A192872 -id:A192872 - cp's OEIS frontend

A192876 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) given in Comments.

1, 1, 4, 9, 31, 94, 309, 989, 3212, 10373, 33595, 108670, 351729, 1138113, 3683172, 11918737, 38570247, 124815294, 403911805, 1307084405, 4229816636, 13687969901, 44295207939, 143342292894, 463865421721, 1501100008249

Offset: 0

Clark Kimberling, Jul 11 2011

nonn

The polynomial p(n,x) is defined by p(0,x) = 1, p(1,x) = x + 1, and p(n,x) = x*p(n-1,x) + 2*(x^2)*p(n-1,x) + 1. See A192872.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,6,-5,-6,4).

Cf. A192872, A192877.

a:=[1,1,4,9,31];; for n in [6..30] do a[n]:=2*a[n-1]+6*a[n-2] - 5*a[n-3]-6*a[n-4]+4*a[n-5]; od; a; # G. C. Greubel, Jan 08 2019

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x-4*x^2)/((1-x)*(1+x-x^2)*(1-2*x-4*x^2)) )); // G. C. Greubel, Jan 08 2019

seq(coeff(series((1-x-4*x^2)/((1-x)*(1+x-x^2)*(1-2*x-4*x^2)),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Jan 08 2019

q = x^2; s = x + 1; z = 26;
p[0, x_] := 1; p[1, x_] := x + 1;
p[n_, x_] := p[n - 1, x]*x + 2*p[n - 2, x]*x^2 + 1;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}] := FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u0 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]  (* A192876 *)
u1 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]  (* A192877 *)
FindLinearRecurrence[u0]
FindLinearRecurrence[u1]
LinearRecurrence[{2,6,-5,-6,4},{1,1,4,9,31},26] (* Ray Chandler, Aug 02 2015 *)

my(x='x+O('x^30)); Vec((1-x-4*x^2)/((1-x)*(1+x-x^2)*(1-2*x-4*x^2) )) \\ G. C. Greubel, Jan 08 2019

((1-x-4*x^2)/((1-x)*(1+x-x^2)*(1-2*x-4*x^2))).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Jan 08 2019

a(n) = 2*a(n-1) + 6*a(n-2) - 5*a(n-3) - 6*a(n-4) + 4*a(n-5).

G.f.: (1-x-4*x^2) / ( (1-x)*(1+x-x^2)*(1-2*x-4*x^2) ). - R. J. Mathar, May 06 2014

A192877 Coefficient of x in the reduction by (x^2->x+1) of the polynomial p(n,x) given in Comments.

Original entry on oeis.org

0, 1, 4, 14, 47, 152, 496, 1601, 5192, 16786, 54351, 175836, 569100, 1841513, 5959484, 19284934, 62407951, 201955408, 653543000, 2114907025, 6843987040, 22147600586, 71671151919, 231932702004, 750550018452, 2428830833977

Offset: 0

Clark Kimberling, Jul 11 2011

nonn

The polynomial p(n,x) is defined by p(0,x) = 1, p(1,x) = x + 1, and p(n,x) = x*p(n-1,x) + 2*(x^2)*p(n-1,x) + 1. See A192872.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,6,-5,-6,4).

Cf. A192872, A192876.

a:=[0,1,4,14,47];; for n in [6..30] do a[n]:=2*a[n-1]+6*a[n-2] -5*a[n-3]-6*a[n-4]+4*a[n-5]; od; a; # G. C. Greubel, Jan 08 2019

m:=30; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!( x*(1+2*x)/((1-x)*(1+x-x^2)*(1-2*x-4*x^2)) )); // G. C. Greubel, Jan 08 2019

(See A192876.)
LinearRecurrence[{2,6,-5,-6,4}, {0,1,4,14,47}, 30] (* G. C. Greubel, Jan 08 2019 *)

my(x='x+O('x^30)); concat([0], Vec(x*(1+2*x)/((1-x)*(1+x-x^2)*(1-2*x-4*x^2)))) \\ G. C. Greubel, Jan 08 2019

(x*(1+2*x)/((1-x)*(1+x-x^2)*(1-2*x-4*x^2))).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Jan 08 2019

a(n) = 2*a(n-1) + 6*a(n-2) - 5*a(n-3) - 6*a(n-4) + 4*a(n-5).

