A007070 -id:A007070 - cp's OEIS frontend

A053538 Triangle: a(n,m) = ways to place p balls in n slots with m in the rightmost p slots, 0<=p<=n, 0<=m<=n, summed over p, a(n,m)= Sum_{k=0..n} binomial(k,m)*binomial(n-k,k-m), (see program line).

1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 5, 4, 1, 1, 8, 10, 7, 5, 1, 1, 13, 18, 16, 9, 6, 1, 1, 21, 33, 31, 23, 11, 7, 1, 1, 34, 59, 62, 47, 31, 13, 8, 1, 1, 55, 105, 119, 101, 66, 40, 15, 9, 1, 1, 89, 185, 227, 205, 151, 88, 50, 17, 10, 1, 1, 144, 324, 426, 414, 321, 213, 113, 61, 19, 11, 1, 1

Offset: 0

Wouter Meeussen, May 23 2001

Riordan array (1/(1-x-x^2), x(1-x)/(1-x-x^2)). Row sums are A000079. Diagonal sums are A006053(n+2). - Paul Barry, Nov 01 2006
Subtriangle of the triangle given by (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 05 2012
Mirror image of triangle in A208342. - Philippe Deléham, Mar 05 2012
A053538 is jointly generated with A076791 as an array of coefficients of polynomials u(n,x):  initially, u(1,x)=v(1,x)=1, for n>1, u(n,x) = x*u(n-1,x) + v(n-1,x) and v(n,x) = u(n-1,x) + v(n-1,x).  See the Mathematica section at A076791. - Clark Kimberling, Mar 08 2012
The matrix inverse starts
    1;
   -1,   1;
   -1,  -1,   1;
    1,  -2,  -1,  1;
    3,   1,  -3, -1,  1;
    1,   6,   1, -4, -1,  1;
   -7,   4,  10,  1, -5, -1,  1;
  -13, -13,   8, 15,  1, -6, -1,  1;
    3, -31, -23, 13, 21,  1, -7, -1, 1; - R. J. Mathar, Mar 15 2013
Also appears to be the number of subsets of {1..n} containing n with k maximal anti-runs of consecutive elements increasing by more than 1. For example, the subset {1,3,6,7,11,12} has maximal anti-runs ((1,3,6),(7,11),(12)) so is counted under a(12,3). For runs instead of anti-runs we get A202064. - Gus Wiseman, Jun 26 2025

			n=4; Table[binomial[k, j]binomial[n-k, k-j], {k, 0, n}, {j, 0, n}] splits {1, 4, 6, 4, 1} into {{1, 0, 0, 0, 0}, {3, 1, 0, 0, 0}, {1, 4, 1, 0, 0}, {0, 0, 3, 1, 0}, {0, 0, 0, 0, 1}} and this gives summed by columns {5, 5, 4, 1, 1}
Triangle begins :
   1;
   1,  1;
   2,  1,  1;
   3,  3,  1, 1;
   5,  5,  4, 1, 1;
   8, 10,  7, 5, 1, 1;
  13, 18, 16, 9, 6, 1, 1;
...
(0, 1, 1, -1, 0, 0, 0, ...) DELTA (1, 0, -1, 1, 0, 0, 0, ...) begins :
  1;
  0,  1;
  0,  1,  1;
  0,  2,  1,  1;
  0,  3,  3,  1, 1;
  0,  5,  5,  4, 1, 1;
  0,  8, 10,  7, 5, 1, 1;
  0, 13, 18, 16, 9, 6, 1, 1;

Alois P. Heinz, Rows n = 0..140, flattened
R. P. Grimaldi, Extraordinary subsets: a generalization, Fib. Quart., 55 (No. 3, 2017), 114-122. See Table 1.

Column k = 1 is A000045.
Row sums are A000079.
Column k = 2 is A010049.
For runs instead of anti-runs we have A202064.
For integer partitions see A268193, strict A384905, runs A116674.
A034839 counts subsets by number of maximal runs.
A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.
A384877 gives lengths of maximal anti-runs in binary indices, firsts A384878.
A384893 counts subsets by number of maximal anti-runs.
Cf. A000071, A001629, A010027, A208342, A384177, A384879, A384889, A384890.

