A163403 -id:A163403 - cp's OEIS frontend

A343177 a(0)=4; if n > 0 is even then a(n) = 2^(n/2+1)+3, otherwise a(n) = 3*(2^((n-1)/2)+1).

4, 6, 7, 9, 11, 15, 19, 27, 35, 51, 67, 99, 131, 195, 259, 387, 515, 771, 1027, 1539, 2051, 3075, 4099, 6147, 8195, 12291, 16387, 24579, 32771, 49155, 65539, 98307, 131075, 196611, 262147, 393219, 524291, 786435, 1048579, 1572867, 2097155, 3145731, 4194307, 6291459

Offset: 0

N. J. A. Sloane, Apr 26 2021

Number of edges along the boundary of the graph G(n) described in A342759.

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

Cf. A342759.

The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent: A029744 (s(n)); A052955 (s(n)-1), A027383 (s(n)-2), A354788 (s(n)-3), A347789 (s(n)-4), A209721 (s(n)+1), A209722 (s(n)+2), A343177 (s(n)+3), A209723 (s(n)+4); A060482, A136252 (minor differences from A354788 at the start); A354785 (3*s(n)), A354789 (3*s(n)-7). The first differences of A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences: A016116, A060546, A117575, A131572, A152166, A158780, A163403, A320770. The bisections of A029744 are A000079 and A007283. - N. J. A. Sloane, Jul 14 2022

f:=n->if n = 0 then 4 elif (n mod 2) = 0 then 2^(n/2+1)+3 else 3*(2^((n-1)/2)+1); fi;
[seq(f(n),n=0..40)];

LinearRecurrence[{1, 2, -2}, {4, 6, 7, 9}, 50] (* or *)
A343177[n_] := Which[n == 0, 4, OddQ[n], 3*(2^((n-1)/2)+1), True, 2^(n/2+1)+3];
Array[A343177, 50, 0] (* Paolo Xausa, Feb 02 2024 *)

G.f.: (4 + 2*x - 7*x^2 - 2*x^3)/((1 - x)*(1 - 2*x^2)). - Stefano Spezia, Feb 04 2023

E.g.f.: 3*cosh(x) + 2*cosh(sqrt(2)*x) + 3*sinh(x) + 3*sinh(sqrt(2)*x)/sqrt(2) - 1. - Stefano Spezia, Jul 25 2024

A354785 Numbers of the form 32^k or 92^k.

Original entry on oeis.org

3, 6, 9, 12, 18, 24, 36, 48, 72, 96, 144, 192, 288, 384, 576, 768, 1152, 1536, 2304, 3072, 4608, 6144, 9216, 12288, 18432, 24576, 36864, 49152, 73728, 98304, 147456, 196608, 294912, 393216, 589824, 786432, 1179648, 1572864, 2359296, 3145728, 4718592, 6291456, 9437184, 12582912, 18874368, 25165824, 37748736, 50331648

Offset: 1

N. J. A. Sloane, Jul 12 2022

Index entries for linear recurrences with constant coefficients, signature (0,2).

The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent: A029744 (s(n)); A052955 (s(n)-1), A027383 (s(n)-2), A354788 (s(n)-3), A347789 (s(n)-4), A209721 (s(n)+1), A209722 (s(n)+2), A343177 (s(n)+3), A209723 (s(n)+4); A060482, A136252 (minor differences from A354788 at the start); A354785 (3*s(n)), A354789 (3*s(n)-7). The first differences of A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences: A016116, A060546, A117575, A131572, A152166, A158780, A163403, A320770. The bisections of A029744 are A000079 and A007283.

seq[max_] := Union[Table[3*2^n, {n, 0, Floor[Log2[max/3]]}], Table[9*2^n, {n, 0, Floor[Log2[max/9]]}]]; seq[10^8] (* Amiram Eldar, Jan 16 2024 *)

Sum_{n>=1} 1/a(n) = 8/9. - Amiram Eldar, Jan 16 2024

G.f.: (3*x^2+6*x+3)/(1-2*x^2). - Georg Fischer, Apr 10 2025

A216218 Square array T, read by antidiagonals: T(n,k) = 0 if n-k>=2 or if k-n>=2, T(1,0) = T(0,0) = T(0,1) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

Original entry on oeis.org

1, 1, 1, 0, 2, 0, 0, 2, 2, 0, 0, 0, 4, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 8, 8, 0, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 16, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Mar 13 2013

With zeros omitted, this is A173862.

			Square array begins:
1, 1, 0, 0,  0,  0,  0, 0, ... row n=0
1, 2, 2, 0,  0,  0,  0, 0, ... row n=1
0, 2, 4, 4,  0,  0,  0, 0, ... row n=2
0, 0, 4, 8,  8,  0,  0, 0, ... row n=3
0, 0, 0, 8, 16, 16,  0, 0, ... row n=4
0, 0, 0, 0, 16, 32, 32, 0, ... row n=5
...

Cf. A000079, A016116, A216216, A216210, A068913

T(n,n) = T(n+1,n) = T(n,n+1) = 2^n = A000079(n).

Sum_{k, 0<=k<=n} T(n-k,k) = A016116(n+1) = A163403(n+1).

A163346 a(n) = 10a(n-1) - 23a(n-2) for n > 1; a(0) = 1, a(1) = 7.

Original entry on oeis.org

1, 7, 47, 309, 2009, 12983, 83623, 537621, 3452881, 22163527, 142219007, 912428949, 5853252329, 37546657463, 240841771063, 1544844588981, 9909085155361, 63559426007047, 407685301497167, 2614986216809589, 16773100233661049, 107586319349989943

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jul 25 2009

Binomial transform of A163350. Fifth binomial transform of A163403.