G.f.: x*(1+2*x) / ( (1-x)*(1+x-x^2)*(1-2*x-4*x^2) ). - R. J. Mathar, May 06 2014

A192880 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) given in Comments.

Original entry on oeis.org

1, 0, 3, 7, 34, 123, 495, 1912, 7501, 29253, 114342, 446545, 1744489, 6814224, 26618619, 103979239, 406172770, 1586623227, 6197795703, 24210320296, 94572284197, 369425778645, 1443080391558, 5637075481729, 22019992977457

Offset: 0

Clark Kimberling, Jul 11 2011

nonn

The polynomial p(n,x) is defined by p(0,x) = 1, p(1,x) = x, and p(n,x) = 2*x*p(n-1,x) + (x^2)*p(n-1,x). See A192872.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,7,2,-1).

Cf. A000045, A002203, A192872, A192882.

a:=[1,0,3,7];; for n in [5..30] do a[n]:=2*a[n-1]+7*a[n-2] +2*a[n-3] -a[n-4]; od; a; # G. C. Greubel, Jan 08 2019

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1+x)*(1-3*x-x^2)/(1-2*x-7*x^2-2*x^3+x^4) )); // G. C. Greubel, Jan 08 2019

(* First program *)
q = x^2; s = x + 1; z = 25;
p[0, x_]:= 1; p[1, x_]:= x;
p[n_, x_]:= 2 p[n-1, x]*x + p[n-2, x]*x^2;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192880 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192882 *)
FindLinearRecurrence[u1]
FindLinearRecurrence[u2]
(* Additional programs *)
LinearRecurrence[{2,7,2,-1}, {1,0,3,7}, 30] (* G. C. Greubel, Jan 08 2019 *)
Table[Fibonacci[n-1]*LucasL[n, 2]/2, {n,0,30}] (* G. C. Greubel, Jul 29 2019 *)

my(x='x+O('x^30)); Vec((1+x)*(1-3*x-x^2)/(1-2*x-7*x^2-2*x^3+x^4)) \\ G. C. Greubel, Jan 08 2019

((1+x)*(1-3*x-x^2)/(1-2*x-7*x^2-2*x^3+x^4)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 08 2019

[fibonacci(n-1)*lucas_number2(n, 2, -1)/2 for n in (0..30)] # G. C. Greubel, Jul 29 2019

a(n) = 2*a(n-1) + 7*a(n-2) + 2*a(n-3) - a(n-4).
G.f.: (1+x)*(1-3*x-x^2) / ( 1-2*x-7*x^2-2*x^3+x^4 ). - R. J. Mathar, May 07 2014
a(n) = Fibonacci(n-1)*Pell-Lucas(n)/2, where Pell-Lucas(n) = A002203(n). - G. C. Greubel, Jul 29 2019

A192882 Coefficient of x in the reduction by (x^2 -> x+1) of the polynomial p(n,x) given in Comments.

Original entry on oeis.org

0, 1, 3, 14, 51, 205, 792, 3107, 12117, 47362, 184965, 722591, 2822544, 11025793, 43069611, 168242270, 657200859, 2567211037, 10028243016, 39173122739, 153021167805, 597743469778, 2334953116653, 9120979734623, 35629097057568

Offset: 0

Clark Kimberling, Jul 11 2011

nonn

The polynomial p(n,x) is defined by p(0,x) = 1, p(1,x) = x, and p(n,x) = 2*x*p(n-1,x) + (x^2)*p(n-1,x). See A192872.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,7,2,-1).

Cf. A000045, A002203, A192872, A192880.

a:=[0,1,3,14];; for n in [5..30] do a[n]:=2*a[n-1]+7*a[n-2] +2*a[n-3] -a[n-4]; od; a; # G. C. Greubel, Jan 08 2019

m:=30; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!( x*(1+x+x^2)/(1-2*x-7*x^2-2*x^3+x^4) )); // G. C. Greubel, Jan 08 2019

(* First program *)
q = x^2; s = x + 1; z = 25;
p[0, x_]:= 1; p[1, x_]:= x;
p[n_, x_]:= 2 p[n-1, x]*x + p[n-2, x]*x^2;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192880 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192882 *)
FindLinearRecurrence[u1]
FindLinearRecurrence[u2]
(* Additional programs *)
LinearRecurrence[{2,7,2,-1}, {0,1,3,14}, 30] (* G. C. Greubel, Jan 08 2019 *)
Table[Fibonacci[n]*LucasL[n, 2]/2, {n,0,30}] (* G. C. Greubel, Jul 29 2019 *)

my(x='x+O('x^30)); concat([0], Vec(x*(1+x+x^2)/(1-2*x-7*x^2-2*x^3 +x^4))) \\ G. C. Greubel, Jan 08 2019