Flat(List([0..12], n-> List([0..n], k-> Sum([0..n], j->  Binomial(j,k)*Binomial(n-j,j-k)) ))); # G. C. Greubel, May 16 2019

[[(&+[Binomial(j,k)*Binomial(n-j,j-k): j in [0..n]]): k in [0..n]]: n in [0..12]]; // G. C. Greubel, May 16 2019

a:= (n, m)-> add(binomial(k, m)*binomial(n-k, k-m), k=0..n):
seq(seq(a(n,m), m=0..n), n=0..12);  # Alois P. Heinz, Sep 19 2013

Table[Sum[Binomial[k, m]*Binomial[n-k, k-m], {k,0,n}], {n,0,12}, {m,0,n}]

{T(n,k) = sum(j=0,n, binomial(j,k)*binomial(n-j,j-k))}; \\ G. C. Greubel, May 16 2019

[[sum(binomial(j,k)*binomial(n-j,j-k) for j in (0..n)) for k in (0..n)] for n in (0..12)] # G. C. Greubel, May 16 2019

From Philippe Deléham, Mar 05 2012: (Start)
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) - T(n-2,k-1), T(0,0) = T(1,0) = T(1,1) = 1 and T(n,k) = 0 if k<0 or if k>n.
G.f.: 1/(1-(1+y)*x-(1-y)*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A077957(n), A000045(n+1), A000079(n), A001906(n+1), A007070(n), A116415(n), A084326(n+1), A190974(n+1), A190978(n+1), A190984(n+1), A190990(n+1), A190872(n+1) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively. (End)

A061594 Number of ways to place 3n nonattacking kings on a 6 X 2n chessboard.

Original entry on oeis.org

1, 32, 408, 3600, 26040, 166368, 976640, 5392704, 28432288, 144605184, 714611200, 3449705600, 16333065216, 76081271168, 349524164224, 1586790140800, 7130144209024, 31752978219904, 140298397039232, 615604372260736

Offset: 0

Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 22 2001

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. E. Knuth, Nonattacking kings on a chessboard, 1994.
Vaclav Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes [_Vaclav Kotesovec_, Feb 06 2010]
H. S. Wilf, The problem of the kings, Elec. J. Combin. 2, 1995.
Index entries for linear recurrences with constant coefficients, signature (19, -148, 604, -1364, 1644, -928, 192).

Column k=3 of A350819.
Cf. A001787, A061593.
Equals 231*A002697(n+1) - 2608*A000302(n) - 384*A000244(n) + 1103*A007070(n-1) + 780*A006012(n+1) + (n+1)*(17*A048580(n) + 12*A007070(n+1)).

a(n)=polcoeff((1+13*x-52*x^2-20*x^3+60*x^4-20*x^5)/((1-3*x)*(1-4*x)^2*(1-4*x+2*x^2)^2)+x*O(x^n),n)

G.f.: (1+13x-52x^2-20x^3+60x^4-20x^5)/((1-3x)(1-4x)^2(1-4x+2x^2)^2).

Explicit formula: (231n-2377)*4^n - 384*3^n + (1953*sqrt(2)/2+1381+(35*sqrt(2)+99/2)*n)*(2+sqrt(2))^n + (1381-1953*sqrt(2)/2+(99/2-35*sqrt(2))*n)*(2-sqrt(2))^n. - Vaclav Kotesovec, Feb 06 2010

Corrected data by Vincenzo Librandi, Oct 12 2011

A116415 a(n) = 5a(n-1) - 3a(n-2).

