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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A274965 G.f. A(x) satisfies: 1 = ...(((((A(x) - x)^(1/2) - x^2)^(1/2) - x^3)^(1/2) - x^4)^(1/2) - x^5)^(1/2) -...- x^n)^(1/2) -..., an infinite series of nested square roots.

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 46, 104, 238, 540, 1228, 2780, 6289, 14180, 31924, 71688, 160694, 359452, 802642, 1788988, 3980916, 8844064, 19618506, 43455324, 96121164, 212331796, 468445180, 1032216460, 2271818652, 4994434788, 10968013396, 24061103888, 52730956193, 115449870424, 252530306764, 551873275488, 1204991320660, 2628810554176, 5730295148952, 12480957518212, 27163290056278
Offset: 0

Views

Author

Paul D. Hanna, Jul 16 2016

Keywords

Comments

Odd terms occur at positions k*2^(k-1) for k>=0.
Limit a(n+1)/a(n) = 2, and A(x) diverges at x=1/2.
A(-1/2) = 1.0891636602638152861240865158090054430536947422594419370337760...
A(2/5) = 4.27983467184471084235872646732512184377478311914374590...
A(1/3) = 2.15485192359458408375371476779655861137906655796801630...
A(x) = 2 at x = 0.32026273178798900824351068844199852911740930864617900985902...

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 20*x^5 + 46*x^6 + 104*x^7 + 238*x^8 + 540*x^9 + 1228*x^10 +...
Illustration of the definition.
R1 = (A(x) - x)^(1/2);
R2 = ((A(x) - x)^(1/2) - x^2)^(1/2);
R3 = (((A(x) - x)^(1/2) - x^2)^(1/2) - x^3)^(1/2);
R4 = ((((A(x) - x)^(1/2) - x^2)^(1/2) - x^3)^(1/2) - x^4)^(1/2);
R5 = (((((A(x) - x)^(1/2) - x^2)^(1/2) - x^3)^(1/2) - x^4)^(1/2) - x^5)^(1/2); ...
where the above series begin:
R1 = 1 + x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 17*x^6 + 36*x^7 + 78*x^8 + 168*x^9 + 364*x^10 + 786*x^11 + 1700*x^12 +...
R2 = 1 + x^3 + 2*x^4 + 4*x^5 + 8*x^6 + 16*x^7 + 33*x^8 + 68*x^9 + 142*x^10 + 296*x^11 + 620*x^12 + 1296*x^13 +...
R3 = 1 + x^4 + 2*x^5 + 4*x^6 + 8*x^7 + 16*x^8 + 32*x^9 + 65*x^10 + 132*x^11 + 270*x^12 + 552*x^13 + 1132*x^14 +...
R4 = 1 + x^5 + 2*x^6 + 4*x^7 + 8*x^8 + 16*x^9 + 32*x^10 + 64*x^11 + 129*x^12 + 260*x^13 + 526*x^14 + 1064*x^15 +...
R5 = 1 + x^6 + 2*x^7 + 4*x^8 + 8*x^9 + 16*x^10 + 32*x^11 + 64*x^12 + 128*x^13 + 257*x^14 + 516*x^15 + 1038*x^16 +...
etc., so that 1 is obtained as a limit.
GENERATING METHOD.
The g.f. of this sequence can be obtained as a limit, as n grows, of the following process: start with 1 + x^n, then square the result and add x^(n-1), then square the result and add x^(n-2), then continue in this way until you reach x^1; this process is illustrated at n=6 as follows:
S6 = 1 + x^6,
S5 = S6^2 + x^5 = 1 + x^5 + 2*x^6 + x^12,
S4 = S5^2 + x^4 = 1 + x^4 + 2*x^5 + 4*x^6 + x^10 + 4*x^11 + 6*x^12 + 2*x^17 +...,
S3 = S4^2 + x^3 = 1 + x^3 + 2*x^4 + 4*x^5 + 8*x^6 + x^8 + 4*x^9 + 14*x^10 +...,
S2 = S3^2 + x^2 = 1 + x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 17*x^6 + 4*x^7 + 14*x^8 + 40*x^9 + 76*x^10 +...,
S1 = S2^2 + x = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 20*x^5 + 46*x^6 + 40*x^7 + 110*x^8 + 220*x^9 + 396*x^10 +...,
which matches the g.f. A(x) up to x^6.
RELATED SERIES.
Note that the bisections are self-convolutions of integer sequences:
sqrt( (A(x) + A(-x))/2 ) = 1 + x^2 + 4*x^4 + 19*x^6 + 92*x^8 + 446*x^10 + 2150*x^12 + 10280*x^14 + 48761*x^16 + 229558*x^18 + 1073278*x^20 + 4986624*x^22 + 23037102*x^24 + 105877968*x^26 + 484337300*x^28 +...+ A275751(n)*x^(2*n) +...
sqrt( x*(A(x) - A(-x))/2 ) = x + 2*x^3 + 8*x^5 + 36*x^7 + 166*x^9 + 770*x^11 + 3574*x^13 + 16560*x^15 + 76516*x^17 + 352498*x^19 + 1619014*x^21 + 7414134*x^23 + 33855996*x^25 + 154181234*x^27 + 700333366*x^29 +...+ A275752(n)*x^(2*n+1) +...

