) = a(n-1) + 2*(n+1), with n > 0, a(0) = 1. - Vincenzo Librandi, Nov 18 2010
For k < n, a(n) = (k+1)*a(n-k) - k*a(n-k-1) + k*(k+1); e.g., a(5) = 41 = 4*11 - 3*5 + 3*4. - Charlie Marion, Jan 13 2011
a(n) = lower right term in M^2, M = the 2 X 2 matrix [1, n; 1, (n+1)]. - Gary W. Adamson, Jun 29 2011
G.f.: (x^2-2*x-1)/(x-1)^3 = G(0) where G(k) = 1 + x*(k+1)*(k+4)/(1 - 1/(1 + (k+1)*(k+4)/G(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 16 2012
Sum_{n>0} 1/a(n) = 1 + Pi*tan(sqrt(5)*Pi/2)/sqrt(5). - Enrique Pérez Herrero, Oct 11 2013
E.g.f.: exp(x) (1+4*x+x^2). - Tom Copeland, Dec 02 2013
a(n) = A005408(A000217(n)). - Tony Foster III, May 31 2016
From Amiram Eldar, Jan 29 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = -Pi*sec(sqrt(5)*Pi/2).
Product_{n>=1} (1 - 1/a(n)) = -Pi*sec(sqrt(5)*Pi/2)/6. (End)
a(5*n+1)/5 = A062786(n+1). - Torlach Rush, Jun 05 2024

Minor edits by N. J. A. Sloane, Jul 04 2010, following suggestions from the Sequence Fans Mailing List

A212500 a(n) is the difference between multiples of 5 with even and odd digit sum in base 4 in interval [0,4^n).

Original entry on oeis.org

1, 4, 7, 36, 65, 340, 615, 3220, 5825, 30500, 55175, 288900, 522625, 2736500, 4950375, 25920500, 46890625, 245522500, 444154375, 2325622500, 4207090625, 22028612500, 39850134375, 208658012500, 377465890625, 1976437062500, 3575408234375

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, May 19 2012

Let the term "transform" mean the operation of summing the products of the numbers in the n-th row of an m-nomial triangle (m-nomial T(n,k)) and the ascending numbers of a sequence. And let T(0,0) be the top entry (0th row, 0th column) in an m-nomial triangle. Then starting with a(1)=1, the bisection of this sequence (1,7,65,615,5825...) is the quadrinomial (4-nomial) transform of A000045 (Fibonacci sequence, F(j)), starting at T(0,0)=1, F(1)=1. - Bob Selcoe, May 24 2014

			Let n=3. In interval [0,4^3) we have 13 multiples of 5,from which in base 4 only three (namely, 35,50,55) have odd digit sums. Thus a(3)=(13-3)-3=7.
From _Bob Selcoe_, May 28 2014: (Start)
n=2: a(5)=65 because T(2,k) {k=0..6} is {1,2,3,4,3,2,1} and {j=1..7} is {1,1,2,3,5,8,13}: 1*1 + 2*1 + 3*2 + 4*3 + 3*5 + 2*8 + 1*13 = 65.
n=3: a(7)=615 because T(3,k) {k=0..9} is {1,3,6,10,12,12,10,6,3,1} and {j=1..10} is {1,1,2,3,5,8,13,21,34,55}: 1*1 + 3*1 + 6*2 + 10*3 + 12*5 + 12*8 + 10*13 + 6*21 + 3*34 + 1*55 = 615. (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Shevelev, On monotonic strengthening of Newman-like phenomenon on (2m+1)-multiples in base 2m, arXiv:0710.3177 [math.NT], 2007.
Index entries for linear recurrences with constant coefficients, signature (0,10,0,-5).

Cf. A038754, A084990, A189334 (bisection).

I:=[1,4,7,36]; [n le 4 select I[n] else 10*Self(n-2)-5*Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jun 17 2014

CoefficientList[Series[-(-1 - 4 x + 3 x^2 + 4 x^3)/(1 - 10 x^2 + 5 x^4), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 17 2014 *)
LinearRecurrence[{0,10,0,-5},{1,4,7,36},30] (* Harvey P. Dale, Apr 07 2019 *)

Vec(-x*(-1-4*x+3*x^2+4*x^3)/(1-10*x^2+5*x^4) + O(x^30)) \\ Michel Marcus, Feb 06 2016