Original entry on oeis.org

1, 5, 22, 95, 409, 1760, 7573, 32585, 140206, 603275, 2595757, 11168960, 48057529, 206780765, 889731238, 3828313895, 16472375761, 70876937120, 304967558317, 1312206980225, 5646132226174, 24294040190195, 104531804272453

Offset: 0

Paul Barry, Feb 13 2006

Row sums of A116414.
Partial sums of A018902. - Greg Dresden and Mulong Xu, Aug 31 2024
Binomial transform of the sequence A006190. - Sergio Falcon, Nov 23 2007
a(n+1) equals the number of words of length n over {0,1,2,3,4} avoiding 01, 02 and 03. - Milan Janjic, Dec 17 2015

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Sergio Falcón, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.
Sergio Falcón and Ángel Plaza, On k-Fibonacci sequences and polynomials and their derivatives, Chaos, Solitons & Fractals, Volume 39, Issue 3, 15 February 2009, Pages 1005-1019.
Index entries for linear recurrences with constant coefficients, signature (5,-3).

Cf. A001906, A007070, A018902, A084326.

I:=[1,5]; [n le 2 select I[n] else 5*Self(n-1)-3*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 16 2015

Join[{a=1,b=5},Table[c=5*b-3*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 18 2011 *)
LinearRecurrence[{5,-3},{1,5},40] (* Harvey P. Dale, Jun 19 2012 *)

Vec(1/(1-5*x+3*x^2) + O(x^100)) \\ Altug Alkan, Dec 17 2015

[lucas_number1(n,5,3) for n in range(1, 24)] # Zerinvary Lajos, Apr 22 2009

G.f.: 1/(1 - 5*x + 3*x^2).
a(n) = Sum_{k=0..n} Sum_{j=0..n} C(n-j,k)*C(k+j,j)*3^j.
a(n) = (1/sqrt(13))*(((5+sqrt(13))/2)^n - ((5-sqrt(13))/2)^n). - Sergio Falcon, Nov 23 2007
If p[i] = (3^i-1)/2, and if A is the Hessenberg matrix of order n defined by: A[i,j] = p[j-i+1], (i <= j), A[i,j] = -1, (i = j+1), and A[i,j] = 0 otherwise. Then, for n >= 1, a(n-1) = det(A). - Milan Janjic, May 08 2010
a(n) = 4*a(n-1) + a(n-2) + a(n-3) + ... + a(0) + 1. These expansions with the partial sums on one side can be generated en masse by taking the g.f. of the partial sum and its partial fraction, 1/(1-x)/(1 - 5*x + 3*x^2) = -1/(1-x)+(2-3*x)/(1 - 5*x + 3*x^2) and reading this as a(0) + a(1) + ... + a(n) = -1 + 2*a(n)- 3*a(n-1). - Gary W. Adamson, Feb 18 2011

A228405 Pellian Array, A(n, k) with numbers m such that 2*m^2 +- 2^k is a square, and their corresponding square roots, read downward by diagonals.

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 2, 2, 3, 5, 0, 2, 4, 7, 12, 4, 4, 6, 10, 17, 29, 0, 4, 8, 14, 24, 41, 70, 8, 8, 12, 20, 34, 58, 99, 169, 0, 8, 16, 28, 48, 82, 140, 239, 408, 16, 16, 24, 40, 68, 116, 198, 338, 577, 985, 0, 16, 32, 56, 96, 164, 280, 478, 816, 1393, 2378