(x*(1+x+x^2)/(1-2*x-7*x^2-2*x^3+x^4)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 08 2019

a(n) = 2*a(n-1) + 7*a(n-2) + 2*a(n-3) - a(n-4).
G.f.: x*(1+x+x^2) / ( 1-2*x-7*x^2-2*x^3+x^4 ). - R. J. Mathar, May 07 2014
a(n) = Fibonacci(n)*Pell-Lucas(n)/2, where Pell-Lucas(n) = A002203(n). - G. C. Greubel, Jul 29 2019

A192905 Coefficient of x in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.

Original entry on oeis.org

0, 1, 3, 8, 25, 79, 248, 777, 2435, 7632, 23921, 74975, 234992, 736529, 2308483, 7235416, 22677769, 71078319, 222778856, 698249753, 2188505347, 6859373216, 21499148257, 67384199871, 211200478176, 661959956001, 2074763216131

Offset: 0

Clark Kimberling, Jul 12 2011

The titular polynomial is defined by p(n,x) = (x^2)*p(n-1,x) + x*p(n-2,x), with p(0,x) = 1, p(1,x) = x. For details, see A192904.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,1,1).

Cf. A192232, A192744, A192904, A192872.

a:=[0,1,3,8];; for n in [5..30] do a[n]:=3*a[n-1]+a[n-3]+a[n-4]; od; a; # G. C. Greubel, Jan 11 2019

m:=30; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!( x*(1-x^2)/(1-3*x-x^3-x^4) )); // G. C. Greubel, Jan 11 2019

(See A192904.)
LinearRecurrence[{3,0,1,1}, {0,1,3,8}, 30] (* G. C. Greubel, Jan 11 2019 *)

my(x='x+O('x^30)); concat([0], Vec(x*(1-x^2)/(1-3*x-x^3-x^4))) \\ G. C. Greubel, Jan 11 2019

(x*(1-x^2)/(1-3*x-x^3-x^4)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 11 2019

a(n) = 3*a(n-1) + a(n-3) + a(n-4).

G.f.: x*(1-x)*(1+x)/(1-3*x-x^3-x^4). - Colin Barker, Aug 31 2012

A192908 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.

Original entry on oeis.org

1, 1, 3, 7, 17, 43, 111, 289, 755, 1975, 5169, 13531, 35423, 92737, 242787, 635623, 1664081, 4356619, 11405775, 29860705, 78176339, 204668311, 535828593, 1402817467, 3672623807, 9615053953, 25172538051, 65902560199

Offset: 0

Clark Kimberling, Jul 12 2011

The titular polynomial is defined by p(n,x) = (x^2)*p(n-1,x) + x*p(n-2,x), with p(0,x) = 1, p(1,x) = x + 1.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Cf. A192232, A192744, A192872, A192904.
Cf. A000045; A052995: 2*Fibonacci(2*n-1) for n>0.
CF. A055588, A097136.

Concatenation([1], List([1..30], n -> 1+2*Fibonacci(2*(n-1)))); # G. C. Greubel, Jan 11 2019

[1] cat [1+2*Fibonacci(2*(n-1)): n in [1..30]]; // G. C. Greubel, Jan 11 2019

u = 1; v = 1; a = 1; b = 1; c = 1; d = 1; e = 0; f = 1;
q = x^2; s = u*x + v; z = 26;
p[0, x_] := a;  p[1, x_] := b*x + c
p[n_, x_] := d*(x^2)*p[n - 1, x] + e*x*p[n - 2, x] + f;
Table[Expand[p[n, x]], {n, 0, 8}]
reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u0 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]    (* A192908 *)
u1 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]    (* A069403 *)
Simplify[FindLinearRecurrence[u0]] (* recurrence for 0-sequence *)
Simplify[FindLinearRecurrence[u1]] (* recurrence for 1-sequence *)
LinearRecurrence[{4,-4,1}, {1,1,3,7}, 30] (* G. C. Greubel, Jan 11 2019 *)

vector(30, n, n--; if(n==0,1,1+2*fibonacci(2*n-2))) \\ G. C. Greubel, Jan 11 2019