For n>=5, a(n) = 10*a(n-2)-5*a(n-4).
a(n) = 0.4*((5+2*sqrt(5))^(n/2)+ (5-2*sqrt(5))^(n/2)) , if n is even, and
a(n) = 0.1*((5+2*sqrt(5))^((n-1)/2)*sqrt(30+10*sqrt(5))+(5-2*sqrt(5))^((n-1)/2)*sqrt(30-10*sqrt(5))), if n is odd.
a(2n+1) = Sum_{j=0..3n+1} fibonacci(j+1)*A008287(n,j). - Bob Selcoe, May 28 2014
G.f.: -x*(-1-4*x+3*x^2+4*x^3) / ( 1-10*x^2+5*x^4 ). - R. J. Mathar, Jun 16 2014

A077415 a(n) = n(n+2)(n-2)/3.

Original entry on oeis.org

0, 5, 16, 35, 64, 105, 160, 231, 320, 429, 560, 715, 896, 1105, 1344, 1615, 1920, 2261, 2640, 3059, 3520, 4025, 4576, 5175, 5824, 6525, 7280, 8091, 8960, 9889, 10880, 11935, 13056, 14245, 15504, 16835, 18240, 19721, 21280, 22919, 24640, 26445

Offset: 2

Wolfdieter Lang, Nov 29 2002

a(n) is the number of independent components of a 3-tensor t(a,b,c) which satisfies t(a,b,c)=t(b,a,c) and sum(t(a,a,c),a=1..n)=0 for all c and t(a,b,c)+t(b,c,a)+t(c,a,b)=0, with a,b,c range 1..n. (3-tensor in n-dimensional space which is symmetric and traceless in one pair of its indices and satisfies the cyclic identity.)
Number of standard tableaux of shape (n-1,2,1) (n>=3). - Emeric Deutsch, May 13 2004
Zero followed by partial sums of A028387, starting at n=1. - Klaus Brockhaus, Oct 21 2008
For n>=4, a(n-1) is the number of permutations of 1,2...,n with the distribution of up (1) - down (0) elements 0...0101 (the first n-4 zeros), or, the same, a(n-1) is up-down coefficient {n,5} (see comment in A060351). - Vladimir Shevelev, Feb 14 2014
For n>=3, a(n) equals the second immanant of the (n-1) X (n-1) tridiagonal matrix with 2's along the main diagonal, and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jan 08 2016

G. C. Greubel, Table of n, a(n) for n = 2..10000
Mark Roger Sepanski, On Divisibility of Convolutions of Central Binomial Coefficients, Electronic Journal of Combinatorics, 21 (1) 2014, #P1.32.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000292, A028387 (first differences), A033275 (partial sums), A060351, A077414, A084990.

[n*(n+2)*(n-2)/3: n in [2..50]]; /* or */ I:=[0,5,16,35]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jan 09 2016

seq((n^3-4*n)/3, n=2..35); # Zerinvary Lajos, Jan 20 2007

Print[Table[Sum[(-1)^i*2^(n-2*i-1)*Binomial[n-i-1, i]*(n-2*i-2), {i, 0, Floor[(n-1)/2]}], {n, 2, 100}]] ;  (* John M. Campbell, Jan 08 2016 *)
LinearRecurrence[{4, -6, 4, -1}, {0, 5, 16, 35}, 50] (* Vincenzo Librandi, Jan 09 2016 *)
Table[n*(n + 2)*(n - 2)/3, {n, 2, 50}] (* G. C. Greubel, Jan 18 2018 *)

{a=0; print1(a,","); for(n=1, 42, print1(a=a+n+(n+1)^2, ","))} \\ Klaus Brockhaus, Oct 21 2008

concat(0, Vec(x^3*(5-4*x+x^2)/(1-x)^4 + O(x^100))) \\ Altug Alkan, Jan 08 2015

a(n) = n*(n+2)*(n-2)/3 = A077414(n) - binomial(n+2,3) = A077414(n) - A000292(n-1).
G.f.: x^3*(5 - 4*x + x^2)/(1-x)^4.
a(n) = A084990(n-1) - 1. - Reinhard Zumkeller, Aug 20 2007
a(n) = Sum_{i=0..floor((n-1)/2)} (-1)^i * 2^(n-2*i-1) * binomial(n-i-1, i) * (n-2*i-2). - John M. Campbell, Jan 08 2016
From Amiram Eldar, Jan 06 2021: (Start)
Sum_{n>=3} 1/a(n) = 11/32.
Sum_{n>=3} (-1)^(n+1)/a(n) = 5/32. (End)
E.g.f.: x*(1 + exp(x)*(x^2 + 3*x - 3)/3). - Stefano Spezia, Mar 06 2024

A200871 T(n,k)=Number of 0..k arrays x(0..n+1) of n+2 elements without any interior element greater than both neighbors or less than both neighbors.