Offset: 0

Richard R. Forberg, Aug 21 2013

The left column, A(n,0), is A000129(n), Pell Numbers.
The top row, A(0,k), is A077957(k) plus an initial 0, which is the inverse binomial transform of A000129.
These may be considered initializing values, or results, depending the perspective taken, since there are several ways to generate the array. See Formula section for details.
The columns of the array hold all values, in sequential order, of numbers m such that 2m^2 + 2^k or  2m^2 - 2^k are squares, and their corresponding square roots in the next column, which then form the "next round" of m values for k+1.
For example A(n,0) are numbers such that 2m^2 +- 1 are squares, the integer square roots of each are in A(n,1), which are then numbers m such that 2m^2 +- 2 are squares, with those square roots in A(n,2), etc.
A(n, k)/A(n,k-2) = 2; A(n,k)/A(n,k-1) converges to sqrt(2) for large n.
A(n,k)/A(n-1,k) converges to 1 + sqrt(2) for large n.
The other columns of this array hold current OEIS sequences as follows:
  A(n,1) = A001333(n);   A(n,2) = A163271(n);  A(n,3) = A002203(n);
Bisections of these column-oriented sequences also appear in the OEIS, corresponding to the even and odd rows of the array, which in turn correspond to the two different recursive square root equations in the formula section below.
Farey fraction denominators interleave columns 0 and 1, and the corresponding numerators interleave columns 1 and 2, for approximating sqrt(2). See A002965 and A119016, respectively.
The other rows of this array hold current OEIS sequences as follows:
A(1,k) = A016116(k);   A(2,k) = A029744(k) less the initial 1;
A(3,k) = A070875(k);    A(4,k) = A091523(k) less the initial 8.
The Pell Numbers (A000219) are the only initializing set of numbers where the two recursive square root equations (see below) produce exclusively integer values, for all iterations of k.  For any other initial values only even iterations (at k = 2, 4, ...) produce integers.
The numbers in this array, especially the first three columns, are also integer square roots of these expressions: floor(m^2/2), floor(m^2/2 + 1), floor (m^2/2 - 1). See A227972 for specific arrangements and relationships. Also: ceiling(m^2/2), ceiling(m^2/2 + 1), ceiling (m^2/2 -1), m^2+1, m^2-1, m^2*(m^2-1)/2, m^2*(m^2-1)/2, in various different arrangements. Many of these involve: A000129(2n)/2 = A001109(n).
A001109 also appears when multiplying adjacent columns:   A(n,k) * A(n,k+1) = (k+1) * A001109(n), for all k.

			With row # as n. and column # as k, and n, k =>0, the array begins:
0,     1,     0,     2,     0,     4,     0,     8, ...
1,     1,     2,     2,     4,     4,     8,     8, ...
2,     3,     4,     6,     8,    12,    16,    24, ...
5,     7,    10,    14,    20,    28,    40,    56, ...
12,   17,    24,    34,    48,    68,    96,   136, ...
29,   41,    58,    82,   116,   164,   232,   328, ...
70,   99,   140,   198,   280,   396,   560,   792, ...
169,  239,  338,   478,   676,   956,  1352,  1912, ...
408,  577,  816,  1154,  1632,  2308,  3264,  4616, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
MacTutor, D'Arcy Thompson on Greek irrationals
D'Arcy Thompson, Excess and Defect: Or the Little More and the Little Less, Mind, New Series, Vol. 38, No. 149 (Jan., 1929), pp. 43-55 (13 pages). See page 50.

Cf. A000219, A077957, A001333, A163271, A002203, A001109, A227972.

Main diagonal is A007070.

If using the left column and top row to initialize: A(n,k) = A(n,k-1) + A(n-1,k-1).
If using only the top row to initialize, then each column for k = i is the binomial transform of A(0,k) restricted to k=> i as input to the transform with an appropriate down shift of index. The inverse binomial transform with a similar condition can produce each row from A000129.
If using only the first two rows to initialize then the Pell equation produces each column, as: A(n,k) = 2*A(n-1, k) + A(n-2, k).
If using only the left column (A000219(n) = Pell Numbers) to initialize then the following two equations will produce each row:
     A(n,k) = sqrt(2*A(n,k-1) + (-2)^(k-1)) for even rows
     A(n,k) = sqrt(2*A(n,k-1) - (-2)^(k-1)) for odd rows.
Interestingly, any portion of the array can also be filled "backwards" given the top row and any column k, using only:  A(n,k-1) = A(n-1,k-1) + A(n-1, k), or if given any column and its column number by rearranging the sqrt recursions above.

