A006497 -id:A006497 - cp's OEIS frontend

A302941 Number of total dominating sets in the 2n-crossed prism graph.

9, 121, 1296, 14161, 154449, 1684804, 18378369, 200477281, 2186871696, 23855111401, 260219353689, 2838557779204, 30963916217529, 337764520613641, 3684445810532496, 40191139395243841, 438418087537149729, 4782407823513403204, 52168067971110285489

Offset: 1

Eric W. Weisstein, Apr 16 2018

Eric Weisstein's World of Mathematics, Crossed Prism Graph
Eric Weisstein's World of Mathematics, Total Dominating Set
Index entries for linear recurrences with constant coefficients, signature (10,10,-1).

Cf. A006497, A287062, A291772, A302946.

Table[2 (-1)^n + ((11 - 3 Sqrt[13])/2)^n + ((11 + 3 Sqrt[13])/2)^n, {n, 20}] // FullSimplify
Table[LucasL[n, 3]^2, {n, 20}]
LucasL[Range[20], 3]^2
LinearRecurrence[{10, 10, -1}, {9, 121, 1296}, 20]
CoefficientList[Series[(9 + 31 x - 4 x^2)/(1 - 10 x - 10 x^2 + x^3), {x, 0, 20}], x]

Vec((9 + 31*x - 4*x^2)/((1 + x)*(1 - 11*x + x^2)) + O(x^30)) \\ Andrew Howroyd, Apr 16 2018

From Andrew Howroyd, Apr 16 2018: (Start)
G.f.: x*(9 + 31*x - 4*x^2)/((1 + x)*(1 - 11*x + x^2)).
a(n) = 10*a(n-1) + 10*a(n-2) - a(n-3) for n > 3.
a(n) = A006497(n)^2. (End)

a(1) and terms a(6) and beyond from Andrew Howroyd, Apr 16 2018

A309852 Array read by antidiagonals: ((z+sqrt(x))/2)^k + ((z-sqrt(x))/2)^k for columns k >= 0 and rows n >= 0, where x = 4*n+1 and y = floor(sqrt(x)) and z = y-1+(y mod 2).

Original entry on oeis.org

2, 1, 2, 1, 1, 2, 1, 3, 3, 2, 1, 4, 9, 3, 2, 1, 7, 27, 11, 3, 2, 1, 11, 81, 36, 13, 3, 2, 1, 18, 243, 119, 45, 15, 5, 2, 1, 29, 729, 393, 161, 54, 25, 5, 2, 1, 47, 2187, 1298, 573, 207, 125, 27, 5, 2, 1, 76, 6561, 4287, 2041, 783, 625, 140, 29, 5, 2

Offset: 0

Charles L. Hohn, Aug 20 2019

One of 4 related arrays (the others being A191347, A191348, and A309853) where the two halves of the main formula approach the integers shown and 0 respectively, and also with A309853 where rows represent various Fibonacci series a(n) = a(n-2)*b + a(n-1)*c where b and c are integers >= 0.

			Array begins:
  2, 1,  1,   1,    1,    1,     1,      1,       1,       1,        1, ...
  2, 1,  3,   4,    7,   11,    18,     29,      47,      76,      123, ...
  2, 3,  9,  27,   81,  243,   729,   2187,    6561,   19683,    59049, ...
  2, 3, 11,  36,  119,  393,  1298,   4287,   14159,   46764,   154451, ...
  2, 3, 13,  45,  161,  573,  2041,   7269,   25889,   92205,   328393, ...
  2, 3, 15,  54,  207,  783,  2970,  11259,   42687,  161838,   613575, ...
  2, 5, 25, 125,  625, 3125, 15625,  78125,  390625, 1953125,  9765625, ...
  2, 5, 27, 140,  727, 3775, 19602, 101785,  528527, 2744420, 14250627, ...
  2, 5, 29, 155,  833, 4475, 24041, 129155,  693857, 3727595, 20025689, ...
  2, 5, 31, 170,  943, 5225, 28954, 160445,  889087, 4926770, 27301111, ...
  2, 5, 33, 185, 1057, 6025, 34353, 195865, 1116737, 6367145, 36302673, ...
  ...