Original entry on oeis.org

6, 17, 10, 36, 37, 16, 65, 94, 77, 26, 106, 195, 236, 163, 42, 161, 356, 567, 602, 343, 68, 232, 595, 1168, 1673, 1528, 723, 110, 321, 932, 2163, 3886, 4917, 3882, 1523, 178, 430, 1389, 3704, 7973, 12890, 14455, 9858, 3209, 288, 561, 1990, 5973, 14932, 29325

Offset: 1

R. H. Hardin Nov 23 2011

Table starts
...6....17.....36......65.....106......161......232.......321.......430
..10....37.....94.....195.....356......595......932......1389......1990
..16....77....236.....567....1168.....2163.....3704......5973......9184
..26...163....602....1673....3886.....7973....14932.....26073.....43066
..42...343...1528....4917...12890....29325....60112....113745....201994
..68...723...3882...14455...42744...107777...241718....495495....945790
.110..1523...9858...42479..141688...395929...971416...2156867...4424298
.178..3209..25038..124851..469726..1454643..3904290...9389377..20696974
.288..6761..63592..366959.1557320..5344795.15693816..40880321..96838448
.466.14245.161514.1078565.5163158.19638715.63085186.177996275.453123270

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

R. H. Hardin, Table of n, a(n) for n = 1..9999

Column 1 is A006355(n+4)

Row 1 is A084990(n+1)

t[0,k_,x_,y_] := 1; t[n_,k_,x_,y_] := t[n,k,x,y] = Sum[If[z <= x <= y || y <= x <= z, t[n-1, k, z, x], 0], {z, k+1}]; t[n_, k_] := Sum[t[n, k, x, y], {x, k+1}, {y, k+1}]; TableForm@ Table[t[n, k], {n, 8}, {k, 8}] (* Giovanni Resta, Mar 05 2014 *)

Empirical for columns:
k=1: a(n) = a(n-1) +a(n-2)
k=2: a(n) = 2*a(n-1) +a(n-4)
k=3: a(n) = 2*a(n-1) +a(n-2) +2*a(n-4) +a(n-5)
k=4: a(n) = 3*a(n-1) -a(n-2) +a(n-3) +4*a(n-4) +a(n-6) +a(n-7)
k=5: a(n) = 3*a(n-1) +a(n-3) +7*a(n-4) +3*a(n-5) +2*a(n-6) +3*a(n-7) +a(n-8)
k=6: a(n) = 4*a(n-1) -3*a(n-2) +4*a(n-3) +9*a(n-4) +7*a(n-6) +6*a(n-7) +a(n-8) +2*a(n-9) +a(n-10)
k=7: a(n) = 4*a(n-1) -2*a(n-2) +4*a(n-3) +15*a(n-4) +6*a(n-5) +12*a(n-6) +16*a(n-7) +7*a(n-8) +5*a(n-9) +4*a(n-10) +a(n-11)
Empirical for rows:
n=1: a(k) = (1/3)*k^3 + 2*k^2 + (8/3)*k + 1
n=2: a(k) = (1/12)*k^4 + (3/2)*k^3 + (47/12)*k^2 + (7/2)*k + 1
n=3: a(k) = (1/60)*k^5 + (3/4)*k^4 + (15/4)*k^3 + (25/4)*k^2 + (127/30)*k + 1
n=4: a(k) = (1/360)*k^6 + (7/24)*k^5 + (197/72)*k^4 + (185/24)*k^3 + (1667/180)*k^2 + 5*k + 1
n=5: a(k) = (1/2520)*k^7 + (17/180)*k^6 + (281/180)*k^5 + (64/9)*k^4 + (4927/360)*k^3 + (2303/180)*k^2 + (604/105)*k + 1
n=6: a(k) = (1/20160)*k^8 + (19/720)*k^7 + (211/288)*k^6 + (1889/360)*k^5 + (44167/2880)*k^4 + (15991/720)*k^3 + (5689/336)*k^2 + (391/60)*k + 1
n=7: a(k) = (1/181440)*k^9 + (131/20160)*k^8 + (8893/30240)*k^7 + (4621/1440)*k^6 + (118933/8640)*k^5 + (83957/2880)*k^4 + (763489/22680)*k^3 + (36343/1680)*k^2 + (9169/1260)*k + 1

A214352 T(n,k)=Number of nXnXn triangular 0..k arrays with no element lying outside the (possibly reversed) range delimited by its sw and se neighbors.