A216210 Square array T read by antidiagonals: T(n,k) = 0 if n-k>=4 or if k-n>=4, T(3,0) = T(2,0) = TT(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 0, 4, 6, 4, 0, 0, 4, 10, 10, 4, 0, 0, 0, 14, 20, 14, 0, 0, 0, 0, 14, 34, 34, 14, 0, 0, 0, 0, 0, 48, 68, 48, 0, 0, 0, 0, 0, 0, 48, 116, 116, 48, 0, 0, 0, 0, 0, 0, 0, 164, 232, 164, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Mar 12 2013

			Square array begins:
1, 1,  1,  1,   0,   0,   0,    0,    0, 0, ... row n=0
1, 2,  3,  4,   4,   0,   0,    0,    0, 0, ... row n=1
1, 3,  6, 10,  14,  14,   0,    0,    0, 0, ... row n=2
1, 4, 10, 20,  34,  48,  48,    0,    0, 0, ... row n=3
0, 4, 14, 34,  68, 116, 164,  164,    0, 0, ... row n=4
0, 0, 14, 48, 116, 232, 396,  560,  560, 0, ... row n=5
0, 0,  0, 48, 164, 396, 792, 1352, 1912, 1912,  row n=6
...

Cf. A006012, A007052, A007070, A068912

T(n,n) = A006012(n).
T(n,n+1) = T(n+1,n) = A007052(n).
T(n,n+2) = T(n,n+3) = T(n+2,n) = T(n+3,n) = A007070(n).
Sum_{k, 0<=k<=n} T(n-k,k) = A068912(n).

A007068 a(n) = a(n-1) + (3+(-1)^n)*a(n-2)/2.

Original entry on oeis.org

1, 3, 4, 10, 14, 34, 48, 116, 164, 396, 560, 1352, 1912, 4616, 6528, 15760, 22288, 53808, 76096, 183712, 259808, 627232, 887040, 2141504, 3028544, 7311552, 10340096, 24963200, 35303296, 85229696, 120532992, 290992384, 411525376, 993510144, 1405035520, 3392055808

Offset: 1

N. J. A. Sloane, Mira Bernstein

First row of spectral array W(sqrt 2).

Row sums of the square of the matrix with general term binomial(floor(n/2),n-k). - Paul Barry, Feb 14 2005

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Aviezri S. Fraenkel and Clark Kimberling, Generalized Wythoff arrays, shuffles and interspersions, Discr. Math. 126 (1-3) (1994) 137-149.
Sean A. Irvine, Walks on Graphs.
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-2).

Cf. A062112, A001882, A007070, A007052, A000034.

a007068 n = a007068_list !! (n-1)
a007068_list = 1 : 3 : zipWith (+)
   (tail a007068_list) (zipWith (*) a000034_list a007068_list)
-- Reinhard Zumkeller, Jan 21 2012

RecurrenceTable[{a[1]==1,a[2]==3,a[n]==a[n-1]+(3+(-1)^n) a[n-2]/2},a[n],{n,40}] (* Harvey P. Dale, Nov 12 2012 *)

a(2n+1) = a(2n)+a(2n-1); a(2n) = a(2n-1)+2*a(2n-2); same recurrence (mod parity) as A001882. - Len Smiley, Feb 05 2001
a(n) = Sum_{k=0..n} Sum_{j=0..n} C(floor(n/2), n-j)*C(floor(j/2), j-k). - Paul Barry, Feb 14 2005
a(n) = 4*a(n-2)-2*a(n-4). G.f.: -x*(1+x)*(2*x^2-2*x-1)/(1-4*x^2+2*x^4). a(2n+1)=A007070(n). a(2n)=A007052(n). [R. J. Mathar, Aug 17 2009]
a(n) = a(n-1) + a(n-2) * A000034(n-1). [Reinhard Zumkeller, Jan 21 2012]

Better description and more terms from Olivier Gérard, Jun 05 2001

A164298 a(n) = ((1+4sqrt(2))(2+sqrt(2))^n + (1-4sqrt(2))(2-sqrt(2))^n)/2.

