A370174 -id:A370174 - cp's OEIS frontend

A218988 Power floor sequence of 2+sqrt(8).

4, 19, 91, 439, 2119, 10231, 49399, 238519, 1151671, 5560759, 26849719, 129641911, 625966519, 3022433719, 14593600951, 70464138679, 340230958519, 1642780388791, 7932045389239, 38299303112119, 184925394005431, 892898788470199, 4311296729902519

Offset: 0

Clark Kimberling, Nov 11 2012

See A214992 for a discussion of power floor sequence and the power floor function, p1(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = 2+sqrt(8), and the limit p1(r) = 3.8983688904482395322594950087206...

See A218989 for the power floor function, p4. For comparison with p1, limit(p4(r)/p1(r) = 4/3.

			a(0) = [r] = 4, where r = 2+sqrt(8).
a(1) = [4*r] = 19; a(2) = [19*r] = 91.

Clark Kimberling, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (5,0,-4).

Cf. A214992, A057087, A086347, A218989.

x = 2 + Sqrt[8]; z = 30; (* z = # terms in sequences *)
f[x_] := Floor[x]; c[x_] := Ceiling[x];
p1[0] = f[x]; p2[0] = f[x]; p3[0] = c[x]; p4[0] = c[x];
p1[n_] := f[x*p1[n - 1]]
p2[n_] := If[Mod[n, 2] == 1, c[x*p2[n - 1]], f[x*p2[n - 1]]]
p3[n_] := If[Mod[n, 2] == 1, f[x*p3[n - 1]], c[x*p3[n - 1]]]
p4[n_] := c[x*p4[n - 1]]
t1 = Table[p1[n], {n, 0, z}]  (* this sequence *)
t2 = Table[p2[n], {n, 0, z}]  (* A057087 *)
t3 = Table[p3[n], {n, 0, z}]  (* A086347 *)
t4 = Table[p4[n], {n, 0, z}]  (* A218989 *)

Vec((4 - x - 4*x^2) / ((1 - x)*(1 - 4*x - 4*x^2)) + O(x^40)) \\ Colin Barker, Nov 13 2017

a(n) = floor(x*a(n-1)), where x=2+sqrt(8), a(0) = floor(x).
a(n) = 5*a(n-1) - 4*a(n-3).
G.f.: (4 - x - 4*x^2) / ((1 - x)*(1 - 4*x - 4*x^2)). [Corrected by Colin Barker, Nov 13 2017]
a(n) = (1/28)*(4 + (54-39*sqrt(2))*(2-2*sqrt(2))^n + (2*(1+sqrt(2)))^n*(54+39*sqrt(2))). - Colin Barker, Nov 13 2017
From Philippe Deléham, Mar 18 2024: (Start)
a(n) = 4*a(n-1) + 4*a(n-2) - 1.
a(n-1) = Sum_{k = 0..n} A370174(n,k)*3^k. (End)

A370175 a(n) = floor(xa(n-1)) for n > 0 where x = (5+3sqrt(5))/2, a(0) = 1.

Original entry on oeis.org

1, 5, 29, 169, 989, 5789, 33889, 198389, 1161389, 6798889, 39801389, 233001389, 1364013889, 7985076389, 46745451389, 273652638889, 1601990451389, 9378215451389, 54901029513889, 321396224826389, 1881486271701389, 11014412482638889, 64479493771701389

Offset: 0

Philippe Deléham, Mar 18 2024

x = A090550 = 1 + 3*phi = 5.854101966..., where phi is the golden ratio.

			a(0) = 1, a(1) = floor(x) = 5 where x = (5+3*sqrt(5))/2.
a(2) = floor(5*x) = 29, a(3) = floor(29*x) = 169.

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,0,-5).

Cf. A057088, A090550, A370174.

NestList[Floor[#*(5 + 3*Sqrt[5])/2] &, 1, 30] (* or *)
LinearRecurrence[{6, 0, -5}, {1, 5, 29}, 30] (* Paolo Xausa, May 25 2024 *)

a(n) = 6*a(n-1) - 5*a(n-3), a(0) = 1, a(1) = 5, a(2) = 29.
a(n) = 5*a(n-1) + 5*a(n-2) - 1.
a(n) = (4*(5-2*sqrt(5))*((5-3*sqrt(5))/2)^n + 4*(5+2*sqrt(5))*((5+3*sqrt(5))/2)^n + 5)/45.
G.f.: (1 - x - x^2)/(1 - 6*x + 5*x^3).
a(n) = Sum_{k = 0..n} A370174(n,k)*4^k.
a(n) = (8*A057088(n) + 4*A057088(n-1) + 1)/9.