Original entry on oeis.org

2, 3, 6, 4, 17, 26, 5, 36, 169, 160, 6, 65, 660, 2853, 1386, 7, 106, 1951, 23554, 80573, 16814, 8, 161, 4822, 127813, 1602092, 3778867, 284724, 9, 232, 10507, 529006, 17790765, 205613460, 293207907, 6715224, 10, 321, 20840, 1807653, 135538054

Offset: 1

R. H. Hardin Jul 13 2012

Table starts
....2.....3.......4........5.........6.........7..........8...........9
....6....17......36.......65.......106.......161........232.........321
...26...169.....660.....1951......4822.....10507......20840.......38421
..160..2853...23554...127813....529006...1807653....5349708....14150061
.1386.80573.1602092.17790765.135538054.790197579.3766437036.15329961031

			Some solutions for n=3 k=3
....0......0......1......1......2......2......2......2......2......3......3
...0.1....3.0....1.1....0.2....2.1....1.2....3.1....0.2....2.0....3.3....3.2
..2.0.3..3.1.0..0.3.0..0.3.2..3.0.2..1.2.0..3.0.2..3.0.2..2.3.0..1.3.1..3.2.3

R. H. Hardin, Table of n, a(n) for n = 1..99

Column 1 is 2*A183278

Row 2 is A084990(n+1)

Empirical: rows n=1..5 are polynomials of degree n(n+1)/2 in k

A250419 T(n,k)=Number of length n+1 0..k arrays with the sum of the minimum of each adjacent pair multiplied by some arrangement of +-1 equal to zero.

Original entry on oeis.org

3, 5, 6, 7, 17, 10, 9, 36, 38, 20, 11, 65, 99, 125, 36, 13, 106, 205, 476, 335, 72, 15, 161, 370, 1351, 1693, 1061, 136, 17, 232, 606, 3154, 5982, 7504, 3069, 272, 19, 321, 927, 6433, 16790, 34415, 29221, 9495, 528, 21, 430, 1345, 11906, 39916, 119364, 169352

Offset: 1

R. H. Hardin, Nov 22 2014

Table starts
....3.....5.......7........9........11........13.........15..........17
....6....17......36.......65.......106.......161........232.........321
...10....38......99......205.......370.......606........927........1345
...20...125.....476.....1351......3154......6433......11906.......20461
...36...335....1693.....5982.....16790.....39916......84094......161350
...72..1061....7504....34415....119364....341011.....845358.....1878315
..136..3069...29221...169352....713260...2399000....6847916....17247435
..272..9495..123242...904695...4620694..18334295...60473968...173147889
..528.28221..492076..4547008..28033122.130350889..493271080..1595410130
.1056.86149.2021436.23448029.174036890.947356115.4110606460.15000578409

			Some solutions for n=5 k=4
..3....0....3....1....3....3....2....1....2....0....0....1....2....4....0....3
..1....2....1....0....4....2....0....3....1....0....0....0....4....0....2....0
..4....0....0....0....1....4....3....2....2....0....1....0....3....2....2....0
..3....2....2....2....4....2....0....0....1....2....1....4....2....1....4....2
..2....3....3....0....4....4....4....3....2....1....4....1....3....4....1....1
..1....2....1....1....3....4....0....3....3....1....0....3....3....0....2....1

R. H. Hardin, Table of n, a(n) for n = 1..313

Column 1 is A005418(n+2)
Row 1 is A004273(n+1)
Row 2 is A084990(n+1)

Empirical for column k:
k=1: a(n) = 2*a(n-1) +2*a(n-2) -4*a(n-3)
k=2: [order 10]
k=3: [order 29]
Empirical for row n:
n=1: a(n) = 2*n + 1
n=2: a(n) = (1/3)*n^3 + 2*n^2 + (8/3)*n + 1
n=3: a(n) = 3*a(n-1) -2*a(n-2) -2*a(n-3) +3*a(n-4) -a(n-5); also a cubic polynomial plus a constant quasipolynomial with period 2
n=4: [linear recurrence of order 10; also a quintic polynomial plus a linear quasipolynomial with period 3]
n=5: [order 17; also a quintic polynomial plus a quadratic quasipolynomial with period 12]

A250646 T(n,k)=Number of length n+1 0..k arrays with the sum of the maximum of each adjacent pair multiplied by some arrangement of +-1 equal to zero.