Original entry on oeis.org

1, 10, 38, 132, 452, 1544, 5272, 18000, 61456, 209824, 716384, 2445888, 8350784, 28511360, 97343872, 332352768, 1134723328, 3874187776, 13227304448, 45160842240, 154188760064, 526433355776, 1797355902976, 6136556900352, 20951515795456, 71532949381120

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 12 2009

Binomial transform of A048696. Second binomial transform of A164587. Inverse binomial transform of A164299.

This sequence is part of a class of sequences defined by the recurrence a(n,m) = 2*(m+1)*a(n-1,m) - ((m+1)^2 - 2)*a(n-2,m) with a(0) = 1 and a(1) = m+9. The generating function is Sum_{n>=0} a(n,m)*x^n = (1 - (m-7)*x)/(1 - 2*(m+1)*x + ((m+1)^2 - 2)*x^2) and has a series expansion in terms of Pell-Lucas numbers defined by a(n, m) = (1/2)*Sum_{k=0..n} binomial(n,k)*m^(n-k)*(5*Q(k) + 4*Q(k-1)). - G. C. Greubel, Mar 12 2021

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-2).

Sequences in the class a(n, m): this sequence (m=1), A164299 (m=2), A164300 (m=3), A164301 (m=4), A164598 (m=5), A164599 (m=6), A081185 (m=7), A164600 (m=8).
Cf. A007070, A016921, A048696, A056236, A112032, A164587, A241204.
Cf. A016116(n+1).

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+4*r)*(2+r)^n+(1-4*r)*(2-r)^n)/2: n in [0..27] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 17 2009

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1+6*x)/(1-4*x+2*x^2) )); // G. C. Greubel, Dec 14 2018

a:=n->((1+4*sqrt(2))*(2+sqrt(2))^n+(1-4*sqrt(2))*(2-sqrt(2))^n)/2: seq(floor(a(n)),n=0..25); # Muniru A Asiru, Dec 15 2018

LinearRecurrence[{4,-2}, {1,10}, 50] (* or *) CoefficientList[Series[(1 + 6*x)/(1 - 4*x + 2*x^2), {x,0,50}], x] (* G. C. Greubel, Sep 12 2017 *)

my(x='x+O('x^50)); Vec((1+6*x)/(1-4*x+2*x^2)) \\ G. C. Greubel, Sep 12 2017

[( (1+6*x)/(1-4*x+2*x^2) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Dec 14 2018; Mar 12 2021

a(n) = 4*a(n-1) - 2*a(n-2) for n > 1; a(0)=1, a(1)=10.
G.f.: (1+6*x)/(1-4*x+2*x^2).
E.g.f.: (cosh(sqrt(2)*x) + 4*sqrt(2)*sinh(sqrt(2)*x))*exp(2*x). - G. C. Greubel, Sep 12 2017
From G. C. Greubel, Mar 12 2021: (Start)
a(n) = A056236(n) + 8*A007070(n-1).
a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 17 2009

A229392 T(n,k)=Number of nXk 0..3 arrays of the sum of the corresponding element, the element to the east and the element to the south in a larger (n+1)X(k+1) 0..1 array.

Original entry on oeis.org

4, 14, 14, 48, 128, 48, 164, 1064, 1064, 164, 560, 8592, 19124, 8592, 560, 1912, 68672, 319340, 319340, 68672, 1912, 6528, 546752, 5212236, 10624396, 5212236, 546752, 6528, 22288, 4346752, 84210828, 345788172, 345788172, 84210828, 4346752, 22288

Offset: 1

R. H. Hardin Sep 21 2013

Table starts
.....4........14...........48.............164................560
....14.......128.........1064............8592..............68672
....48......1064........19124..........319340............5212236
...164......8592.......319340........10624396..........345788172
...560.....68672......5212236.......345788172........22494002188
..1912....546752.....84210828.....11156280332......1451228983308
..6528...4346752...1353901580....358453456908.....93250181644300
.22288..34537984..21715025932..11493734735884...5979900142878732
.76096.274370048.347864379404.368171037655052.383094040360124428