Row 2 is A000032, row 3 (except the first term) is A000244, row 4 is A006497, row 5 is A206776, row 6 is A172012, row 7 (except the first term) is A000351, row 8 is A087130.

Cf. A191347, A191348, and A309853.

T(n, k) = my(x = 4*n+1, y = sqrtint(x), z = y-1+(y % 2)); round(((z+sqrt(x))/2)^k + ((z-sqrt(x))/2)^k);
matrix(9,9, n, k, T(n-1,k-1)) \\ Michel Marcus, Aug 22 2019

T(n, k) = my(x = 4*n+1, y = sqrtint(x), z=y-1+(y % 2)); v=if(k==0, 2, k==1, z, mapget(m2, n)*((x-z^2)/4) + mapget(m1, n)*z); mapput(m2, n, if(mapisdefined(m1, n), mapget(m1, n), 0)); mapput(m1, n, v); v;
m1=Map(); m2=Map(); matrix(9, 9, n, k, T(n-1, k-1)) \\ Charles L. Hohn, Aug 26 2019

For each row n>=0 let x = 4*n+1, y = floor(sqrt(x)), T(n,0)=2, and T(n,1)=y-1+(y % 2), then for each column k>=2: T(n, k-2)*((x-T(n, 1)^2)/4) + T(n, k-1)*T(n, 1). - Charles L. Hohn, Aug 23 2019

A332060 a(n) = 3*a(n-1) + a(n-2) after initial values a(0..5) = (0, 1, 2, 3, 5, 13).

Original entry on oeis.org

0, 1, 2, 3, 5, 13, 44, 145, 479, 1582, 5225, 17257, 56996, 188245, 621731, 2053438, 6782045, 22399573, 73980764, 244341865, 807006359, 2665360942, 8803089185, 29074628497, 96026974676, 317155552525, 1047493632251, 3459636449278, 11426402980085

Offset: 0

M. F. Hasler, Mar 04 2020

nonn

These numbers arise as borders of intervals [a(n), a(n+1)] = [b(k)=k, b(m)=m] with m := b(k) + b(k-1) after the "holes" between the borders have been filled according to b(k+1) = b(k) + b(m) and b(m-1) = b(k+1) + b(n) for any such interval of length m - k > 2, i.e., starting from k = 5, m = 13.

The initial terms correspond to intervals of length <= 2 with only 0 or 1 "holes" to fill: In the first case we have the same recursion rule as for the Fibonacci sequence, and when there's one hole (between 3 and 5) the next Fibonacci number b(k+1) = b(k) + b(m) = 3 + 5 = 8 gets filled in there, and the next border is m = b(k) + b(k-1) = 5 + 8 = 13. See Example for more.

			The initial a(0) is conventional. (One could also choose a(0) = 1 to have a(2) = a(1) + a(0) as for the next two terms, but this wouldn't correspond to b(m) = b(k) + b(k-1), either.)
We start with [a(1), a(2)] = [1, 2].
No gap or "hole" here to fill, so the next interval [a(2), a(3)] has upper bound a(3) = a(2) + 1 = 3, where 1 is the element just left to the right border a(2).
Again, no gap or hole to fill in [2, 3], so the next interval has upper bound a(4) = a(3) + 2 = 5, where 2 is the element just left to the right border a(3).
Now there's a hole in (3, 5), at position 4, which is filled with 3 + 5 = 8, so the next upper bound is a(5) = a(4) + 8 = 5 + 8 = 13: here the number 8 was the element left to the right border 5.
Now there are several holes in (5, 13). The leftmost one (position 6) is filled with 5 + 13 = 18, and the rightmost (position 12) is filled with 18 + 13 = 31. So the next interval [a(5), a(6)] has upper bound a(6) = a(5) + 31 = 44.

Index entries for linear recurrences with constant coefficients, signature (3,1).

