A026937 -id:A026937 - cp's OEIS frontend

A364553 Number of edges in the n-Pell graph.

0, 1, 5, 18, 58, 175, 507, 1428, 3940, 10701, 28705, 76230, 200766, 525083, 1365175, 3531240, 9093512, 23325785, 59625981, 151947066, 386139650, 978834759, 2475645491, 6248406780, 15740857452, 39585199525, 99389810585, 249177006702, 623846750086, 1559888545075

Offset: 0

Eric W. Weisstein, Jul 28 2023

For n > 0, also the number of maximum and maximal cliques in the n-Pell graph.

E. Munarini, Pell Graphs, Disc. Math., 342 (2019), 2415-2428.
Eric Weisstein's World of Mathematics, Edge Count
Eric Weisstein's World of Mathematics, Maximal Clique
Eric Weisstein's World of Mathematics, Maximum Clique
Eric Weisstein's World of Mathematics, Pell Graph
Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,-1).

Cf. A000129, A001333, A008288, A026937, A093967, A364636, row sums of A364361.

A364553 := n -> (n/8)*((2 + sqrt(2))*(1 + sqrt(2))^n - (sqrt(2) - 2)*(1 - sqrt(2))^n): seq(simplify(A364553(n)), n=0..29); # Peter Luschny, Jul 30 2023

Table[n Fibonacci[n + 1, 2]/2, {n, 0, 20}]
Table[n (Fibonacci[n, 2] + (-I)^n ChebyshevT[n, I])/2, {n, 0, 20}]
Table[With[{s = Sqrt[2]}, n ((s + 2) (1 + s)^n - (s - 2) (1 - s)^n)/8], {n, 0, 20}] // Expand
LinearRecurrence[{4, -2, -4, -1}, {0, 1, 5, 18}, 20]
CoefficientList[Series[x (1 + x)/(-1 + 2 x + x^2)^2, {x, 0, 20}], x]

# Using function 'delannoy_row' from A008288.
def A364553(n:int) -> int:
    return sum(k * delannoy_row(n)[k] for k in range(n + 1))
print([A364553(n) for n in range(30)])  # Peter Luschny, Jul 30 2023

a(n) = n*(A000129(n) + A001333(n))/2.
a(n) = n*A000129(n+1)/2.
a(n) = 4*a(n-1) - 2*a(n-2) - 4*a(n-3) - a(n-4).
G.f.: x*(1+x)/(-1+2*x+x^2)^2.
From Peter Luschny, Jul 31 2023:  (Start)
a(n) = (n/8)*((2 + sqrt(2))*(1 + sqrt(2))^n - (sqrt(2) - 2)*(1 - sqrt(2))^n).
With this formula, the sequence can be continued to the left half of the number line: a(-n) = -(-1)^n*A026937(n-2) for n >= 0.
a(n) = (A093967(n) + A364636(n)) / 2.
a(n) = Sum_{k=0..n} k * A008288(n, k).  (End)

A088210 Numerators of convergents of the continued fraction with the n+1 partial quotients: [2;2,2,...(n 2's)...,2,n+1], starting with [1], [2;2], [2;2,3], [2;2,2,4], ...

Original entry on oeis.org

1, 5, 17, 53, 157, 449, 1253, 3433, 9273, 24765, 65529, 172061, 448853, 1164409, 3006157, 7728337, 19794545, 50532469, 128621281, 326513669, 826887693, 2089505841, 5269572021, 13265211961, 33336792745, 83648953133, 209591807177

Offset: 0

Paul D. Hanna, Sep 23 2003

Denominators are A088211. Partial sums form A054459. Second differences form A026937.

			a(3)/A088211(3) = [2;2,2,4] = 53/22.

R. P. Grimaldi, Ternary strings with no consecutive 0's and no consecutive 1's, Congressus Numerantium, 205 (2011), 129-149. (See the foot of page 136.)

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,-1).

Cf. A000129, A088211, A054459, A026937.