Links

Crossrefs

Cf. A275691, A275751, A275752.
Row sums of triangle A275670.

Programs

  • PARI
    {a(n) = my(A=1 +x*O(x^n)); for(k=0,n, A = A^2 + x^(n+1-k)); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f.: A(x) = G(x,1), where G(x,y) = x*y + G(x,x*y)^2 is the g.f. of A275670.
G.f.: A(x) = F(x)^2 + x, where F(x) is the g.f. of A275691.

A275670 G.f. A(x,y) satisfies: A(x,y) = x*y + A(x,x*y)^2, with A(0,y) = 1.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 4, 0, 8, 1, 0, 16, 4, 0, 32, 14, 0, 64, 40, 0, 128, 108, 2, 0, 256, 272, 12, 0, 512, 664, 52, 0, 1024, 1568, 188, 0, 2048, 3632, 608, 1, 0, 4096, 8256, 1816, 12, 0, 8192, 18528, 5128, 76, 0, 16384, 41088, 13856, 360, 0, 32768, 90304, 36176, 1446, 0, 65536, 196864, 91856, 5192, 4, 0, 131072, 426368, 227968, 17192, 42, 0, 262144, 918016, 555040, 53504, 284, 0, 524288, 1966848, 1329696, 158588, 1496, 0, 1048576, 4195328, 3141632, 451824, 6704, 0, 2097152, 8914432, 7334208, 1245936, 26772, 6
Offset: 0

Views

Author

Paul D. Hanna, Aug 04 2016

Keywords

Comments

Compare g.f. to G(x,y) = x*y + G(x*y,y)^2 with G(0,y) = 0, which generates triangle A138157.
Apparently, the g.f. of column n equals y^n*x^A033156(n) * P(n,x)/Q(n,x), where:
Q(n,x) = Product_{k=1..n} (1 - 2*x^k)^floor(n/k),
and P(n,x) is of degree A024916(n) - A033156(n).