[1]+[1+2*fibonacci(2*(n-1)) for n in (1..30)] # G. C. Greubel, Jan 11 2019

a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3) for n>3.
G.f.: 1 + x*(1 - x - x^2)/((1 - x)*(1 - 3*x + x^2)). - R. J. Mathar, Jul 13 2011
a(n) = 2*Fibonacci(2*n-2) + 1 for n>0, a(0)=1. - Bruno Berselli, Dec 27 2016
a(n) = -1 + 3*a(n-1) - a(n-2) with a(1) = 1 and a(2) = 3. Cf. A055588 and A097136. - Peter Bala, Nov 12 2017

A192909 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.

Original entry on oeis.org

1, 1, 3, 9, 27, 83, 259, 811, 2541, 7963, 24957, 78221, 245165, 768413, 2408415, 7548629, 23659463, 74155215, 232422687, 728476151, 2283243129, 7156307287, 22429820697, 70301181369, 220343094521, 690615411545, 2164577236699

Offset: 0

Clark Kimberling, Jul 12 2011

nonn

The titular polynomial is defined by p(n,x) = (x^2)*p(n-1,x) + x*p(n-2,x) + 1, with p(0,x) = 1, p(1,x) = x + 1.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-3,1,0,-1).

Cf. A192232, A192744, A192872, A192904, A192910.

a:=[1,1,3,9,27];; for n in [6..30] do a[n]:=4*a[n-1]-3*a[n-2] +a[n-3]-a[n-5]; od; a; # G. C. Greubel, Jan 11 2019

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (x^2-x+1)*(x^2+2*x-1)/((1-x)*(x^4+x^3+3*x-1)) )); // G. C. Greubel, Jan 11 2019

u = 1; v = 1; a = 1; b = 1; c = 1; d = 1; e = 1; f = 1;
q = x^2; s = u*x + v; z = 24;
p[0, x_] := a;  p[1, x_] := b*x + c
p[n_, x_] := d*(x^2)*p[n - 1, x] + e*x*p[n - 2, x] + f;
Table[Expand[p[n, x]], {n, 0, 8}]
reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u0 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192909 *)
u1 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192910 *)
Simplify[FindLinearRecurrence[u0]]
Simplify[FindLinearRecurrence[u1]]
LinearRecurrence[{4,-3,1,0,-1}, {1,1,3,9,27}, 30] (* G. C. Greubel, Jan 11 2019 *)

my(x='x+O('x^30)); Vec((x^2-x+1)*(x^2+2*x-1)/((1-x)*(x^4+x^3+3*x -1))) \\ G. C. Greubel, Jan 11 2019

((x^2-x+1)*(x^2+2*x-1)/((1-x)*(x^4+x^3+3*x-1))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 11 2019

a(n) = 4*a(n-1) - 3*a(n-2) + a(n-3) - a(n-5).

G.f.: (x^2-x+1)*(x^2+2*x-1) / ( (1-x)*(x^4+x^3+3*x-1) ). - R. J. Mathar, Jul 13 2011

A192910 Coefficient of x in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.

Original entry on oeis.org

0, 1, 4, 13, 42, 133, 418, 1311, 4110, 12883, 40380, 126563, 396684, 1243317, 3896896, 12213937, 38281814, 119985657, 376067806, 1178699171, 3694364986, 11579148423, 36292212248, 113749700903, 356522616120, 1117439209033, 3502359540252

Offset: 0

Clark Kimberling, Jul 12 2011

nonn

The titular polynomial is defined by p(n,x) = (x^2)*p(n-1,x) + x*p(n-2,x) + 1, with p(0,x) = 1, p(1,x) = x + 1.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-3,1,0,-1).

Cf. A192232, A192744, A192872, A192904, A192909.

a:=[0,1,4,13,42];; for n in [6..30] do a[n]:=4*a[n-1]-3*a[n-2] + a[n-3]-a[n-5]; od; a; # G. C. Greubel, Jan 12 2019

m:=30; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!( x*(1+x)*(1-x+x^2)/((1-x)*(1-3*x-x^3-x^4)) )); // G. C. Greubel, Jan 12 2019

(See A192909.)
LinearRecurrence[{4,-3,1,0,-1}, {0,1,4,13,42}, 30] (* G. C. Greubel, Jan 12 2019 *)

my(x='x+O('x^30)); concat([0], Vec(x*(1+x)*(1-x+x^2)/((1-x)*(1-3*x -x^3-x^4)))) \\ G. C. Greubel, Jan 12 2019

(x*(1+x)*(1-x+x^2)/((1-x)*(1-3*x-x^3-x^4))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 12 2019

a(n) = 4*a(n-1) - 3*a(n-2) + a(n-3) - a(n-5).