A370176 a(n) = floor(x*a(n-1)) for n > 0 where x = 3+sqrt(15), a(0) = 1.

Original entry on oeis.org

1, 6, 41, 281, 1931, 13271, 91211, 626891, 4308611, 29613011, 203529731, 1398856451, 9614317091, 66079041251, 454160150051, 3121435147811, 21453571787171, 147450041609891, 1013421680382371, 6965230331953571, 47871912074015651, 329022854435815331, 2261368599058985891

Offset: 0

Philippe Deléham, Mar 19 2024

x = A092294 = 3+sqrt(15) = 6.872983346...

			a(0) = 1;
a(1) = floor(x) = 6 where x = 3+sqrt(15);
a(2) = floor(6*x) = 41;
a(3) = floor(41*x) = 281.

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,0,-6).

Cf. A057089, A092294, A370174.

NestList[Floor[(Sqrt[15]+3)*#] &, 1, 25] (* or *)
LinearRecurrence[{7, 0, -6}, {1, 6, 41}, 25] (* Paolo Xausa, Mar 31 2024 *)

a(n) = 7*a(n-1) - 6*a(n-3), a(0) = 1, a(1) = 6, a(2) = 41.
a(n) = 6*a(n-1) + 6*a(n-2) - 1.
a(n) = ((30-7*sqrt(15))*(3-sqrt(15))^n + (30+7*sqrt(15))*(3+sqrt(15))^n + 6)/66.
G.f.: (1-x-x^2)/(1-7*x+6*x^3).
a(n) = Sum_{k = 0..n} A370174(n,k)*5^k.
a(n) = (10*A057089(n) + 5*A057089(n-1) + 1)/11.

A371300 Triangle read by rows: Riordan array (1/(1 - x), (1 + x)/(1 - x - x^2)).

Original entry on oeis.org

1, 1, 2, 1, 5, 4, 1, 10, 16, 8, 1, 18, 45, 44, 16, 1, 31, 107, 158, 112, 32, 1, 52, 232, 461, 488, 272, 64, 1, 86, 474, 1190, 1680, 1392, 640, 128, 1, 141, 930, 2831, 5009, 5512, 3760, 1472, 256, 1, 230, 1772, 6355, 13541, 18602, 16816, 9760, 3328, 512

Offset: 0

Peter Luschny, Mar 18 2024

			Triangle begins:
  [0] 1;
  [1] 1,  2;
  [2] 1,  5,   4;
  [3] 1, 10,  16,    8;
  [4] 1, 18,  45,   44,   16;
  [5] 1, 31, 107,  158,  112,   32;
  [6] 1, 52, 232,  461,  488,  272,   64;
  [7] 1, 86, 474, 1190, 1680, 1392,  640,  128;

Cf. A371301 (row sums), A370174, A256893.

T := proc(n, k) option remember; if k > n or k < 0 then 0 elif k = 0 then 1 else
2*T(n-1, k-1) + T(n-1, k) + T(n-2, k-1) + T(n-2, k) fi end:
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;  # Peter Luschny, Apr 22 2024

# using function riordan_array from A256893
riordan_array(1/(1 - x), (1 + x)/(1 - x - x^2), 8)

T(n, k) = 2*T(n-1, k-1) + T(n-1, k) + T(n-2, k-1) + T(n-2, k), T(n, k) = 0 if k > n or if k < 0, T(n, 0) = 1. - Philippe Deléham , Apr 22 2024

A370177 a(n) = floor(x*a(n-1)) for n > 0 where x = (7 + sqrt(77))/2, a(0) = 1.

Original entry on oeis.org

1, 7, 55, 433, 3415, 26935, 212449, 1675687, 13216951, 104248465, 822257911, 6485544631, 51154617793, 403481136967, 3182450283319, 25101519942001, 197987791577239, 1561625180634679, 12317290805483425, 97152411902826727, 766287918958171063, 6044082316026984529

Offset: 0

Philippe Deléham, Mar 29 2024

x = (7+sqrt(77))/2 = A092290 = 7.88748219...