LinearRecurrence[{4, -2, -4, -1}, {1, 5, 17, 53}, 30] (* Paolo Xausa, Feb 08 2024 *)

G.f.: (1+x)(1-x^2)/(1-2*x-x^2)^2.

a(n) = A000129(n) + (n+1)*A000129(n+1) where A000129 are the Pell numbers. [Corrected by Paolo Xausa, Feb 08 2024]

A164981 A triangle with Pell numbers in the first column.

Original entry on oeis.org

1, 2, 1, 5, 3, 1, 12, 10, 4, 1, 29, 30, 16, 5, 1, 70, 87, 56, 23, 6, 1, 169, 245, 185, 91, 31, 7, 1, 408, 676, 584, 334, 136, 40, 8, 1, 985, 1836, 1784, 1158, 546, 192, 50, 9, 1, 2378, 4925, 5312, 3850, 2052, 834, 260, 61, 10, 1, 5741, 13079, 15497, 12386, 7342, 3366, 1212, 341, 73, 11, 1

Offset: 1

Mark Dols, Sep 03 2009

Rows sum up to A000244 (powers of 3), diagonals to A001654 (golden rectangles).
Up to reflection at the vertical axis, the triangle of numbers given here coincides with the triangle given in A210557, i.e. the numbers are the same just read row-wise in the opposite direction. [Christine Bessenrodt, Jul 20 2012]
Subtriangle of (0, 2, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 10 2013

			Triangle begins
  1
  2,1
  5,3,1
  12,10,4,1
  29,30,16,5,1
  70,87,56,23,6,1
  169,245,185,91,31,7,1
  ...
From _Philippe Deléham_, Oct 10 2013: (Start)
Triangle (0, 2, 1/2, -1/2, 0, 0, ...) DELTA (1, 0, -1/2, 1/2, 0, 0, ...):
  1
  0, 1
  0, 2, 1
  0, 5, 3, 1
  0, 12, 10, 4, 1
  0, 29, 30, 16, 5, 1
  0, 70, 87, 56, 23, 6, 1
  0, 169, 245, 185, 91, 31, 7, 1
  ... (End)

Cf. A000244, A001654, A164975, A164976, A210557.

A164981 := proc(n,k) option remember; if n <1 or k<1 or k>n then 0; elif n = 1 then 1; else 2*procname(n-1,k)+procname(n-1,k-1)+procname(n-2,k)-procname(n-2,k-1) ; end if; end proc:

T[n_, k_] := T[n, k] = Which[n < 1 || k < 1 || k > n, 0, n == 1, 1, True, 2*T[n-1, k] + T[n-1, k-1] + T[n-2, k] - T[n-2, k-1]];
Table[T[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 06 2023 *)

T(n,k) = if ((n==1) && (k==1), return(1)); if ((n<=0) || (k<=0) || (nMichel Marcus, Feb 01 2023

From R. J. Mathar, Jan 27 2011: (Start)
T(1,1) =1. T(n,k)=0 if n<1 or k<1 or k>n. T(n,k) = 2*T(n-1,k)+T(n-1,k-1)+T(n-2,k)-T(n-2,k-1) otherwise.
T(n,1) = A000129(n).
T(n,n-1) = n.
T(n,n-2) = A052905(n-2).
T(n,2) = A026937(n-2). (End)
G.f. x*y/(1-2*x-x^2+x^2*y-x*y). - R. J. Mathar, Aug 11 2015

Rows 10-11 from Michel Marcus, Feb 01 2023

A103415 Triangle, read by rows, T(n,k) = A000129(n+1) - Sum_{j=1..k} t(n+1, j), where t(n, k) is defined in the formula section.