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

R. H. Hardin, Table of n, a(n) for n = 1..287

Column 1 is A007070

Empirical for column k:
k=1: a(n) = 4*a(n-1) -2*a(n-2)
k=2: a(n) = 12*a(n-1) -36*a(n-2) +32*a(n-3) -16*a(n-4)
k=3: a(n) = 25*a(n-1) -152*a(n-2) +144*a(n-3) -368*a(n-4) +1888*a(n-5) -1536*a(n-6) for n>9
k=4: a(n) = 49*a(n-1) -560*a(n-2) +544*a(n-3) -1568*a(n-4) +17920*a(n-5) -16384*a(n-6) for n>9
k=5: a(n) = 101*a(n-1) -2532*a(n-2) +10624*a(n-3) -8192*a(n-4) for n>7
k=6: a(n) = 193*a(n-1) -8384*a(n-2) +8320*a(n-3) -24704*a(n-4) +1073152*a(n-5) -1048576*a(n-6) for n>9
k=7: a(n) = 385*a(n-1) -33152*a(n-2) +33024*a(n-3) -98560*a(n-4) +8486912*a(n-5) -8388608*a(n-6) for n>9
k=8: a(n) = 777*a(n-1) -137992*a(n-2) +1185792*a(n-3) -1048576*a(n-4) for n>7
k=9: a(n) = 1537*a(n-1) -525824*a(n-2) +525312*a(n-3) -1573888*a(n-4) +538443776*a(n-5) -536870912*a(n-6) for n>9
k=10: a(n) = 3073*a(n-1) -2100224*a(n-2) +2099200*a(n-3) -6293504*a(n-4) +4301258752*a(n-5) -4294967296*a(n-6) for n>9
k=11: a(n) = 6161*a(n-1) -8493072*a(n-2) +142704640*a(n-3) -134217728*a(n-4) for n>7

A035344 Expansion of 1/((1 - x)(1 - 4x + 2 * x^2)).

Original entry on oeis.org

1, 5, 19, 67, 231, 791, 2703, 9231, 31519, 107615, 367423, 1254463, 4283007, 14623103, 49926399, 170459391, 581984767, 1987020287, 6784111615, 23162405887, 79081400319, 270000789503, 921840357375, 3147359850495, 10745758687231, 36688315047935, 125261742817279

Offset: 0

N. J. A. Sloane

S. Bilotta, E. Pergola, R. Pinzani, and S. Rinaldi, Recurrence Relations, Succession Rules, and the Positivity Problem, in Language and Automata Theory and Applications, 9th International Conference, LATA 2015, Nice, France, March 2-6, 2015, Proceedings, Pages 499-510, Lecture Notes Comp. Sci. Vol. 8977.

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Stefano Bilotta, Elisa Pergola, Renzo Pinzani, and Simone Rinaldi, Recurrence relations versus succession rules, arXiv preprint arXiv:1301.2967 [cs.DM], 2013.
Sela Fried, Even-up words and their variants, arXiv:2505.14196 [math.CO], 2025. See p. 8.
László Németh, Hyperbolic Pascal pyramid, arXiv:1511.02067 [math.CO], 2015 (2nd line of Table 2 is 6*a(n-3)).
László Németh, Pascal pyramid in the space H^2 x R, arXiv:1701.06022 [math.CO], 2017 (2nd line of Table is 2*a(n-3)).
Index entries for linear recurrences with constant coefficients, signature (5,-6,2).

Partial sums of A007070.

Cf. A000129, A007052.

a[ -1]:=0:a[0]:=1:for n from 1 to 50 do a[n]:=4*a[n-1]-2*a[n-2]+1 od: seq(a[n],n=0..50); # after Miklos Kristof

Join[{a=1,b=5},Table[c=4*b-2*a+1;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 06 2011 *)
CoefficientList[Series[1/((1-x)(1-4x+2x^2)),{x,0,30}],x] (* or *) LinearRecurrence[ {5,-6,2},{1,5,19},30] (* Harvey P. Dale, Mar 28 2016 *)

