A093145 -id:A093145 - cp's OEIS frontend

A163146 a(n) = 12a(n-1)-31a(n-2) for n > 1; a(0) = 1, a(1) = 11.

1, 11, 101, 871, 7321, 60851, 503261, 4152751, 34231921, 282047771, 2323383701, 19137123511, 157620587401, 1298196219971, 10692116430221, 88061314343551, 725280162785761, 5973461208779051, 49197849458990021, 405196896035729671

Offset: 0

Klaus Brockhaus, Jul 21 2009

nonn

Binomial transform of A093145 without initial 0. Inverse binomial transform of A163147.

Index entries for linear recurrences with constant coefficients, signature (12,-31).

Cf. A093145, A163147.

[ n le 2 select 10*n-9 else 12*Self(n-1)-31*Self(n-2): n in [1..20] ];

LinearRecurrence[{12,-31},{1,11},20] (* Harvey P. Dale, Apr 15 2019 *)

a(n) = ((1+sqrt(5))*(6+sqrt(5))^n+(1-sqrt(5))*(6-sqrt(5))^n)/2.

G.f.: (1-x)/(1-12*x+31*x^2).

A361290 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..floor((n-1)/2)} k^(n-1-j) * binomial(n,2*j+1).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 2, 0, 0, 1, 4, 4, 0, 0, 1, 6, 14, 8, 0, 0, 1, 8, 30, 48, 16, 0, 0, 1, 10, 52, 144, 164, 32, 0, 0, 1, 12, 80, 320, 684, 560, 64, 0, 0, 1, 14, 114, 600, 1936, 3240, 1912, 128, 0, 0, 1, 16, 154, 1008, 4400, 11648, 15336, 6528, 256, 0

Offset: 0

Seiichi Manyama, Mar 11 2023

			Square array begins:
  0,  0,   0,   0,    0,    0, ...
  1,  1,   1,   1,    1 ,   1, ...
  0,  2,   4,   6,    8,   10, ...
  0,  4,  14,  30,   52,   80, ...
  0,  8,  48, 144,  320,  600, ...
  0, 16, 164, 684, 1936, 4400, ...

Column k=1..10 give A131577, A007070(n-1), A030192(n-1), A016129(n-1), A093145, A154237, A154248, A154348(n-1), A016175(n-1), A361293.
Main diagonal gives A360766.
Cf. A361432.

T(n, k) = polcoef(lift(Mod('x, ('x-k)^2-k)^n), 1);

T(0,k) = 0, T(1,k) = 1; T(n,k) = 2 * k * T(n-1,k) - (k-1) * k * T(n-2,k).
T(n,k) = ((k + sqrt(k))^n - (k - sqrt(k))^n)/(2 * sqrt(k)) for k > 0.
G.f. of column k: x/(1 - 2 * k * x + (k-1) * k * x^2).
E.g.f. of column k: exp(k * x) * sinh(sqrt(k) * x) / sqrt(k) for k > 0.

A162769 a(n) = ((1+sqrt(5))(4+sqrt(5))^n + (1-sqrt(5))(4-sqrt(5))^n)/2.

Original entry on oeis.org

1, 9, 61, 389, 2441, 15249, 95141, 593389, 3700561, 23077209, 143911501, 897442709, 5596515161, 34900251489, 217640345141, 1357219994749, 8463716161441, 52780309349289, 329141597018461, 2052549373305509

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jul 13 2009

Binomial transform of A082762. Fourth binomial transform of A162962. Inverse binomial transform of A093145 without initial 0.

Index entries for linear recurrences with constant coefficients, signature (8,-11).