Original entry on oeis.org

1, 2, 1, 5, 4, 1, 12, 11, 6, 1, 29, 28, 21, 8, 1, 70, 69, 60, 35, 10, 1, 169, 168, 157, 116, 53, 12, 1, 408, 407, 394, 333, 204, 75, 14, 1, 985, 984, 969, 884, 653, 332, 101, 16, 1, 2378, 2377, 2360, 2247, 1870, 1189, 508, 131, 18, 1, 5741, 5740, 5721, 5576, 5001, 3712, 2029, 740, 165, 20, 1

Offset: 0

Lambert Klasen (lambert.klasen(AT)gmx.net) and Gary W. Adamson, Feb 04 2005

Triangle is generated from the product A*B of the infinite lower triangular matrices A = A008288(n,k) and B =
  1;
  1 1;
  1 1 1;
  1 1 1 1; ...
Determinant(A*B) = 1 for all n.
Absolute values of coefficients of characteristic polynomials of n-th matrix are the (n+1)-th row of A007318 (Pascal's triangle). As they are:
  x^1 - 1;
  x^2 - 2*x^1 +  1;
  x^3 - 3*x^2 +  3*x^1 -  1;
  x^4 - 4*x^3 +  6*x^2 -  4*x^1 + 1;
  x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x^1 - 1.

			Triangle begins as:
    1;
    2,   1;
    5,   4,   1;
   12,  11,   6,   1;
   29,  28,  21,   8,   1;
   70,  69,  60,  35,  10,  1;
  169, 168, 157, 116,  53, 12,  1;
  408, 407, 394, 333, 204, 75, 14, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000129, A002203, A005409, A007318, A008288, A026937, A093328, A103416.

t[n_, k_]:= If[k==0, (2*Boole[n<2] + LucasL[n-1, 2]*Boole[n>1])/2, Binomial[n-1, k-1]*Hypergeometric2F1[1-k, k-n, 1-n, -1]];
st[n_, k_]:= Sum[t[n+1, j], {j,k}];
T[n_, k_]:= Fibonacci[n+1, 2] - st[n, k];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, May 25 2021 *)

Pell(n) = if( n<2, n, 2*Pell(n-1) + Pell(n-2) );
t(n, k) = if(n<3, 1, if(k==1||k==n, 1, t(n-1,k) + t(n-1,k-1) + t(n-2,k-1) ));
st(n, k) = sum(i=1, k, t(n+1,i));
T(n, k) = Pell(n+1) - st(n,k);
for(n=0, 10, for(k=0, n, print1(T(n,k), ",")); print()) \\ modified by G. C. Greubel, May 25 2021

@CachedFunction
def t(n,k): return 1 if (n<3) else 1 if (k==1 or k==n) else t(n-1,k) + t(n-1,k-1) + t(n-2,k-1)
def st(n,k): return sum(t(n+1, j) for j in (1..k))
def T(n,k): return lucas_number1(n+1,2,-1) - st(n,k)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 25 2021

T(n, k) = Pell(n+1) - ST(n, k), where ST(n, k) = Sum_{j=1..k} t(n+1, j), t(n, k) = t(n-1,k) + t(n-1,k-1) + t(n-2,k-1), t(n, 1) = t(n, n) = 1 and t(0, k) = t(1, k) = t(2, k) = 1.
T(n, 0) = A000129(n+1).
T(n, 1) = A005409(n) = A000129(n) - 1.
Sum_{k=0..n} T(n, k) = A026937(n).
From G. C. Greubel, May 25 2021: (Start)
T(n, k) = A000129(n+1) - st(n,k), where st(n, k) = Sum_{j=1..k} t(n+1, j), t(n, k) = A008288(n-1, k-1) for n >= 1 and k >= 1, and t(n, 0) = (1/2)*(2*[n<2] + A002203(n-1)*[n>1]).
T(n, n) = A000012(n).
T(n, n-1) = A005843(n+1).
T(n, n-2) = A093328(n-1).
T(n, n-3) = (4/3)*((n-3)^3 + 5*(n-3) + 3).
T(n, n-4) = (1/3)*(2*(n-4)^2 + 22*(n-4)^2 + 22*(n-4) + 39). (End)

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A364553 Number of edges in the n-Pell graph.

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A088210 Numerators of convergents of the continued fraction with the n+1 partial quotients: [2;2,2,...(n 2's)...,2,n+1], starting with [1], [2;2], [2;2,3], [2;2,2,4], ...

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A164981 A triangle with Pell numbers in the first column.

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A103415 Triangle, read by rows, T(n,k) = A000129(n+1) - Sum_{j=1..k} t(n+1, j), where t(n, k) is defined in the formula section.

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