Cf. A000045 (Fibonacci numbers F(n+1) = F(n) + F(n-1)); A006190, A052924, A006497 (a(n+1) = 3*a(n) + a(n-1)); A000129 (Pell numbers a(n+1) = 2*a(n) + a(n-1)).

for(n=1+#a=[0,1,2,3,5,13],#a=Vec(a,30),a[n]=a[n-1]*3+a[n-2]);a \\ Remove initial 0 to get a[1] = 1 etc.
apply( {A332060(n)=if(n>3,[5,13]*([0,1;1,3]^(n-4))[,1],n)}, [0..20])

G.f.: x*(1 - x - 4*x^2 - 6*x^3 - 5*x^4)/(1 - 3*x - x^2).

A338078 Odd composite integers m such that A085447(m) == 6 (mod m).

Original entry on oeis.org

57, 185, 385, 481, 629, 721, 779, 1121, 1441, 1729, 2419, 2737, 5665, 6721, 7471, 8401, 9361, 10465, 10561, 11285, 11521, 11859, 12257, 13585, 14705, 15281, 16321, 16583, 18849, 24721, 25345, 25441, 25593, 30745, 33649, 35219, 36481, 36581, 37949, 38665, 39169

Offset: 1

Ovidiu Bagdasar, Oct 08 2020

nonn

If p is a prime, then A085447(p)==6 (mod p).
This sequence contains the odd composite integers for which the congruence holds.
The generalized Pell-Lucas sequence of integer parameters (a,b) defined by V(m+2)=a*V(m+1)-b*V(m) and V(0)=2, V(1)=a, satisfy the identity V(p)==a (mod p) whenever p is prime and b=-1,1.
For a=6, b=-1, V(m) recovers A085447(m).

D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021)

Cf. A006497, A005845 (a=1), A330276 (a=2), A335669 (a=3), A335670 (a=4), A335671 (a=5).

Select[Range[3, 20000, 2], CompositeQ[#] && Divisible[LucasL[#, 6] - 6, #] &]

More terms from Amiram Eldar, Oct 09 2020

A338079 Odd composite integers m such that A086902(m) == 7 (mod m).

Original entry on oeis.org

25, 51, 91, 161, 265, 325, 425, 561, 791, 1105, 1113, 1325, 1633, 1921, 1961, 2001, 2465, 2599, 2651, 2737, 3445, 4081, 4505, 4929, 7345, 7685, 8449, 9361, 10325, 10465, 10825, 11285, 11713, 12025, 12291, 13021, 15457, 17111, 18193, 18881, 18921, 19307

Offset: 1

Ovidiu Bagdasar, Oct 08 2020

nonn

If p is a prime, then A086902(p)==7 (mod p).
This sequence contains the odd composite integers for which the congruence holds.
The generalized Pell-Lucas sequence of integer parameters (a,b) defined by V(m+2)=a*V(m+1)-b*V(m) and V(0)=2, V(1)=a, satisfy the identity V(p)==a (mod p) whenever p is prime and b=-1,1.
For a=7, b=-1, V(m) recovers A086902(m).

D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021)

Cf. A006497, A005845 (a=1), A330276 (a=2), A335669 (a=3), A335670 (a=4), A335671 (a=5), A338078 (a=6).

Select[Range[3, 20000, 2], CompositeQ[#] && Divisible[LucasL[#, 7] - 7, #] &]

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A302941 Number of total dominating sets in the 2n-crossed prism graph.

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A309852 Array read by antidiagonals: ((z+sqrt(x))/2)^k + ((z-sqrt(x))/2)^k for columns k >= 0 and rows n >= 0, where x = 4*n+1 and y = floor(sqrt(x)) and z = y-1+(y mod 2).

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A332060 a(n) = 3*a(n-1) + a(n-2) after initial values a(0..5) = (0, 1, 2, 3, 5, 13).

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A338078 Odd composite integers m such that A085447(m) == 6 (mod m).

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A338079 Odd composite integers m such that A086902(m) == 7 (mod m).

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Mathematica