Cf. A082762, A162962, A093145.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-5); 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 19 2009

f[n_] := Block[{s = Sqrt@ 5}, Simplify[((1 + s)(4 + s)^n + (1 - s)(4 - s)^n)/2]]; Array[f, 21, 0] (* Or *)
a[n_] := 8 a[n - 1] - 11 a[n - 2]; a[0] = 1; a[1] = 9; Array[a, 22, 0] (* Or *)
CoefficientList[Series[(1 + x)/(1 - 8 x + 11 x^2), {x, 0, 21}], x] (* Robert G. Wilson v *)

a(n) = 8*a(n-1) - 11*a(n-2) for n > 1; a(0) = 1, a(1) = 9.
G.f.: (1+x)/(1-8*x+11*x^2).
a(n) = A091870(n)+A091870(n+1). - R. J. Mathar, Feb 04 2021

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 19 2009

A162962 a(n) = 5*a(n-2) for n > 2; a(1) = 1, a(2) = 5.

Original entry on oeis.org

1, 5, 5, 25, 25, 125, 125, 625, 625, 3125, 3125, 15625, 15625, 78125, 78125, 390625, 390625, 1953125, 1953125, 9765625, 9765625, 48828125, 48828125, 244140625, 244140625, 1220703125, 1220703125, 6103515625, 6103515625, 30517578125

Offset: 1

Klaus Brockhaus, Jul 19 2009

Apparently a(n) = A074872(n+1), a(n) = A056451(n-1) for n > 1.

Binomial transform is A084057 without initial 1, second binomial transform is A048876, third binomial transform is A082762, fourth binomial transform is A162769, fifth binomial transform is A093145 without initial 0.

Index entries for linear recurrences with constant coefficients, signature (0,5).

Cf. A000351 (powers of 5), A074872 (powers of 5 repeated), A056451 (5^floor((n+1)/2)), A084057, A048876, A082762, A162769, A093145.

[ n le 2 select 4*n-3 else 5*Self(n-2): n in [1..30] ];

LinearRecurrence[{0,5},{1,5},30] (* Harvey P. Dale, Mar 18 2023 *)

a(n) = 5^((1/4)*(2*n-1+(-1)^n)).

G.f.: x*(1+5*x)/(1-5*x^2).

A360766 a(0) = 0; a(n) = ( (n + sqrt(n))^n - (n - sqrt(n))^n )/(2 * sqrt(n)).

Original entry on oeis.org

0, 1, 4, 30, 320, 4400, 73872, 1462552, 33325056, 858283776, 24641000000, 779935205984, 26972930949120, 1011642325897216, 40890444454377728, 1771640957790000000, 81896889467638120448, 4022826671022707900416, 209224123984489179202560

Offset: 0

Seiichi Manyama, Mar 11 2023

Main diagonal of A361290.
Cf. A007070, A030192, A093145, A154237.
Cf. A084062.

a(n) = polcoeff(lift(Mod('x, ('x-n)^2-n)^n), 1); \\ Kevin Ryde, Mar 16 2023

a(n) = Sum_{k=0..floor((n-1)/2)} n^(n-1-k) * binomial(n,2*k+1).
a(n) = [x^n] x/(1 - 2*n*x + (n-1)*n*x^2).
a(n) = n! * [x^n]  exp(n * x) * sinh(sqrt(n) * x) / sqrt(n) for n > 0.

cp's OEIS Frontend

A163146 a(n) = 12*a(n-1)-31*a(n-2) for n > 1; a(0) = 1, a(1) = 11.

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A361290 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..floor((n-1)/2)} k^(n-1-j) * binomial(n,2*j+1).

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A162769 a(n) = ((1+sqrt(5))*(4+sqrt(5))^n + (1-sqrt(5))*(4-sqrt(5))^n)/2.

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A162962 a(n) = 5*a(n-2) for n > 2; a(1) = 1, a(2) = 5.

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A360766 a(0) = 0; a(n) = ( (n + sqrt(n))^n - (n - sqrt(n))^n )/(2 * sqrt(n)).

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A163146 a(n) = 12a(n-1)-31a(n-2) for n > 1; a(0) = 1, a(1) = 11.

A162769 a(n) = ((1+sqrt(5))(4+sqrt(5))^n + (1-sqrt(5))(4-sqrt(5))^n)/2.