Cf. A057090, A092290, A370174.

a(n) = 8*a(n-1) - 7*a(n-3) for n > 2, a(0) = 1, a(1) = 7, a(2) = 55.
G.f.: (1-x-x^2)/(1-8*x+7*x^3).
a(n) = 7*a(n-1) + 7*a(n-2) - 1.
a(n) = (12*A057090(n) + 6*A057090(n-1) + 1)/13.
a(n) = (6*(77+8*sqrt(77))*((7+sqrt(77))/2)^n + 6*(77-8*sqrt(77))*((7-sqrt(77))/2)^n + 77)/1001.
a(n) = Sum_{k = 0..n} A370174(n,k)*6^k.

A370178 a(n) = floor(xa(n-1)) for n > 0 where x = 4 + 2sqrt(6), a(0) = 1.

Original entry on oeis.org

1, 8, 71, 631, 5615, 49967, 444655, 3956975, 35213039, 313360111, 2788585199, 24815562479, 220833181423, 1965189951215, 17488185061103, 155627000098543, 1384921481277167, 12324387851005679, 109674474658262767, 975990900074147567, 8685322997859282671

Offset: 0

Philippe Deléham, Apr 02 2024

Index entries for linear recurrences with constant coefficients, signature (9,0,-8).

Cf. A057091, A090654 (x value), A370174.

LinearRecurrence[{9,0,-8},{1,8,71},21] (* James C. McMahon, Apr 21 2024 *)

a(n) = 9*a(n-1) - 8*a(n-3) for n>2, a(0) = 1, a(1) = 8, a(2) = 71.
a(n) = 8*a(n-1) + 8*a(n-2) - 1.
G.f.: (1-x-x^2)/((1-x)*(1-8*x-8*x^2)).
a(n) = Sum_{k=0..n} A370174(n,k)*7^k.
a(n) = (7*(8-3*sqrt(6))*(4-2*sqrt(6))^n + 7*(8+3*sqrt(6))*(4+2*sqrt(6))^n + 8)/120.
a(n) = (14*A057091(n) + 7*A057091(n-1) + 1)/15.

A370179 a(n) = floor(xa(n-1)) for n > 0 where x = (9 + 3sqrt(13))/2, a(0) = 1.

Original entry on oeis.org

1, 9, 89, 881, 8729, 86489, 856961, 8491049, 84132089, 833608241, 8259662969, 81839440889, 810891934721, 8034582380489, 79609268836889, 788794660956401, 7815635368139609, 77439870261864089, 767299550670033281, 7602654788387076329, 75329589051513986489

Offset: 0

Philippe Deléham, Apr 24 2024

Index entries for linear recurrences with constant coefficients, signature (10,0,-9).

Cf. A057092, A090655 (x value), A370174.

LinearRecurrence[{10,0,-9},{1,9,89},21] (* Stefano Spezia, Apr 24 2024 *)

a(n) = 10*a(n-1) - 9*a(n-3) for n > 2, a(0) = 1, a(1) = 9, a(2) = 89.
a(n) = 9*a(n-1) + 9*a(n-2) - 1.
G.f.: (1-x-x^2)/((1-x)*(1-9*x-9*x^2)).
a(n) = Sum_{k = 0..n} A370174(n,k)*8^k.
a(n) = (16*A057092(n) + 8*A057092(n-1) + 1)/17.

cp's OEIS Frontend

A218988 Power floor sequence of 2+sqrt(8).

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A370175 a(n) = floor(x*a(n-1)) for n > 0 where x = (5+3*sqrt(5))/2, a(0) = 1.

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A370176 a(n) = floor(x*a(n-1)) for n > 0 where x = 3+sqrt(15), a(0) = 1.

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A371300 Triangle read by rows: Riordan array (1/(1 - x), (1 + x)/(1 - x - x^2)).

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A370177 a(n) = floor(x*a(n-1)) for n > 0 where x = (7 + sqrt(77))/2, a(0) = 1.

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A370178 a(n) = floor(x*a(n-1)) for n > 0 where x = 4 + 2*sqrt(6), a(0) = 1.

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A370179 a(n) = floor(x*a(n-1)) for n > 0 where x = (9 + 3*sqrt(13))/2, a(0) = 1.

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A370175 a(n) = floor(xa(n-1)) for n > 0 where x = (5+3sqrt(5))/2, a(0) = 1.

A370178 a(n) = floor(xa(n-1)) for n > 0 where x = 4 + 2sqrt(6), a(0) = 1.

A370179 a(n) = floor(xa(n-1)) for n > 0 where x = (9 + 3sqrt(13))/2, a(0) = 1.