Vec(1/((1-x)*(1-4*x+2*x^2))+O(x^99)) \\ Charles R Greathouse IV, Sep 24 2012

a(n) = 2*A007052(n)-1. The sequence 0, 0, 1, 5, 19, ... is the binomial transform of the Pell numbers A000129, preceded by an additional 0. a(n) = (1 + 1/sqrt(2))(2 + sqrt(2))^n + (1 - 1/sqrt(2))(2 - sqrt(2))^n - 1. - Paul Barry, Jul 16 2003
a(-1)=0, a(0)=1, a(n) = 4*a(n-1) - 2*a(n-2) + 1. - Miklos Kristof, Mar 09 2005
E.g.f.: exp(2*x)*(2*cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x)) - cosh(x) - sinh(x). - Stefano Spezia, May 20 2024

a(23) onwards from Andrew Howroyd, Jan 28 2024

A062112 a(0)=0; a(1)=1; a(n) = a(n-1) + (3 + (-1)^n)*a(n-2)/2.

Original entry on oeis.org

0, 1, 1, 2, 4, 6, 14, 20, 48, 68, 164, 232, 560, 792, 1912, 2704, 6528, 9232, 22288, 31520, 76096, 107616, 259808, 367424, 887040, 1254464, 3028544, 4283008, 10340096, 14623104, 35303296, 49926400, 120532992, 170459392, 411525376

Offset: 0

Olivier Gérard, Jun 05 2001

			a(4) = a(3) + 2*a(2) = 2 + 2 = 4.

Harry J. Smith, Table of n, a(n) for n = 0..200
Sean A. Irvine, Walks on Graphs.
Index entries for linear recurrences with constant coefficients, signature (0, 4, 0, -2).

Cf. A006012, A007068, A007070.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1+x-2*x^2)/(1-4*x^2+2*x^4))); // G. C. Greubel, Oct 16 2018

RecurrenceTable[{a[0]==0,a[1]==1,a[n]==a[n-1]+(3+(-1)^n) (a[n-2])/2},a,{n,40}] (* or *) LinearRecurrence[{0,4,0,-2},{0,1,1,2},40] (* Harvey P. Dale, May 24 2013 *)

{ for (n=0, 200, if (n>1, a=a1 + (3 + (-1)^n)*a2/2; a2=a1; a1=a, if (n==0, a=a2=0, a=a1=1)); write("b062112.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 01 2009

a(2*n) = A007070(n+1).
a(2*n+1) = A006012(n).
G.f.: x*(1+x-2*x^2)/(1-4*x^2+2*x^4).
a(n) = 4*a(n-2) - 2*a(n-4), a(0)=0, a(1)=1, a(2)=1, a(3)=2. - Harvey P. Dale, May 24 2013

cp's OEIS Frontend

A053538 Triangle: a(n,m) = ways to place p balls in n slots with m in the rightmost p slots, 0<=p<=n, 0<=m<=n, summed over p, a(n,m)= Sum_{k=0..n} binomial(k,m)*binomial(n-k,k-m), (see program line).

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A061594 Number of ways to place 3n nonattacking kings on a 6 X 2n chessboard.

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A116415 a(n) = 5*a(n-1) - 3*a(n-2).

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A228405 Pellian Array, A(n, k) with numbers m such that 2*m^2 +- 2^k is a square, and their corresponding square roots, read downward by diagonals.

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A216210 Square array T read by antidiagonals: T(n,k) = 0 if n-k>=4 or if k-n>=4, T(3,0) = T(2,0) = TT(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

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A007068 a(n) = a(n-1) + (3+(-1)^n)*a(n-2)/2.

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A164298 a(n) = ((1+4*sqrt(2))*(2+sqrt(2))^n + (1-4*sqrt(2))*(2-sqrt(2))^n)/2.

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A116415 a(n) = 5a(n-1) - 3a(n-2).

A164298 a(n) = ((1+4sqrt(2))(2+sqrt(2))^n + (1-4sqrt(2))(2-sqrt(2))^n)/2.

A035344 Expansion of 1/((1 - x)(1 - 4x + 2 * x^2)).