G.f.: x*(1+x)*(1-x+x^2)/((1-x)*(1-3*x-x^3-x^4)). - R. J. Mathar, Jul 13 2011

A194032 Natural interspersion of the squares (1,4,9,16,25,...), a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 4, 2, 9, 5, 3, 16, 10, 6, 7, 25, 17, 11, 12, 8, 36, 26, 18, 19, 13, 14, 49, 37, 27, 28, 20, 21, 15, 64, 50, 38, 39, 29, 30, 22, 23, 81, 65, 51, 52, 40, 41, 31, 32, 24, 100, 82, 66, 67, 53, 54, 42, 43, 33, 34, 121, 101, 83, 84, 68, 69, 55, 56, 44, 45

Offset: 1

Clark Kimberling, Aug 12 2011

See A194029 for definitions of natural fractal sequence and natural interspersion. Every positive integer occurs exactly once (and every pair of rows intersperse), so that as a sequence, A194032 is a permutation of the positive integers; its inverse is A194033.

			Northwest corner:
  1...4...9...16...25
  2...5...10..17...26
  3...6...11..18...27
  7...12..19..28...39
  8...13..20..29...40

Zhuorui He, Table of n, a(n) for n = 1..11325

Cf. A000290, A071797, A194033, A192872.

z = 30;
c[k_] := k^2;
c = Table[c[k], {k, 1, z}]  (* A000290 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]] (* A071797 *)
f = Table[f[n], {n, 1, 255}]
r[n_] := Flatten[Position[f, n]]
t[n_, k_] := r[n][[k]]
TableForm[Table[t[n, k], {n, 1, 7}, {k, 1, 7}]]
p = Flatten[Table[t[k, n - k + 1], {n, 1, 14}, {k, 1, n}]] (* A194032 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 70}]] (* A194033 *)

T(n, k) = (k + max(floor(n/2)-1,0))^2 + n - 1. - Zhuorui He, Jul 08 2025

A194036 Natural interspersion of A028872, a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 6, 2, 13, 7, 3, 22, 14, 8, 4, 33, 23, 15, 9, 5, 46, 34, 24, 16, 10, 11, 61, 47, 35, 25, 17, 18, 12, 78, 62, 48, 36, 26, 27, 19, 20, 97, 79, 63, 49, 37, 38, 28, 29, 21, 118, 98, 80, 64, 50, 51, 39, 40, 30, 31, 141, 119, 99, 81, 65, 66, 52, 53, 41, 42, 32, 166, 142

Offset: 1

Clark Kimberling, Aug 12 2011

See A194029 for definitions of natural fractal sequence and natural interspersion. Every positive integer occurs exactly once (and every pair of rows intersperse), so that as a sequence, A194036 is a permutation of the positive integers; its inverse is A194037.

			Northwest corner:
1...6...13...22...33
2...7...14...23...34
3...8...15...24...35
4...9...16...25...36
5...10..17...26...37
11..18..27...38...51

Cf. A192872, A194037.

z = 30;
c[k_] := k^2 + 2 k - 2;
c = Table[c[k], {k, 1, z}]  (* A028872 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 255}]  (* A071797 *)
r[n_] := Flatten[Position[f, n]]
t[n_, k_] := r[n][[k]]
TableForm[Table[t[n, k], {n, 1, 7}, {k, 1, 7}]]
p = Flatten[Table[t[k, n - k + 1], {n, 1, 13}, {k, 1, n}]]  (* A194036 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 70}]]  (* A194037 *)

cp's OEIS Frontend

A192876 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) given in Comments.

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A192877 Coefficient of x in the reduction by (x^2->x+1) of the polynomial p(n,x) given in Comments.

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A192880 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) given in Comments.

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A192882 Coefficient of x in the reduction by (x^2 -> x+1) of the polynomial p(n,x) given in Comments.

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A192905 Coefficient of x in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.

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A192908 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.

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