Matthew House, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-23).

Cf. A163350, A163403.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+r)*(5+r)^n+(1-r)*(5-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 26 2009

CoefficientList[Series[(1 - 3 x)/(1 - 10 x + 23 x^2), {x, 0, 21}], x] (* Michael De Vlieger, Jun 30 2016 *)
LinearRecurrence[{10,-23}, {1, 7}, 50] (* G. C. Greubel, Dec 19 2016 *)

Vec((1-3*x)/(1-10*x+23*x^2) + O(x^99)) \\ Altug Alkan, Jul 05 2016

a(n) = 10*a(n-1)-23*a(n-2) for n > 1; a(0) = 1, a(1) = 7.
a(n) = ((1+sqrt(2))*(5+sqrt(2))^n + (1-sqrt(2))*(5-sqrt(2))^n)/2.
G.f.: (1-3*x)/(1-10*x+23*x^2).
E.g.f.: (sqrt(2)*sinh(sqrt(2)*x) + cosh(sqrt(2)*x))*exp(5*x). - Ilya Gutkovskiy, Jun 30 2016

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 26 2009

New name from G. C. Greubel, Dec 19 2016

A163350 a(n) = 8a(n-1) - 14a(n-2) for n > 1; a(0) = 1, a(1) = 6.

Original entry on oeis.org

1, 6, 34, 188, 1028, 5592, 30344, 164464, 890896, 4824672, 26124832, 141453248, 765878336, 4146681216, 22451153024, 121555687168, 658129355008, 3563255219712, 19292230787584, 104452273224704, 565526954771456

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jul 25 2009

Binomial transform of A102285. Fourth binomial transform of A163403. Inverse binomial transform of A163346.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8, -14).

Cf. A102285, A163403, A163346.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+r)*(4+r)^n+(1-r)*(4-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 26 2009

LinearRecurrence[{8,-14},{1,6},30] (* Harvey P. Dale, May 08 2014 *)

Vec((1-2*x)/(1-8*x+14*x^2) + O(x^50)) \\ G. C. Greubel, Dec 19 2016

a(n) = 8*a(n-1) - 14*a(n-2) for n > 1; a(0) = 1, a(1) = 6.
a(n) = ((1+sqrt(2))*(4+sqrt(2))^n+(1-sqrt(2))*(4-sqrt(2))^n)/2.
G.f.: (1-2*x)/(1-8*x+14*x^2).
E.g.f.: exp(4*x)*( cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Dec 19 2016

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 26 2009

New name from G. C. Greubel, Dec 19 2016

A163448 a(n) = 20a(n-1) - 98a(n-2) for n > 1; a(0) = 1, a(1) = 12.

Original entry on oeis.org

1, 12, 142, 1664, 19364, 224208, 2586488, 29757376, 341671696, 3917211072, 44860395232, 513321219584, 5870105658944, 67096633659648, 766662318616448, 8757776273683456, 100022618249257216, 1142190290164165632

Offset: 0

Klaus Brockhaus, Jul 27 2009

nonn

Binomial transform of A163447. Tenth binomial transform of A163403.

G. C. Greubel, Table of n, a(n) for n = 0..940 (first 100 terms from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (20, -98).

Cf. A163403, A163447.

[ n le 2 select 11*n-10 else 20*Self(n-1)-98*Self(n-2): n in [1..18] ];

LinearRecurrence[{20,-98},{1,12},30] (* or *) With[{ms=10-Sqrt[2], ps=10+ Sqrt[2]},Table[Simplify[(31ms^n-41Sqrt[2](ms^n)+49ps^n+49Sqrt[2] ps^n)/ (98ps)],{n,20}]] (* Harvey P. Dale, Nov 14 2011 *)

Vec((1-8*x)/(1-20*x+98*x^2) + O(x^50)) \\ G. C. Greubel, Dec 24 2016

a(n) = ((1+sqrt(2))*(10+sqrt(2))^n + (1-sqrt(2))*(10-sqrt(2))^n)/2.
G.f.: (1-8*x)/(1-20*x+98*x^2).
a(n) = (31*(10-sqrt(2))^n - 41*sqrt(2)*(10-sqrt(2))^n + 49*(10+sqrt(2))^n + 49*sqrt(2)*(10+sqrt(2))^n)/(98*(10+sqrt(2))). - Harvey P. Dale, Nov 14 2011
E.g.f.: exp(10*x)*( cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Dec 24 2016

cp's OEIS Frontend

A343177 a(0)=4; if n > 0 is even then a(n) = 2^(n/2+1)+3, otherwise a(n) = 3*(2^((n-1)/2)+1).

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A354785 Numbers of the form 3*2^k or 9*2^k.

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A216218 Square array T, read by antidiagonals: T(n,k) = 0 if n-k>=2 or if k-n>=2, T(1,0) = T(0,0) = T(0,1) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

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A163346 a(n) = 10*a(n-1) - 23*a(n-2) for n > 1; a(0) = 1, a(1) = 7.

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A163350 a(n) = 8*a(n-1) - 14*a(n-2) for n > 1; a(0) = 1, a(1) = 6.

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A163448 a(n) = 20*a(n-1) - 98*a(n-2) for n > 1; a(0) = 1, a(1) = 12.

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Author

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Comments

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Programs

Magma

Mathematica

PARI

Formula

A354785 Numbers of the form 32^k or 92^k.

A163346 a(n) = 10a(n-1) - 23a(n-2) for n > 1; a(0) = 1, a(1) = 7.

A163350 a(n) = 8a(n-1) - 14a(n-2) for n > 1; a(0) = 1, a(1) = 6.

A163448 a(n) = 20a(n-1) - 98a(n-2) for n > 1; a(0) = 1, a(1) = 12.