Original entry on oeis.org

1, 1, 6, 1, 17, 6, 1, 36, 23, 20, 1, 65, 44, 125, 28, 1, 106, 89, 476, 280, 72, 1, 161, 134, 1293, 1424, 1061, 120, 1, 232, 219, 2954, 4853, 7696, 2870, 272, 1, 321, 296, 5901, 12473, 34441, 28238, 9495, 496, 1, 430, 433, 10766, 28379, 120114, 163043, 126482

Offset: 1

R. H. Hardin, Nov 26 2014

Table starts
......1.........1............1.............1...............1...............1
......6........17...........36............65.............106.............161
......6........23...........44............89.............134.............219
.....20.......125..........476..........1293............2954............5901
.....28.......280.........1424..........4853...........12473...........28379
.....72......1061.........7696.........34441..........120114..........332827
....120......2870........28238........163043..........677505.........2225195
....272......9495.......126482........915663.........4749950........18399217
....496.....27507.......491943.......4537317........28200435.......129137886
...1056.....86149......2059700......23671551.......177863786.......953809557
...2016....255704......8161068.....118358549......1063874048......6704767056
...4160....782393.....33268124.....601565301......6491819162.....47777146765
...8128...2341381....132637221....3011330309.....38892883673....335147823244
..16512...7090347....534771362...15155615651....234724691398...2360792885729
..32640..21271463...2136620867...75845220727...1407192408230..16540740396740
..65792..64109181...8574987528..380253505733...8460554956974.116054610957529
.130816.192439733..34285733053.1902264449049..50741165814612.812699929957712
.262656.578665211.137334914170.9522274036139.304671802762820

			Some solutions for n=5 k=4
..4....2....1....4....0....1....1....2....0....2....1....2....2....4....4....3
..1....1....1....1....2....2....0....2....0....1....0....1....2....0....1....3
..1....0....1....1....2....0....0....1....1....0....2....0....2....2....0....1
..0....1....0....0....0....0....1....1....2....4....3....4....1....3....2....2
..1....1....1....3....1....4....3....0....1....3....3....3....1....0....1....4
..3....0....4....3....3....3....3....0....0....2....0....3....3....2....0....4

R. H. Hardin, Table of n, a(n) for n = 1..267

Column 1 is A113979(n+2)

Row 2 is A084990(n+1)

Empirical for column k:
k=1: a(n) = 2*a(n-1) +2*a(n-2) -4*a(n-3)
k=2: [order 10]
k=3: [order 24] for n>25
Empirical for row n:
n=1: a(n) = a(n-1)
n=2: a(n) = (1/3)*n^3 + 2*n^2 + (8/3)*n + 1
n=3: a(n) = a(n-1) +3*a(n-2) -3*a(n-3) -3*a(n-4) +3*a(n-5) +a(n-6) -a(n-7); also a polynomial of degree 3 plus a quasipolynomial of degree 2 with period 2
n=4: [order 14; also a polynomial of degree 5 plus a quasipolynomial of degree 2 with period 6]
n=5: [order 25; also a polynomial of degree 5 plus a quasipolynomial of degree 4 with period 12]

cp's OEIS Frontend

A028387 a(n) = n + (n+1)^2.

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A212500 a(n) is the difference between multiples of 5 with even and odd digit sum in base 4 in interval [0,4^n).

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A077415 a(n) = n*(n+2)*(n-2)/3.

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A200871 T(n,k)=Number of 0..k arrays x(0..n+1) of n+2 elements without any interior element greater than both neighbors or less than both neighbors.

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A214352 T(n,k)=Number of nXnXn triangular 0..k arrays with no element lying outside the (possibly reversed) range delimited by its sw and se neighbors.

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A250419 T(n,k)=Number of length n+1 0..k arrays with the sum of the minimum of each adjacent pair multiplied by some arrangement of +-1 equal to zero.

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A250646 T(n,k)=Number of length n+1 0..k arrays with the sum of the maximum of each adjacent pair multiplied by some arrangement of +-1 equal to zero.

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A077415 a(n) = n(n+2)(n-2)/3.