Examples

			G.f.: A(x,y) = 1 + y*x + 2*y*x^2 + 4*y*x^3 + (y^2 + 8*y)*x^4 + (4*y^2 + 16*y)*x^5 + (14*y^2 + 32*y)*x^6 + (40*y^2 + 64*y)*x^7 + (2*y^3 + 108*y^2 + 128*y)*x^8 + (12*y^3 + 272*y^2 + 256*y)*x^9 + (52*y^3 + 664*y^2 + 512*y)*x^10 + (188*y^3 + 1568*y^2 + 1024*y)*x^11 + (y^4 + 608*y^3 + 3632*y^2 + 2048*y)*x^12 +...
such that A(x,y) = x*y + A(x,x*y)^2, with A(0,y) = 1; further,
A(x,y) = x*y + ( x^2*y + A(x,x^2*y)^2 )^2,
A(x,y) = x*y + ( x^2*y + ( x^3*y + A(x,x^3*y)^2 )^2 )^2, etc.
This table of coefficients in g.f. A(x,y) begins:
1;
0, 1;
0, 2;
0, 4;
0, 8, 1;
0, 16, 4;
0, 32, 14;
0, 64, 40;
0, 128, 108, 2;
0, 256, 272, 12;
0, 512, 664, 52;
0, 1024, 1568, 188;
0, 2048, 3632, 608, 1;
0, 4096, 8256, 1816, 12;
0, 8192, 18528, 5128, 76;
0, 16384, 41088, 13856, 360;
0, 32768, 90304, 36176, 1446;
0, 65536, 196864, 91856, 5192, 4;
0, 131072, 426368, 227968, 17192, 42;
0, 262144, 918016, 555040, 53504, 284;
0, 524288, 1966848, 1329696, 158588, 1496;
0, 1048576, 4195328, 3141632, 451824, 6704;
0, 2097152, 8914432, 7334208, 1245936, 26772, 6;
0, 4194304, 18876416, 16943680, 3342784, 98060, 80;
0, 8388608, 39848960, 38785536, 8761720, 335704, 636;
0, 16777216, 83890176, 88063616, 22508448, 1088496, 3844;
0, 33554432, 176166912, 198506624, 56822624, 3375096, 19492;
0, 67108864, 369106944, 444562432, 141270272, 10080760, 87184, 4;
0, 134217728, 771764224, 989807872, 346507120, 29167000, 354628, 80;
0, 268435456, 1610629120, 2192154880, 839762496, 82113648, 1338376, 812;
0, 536870912, 3355467776, 4831741952, 2013427136, 225746384, 4753320, 5916;
0, 1073741824, 6979354624, 10603063808, 4781027584, 607828752, 16052296, 35000;
0, 2147483648, 14495563776, 23174734336, 11254280416, 1606760304, 51954808, 178904, 1; ...
Row polynomials begin:
n=0: 1;
n=1: y;
n=2: 2*y;
n=3: 4*y;
n=4: 8*y + y^2;
n=5: 16*y + 4*y^2;
n=6: 32*y + 14*y^2;
n=7: 64*y + 40*y^2;
n=8: 128*y + 108*y^2 + 2*y^3;
n=9: 256*y + 272*y^2 + 12*y^3;
n=10: 512*y + 664*y^2 + 52*y^3;
n=11: 1024*y + 1568*y^2 + 188*y^3;
n=12: 2048*y + 3632*y^2 + 608*y^3 + y^4;
n=13: 4096*y + 8256*y^2 + 1816*y^3 + 12*y^4;
n=14: 8192*y + 18528*y^2 + 5128*y^3 + 76*y^4;
n=15: 16384*y + 41088*y^2 + 13856*y^3 + 360*y^4;
n=16: 32768*y + 90304*y^2 + 36176*y^3 + 1446*y^4;
n=17: 65536*y + 196864*y^2 + 91856*y^3 + 5192*y^4 + 4*y^5; ...
the first row in which y^m appears is given by n = A033156(m), where A033156 begins:
[1, 4, 8, 12, 17, 22, 27, 32, 38, 44, 50, 56, 62, 68, 74, 80, 87, 94, 101, 108, 115, 122, 129, 136, 143, 150, 157, 164, 171, 178, 185, 192, 200, ...].
Generating functions of initial columns.
G.f. of column 0: 1
G.f. of column 1: y*x/(1-2*x).
G.f. of column 2: y^2*x^4/((1-2*x)^2*(1-2*x^2)).
G.f. of column 3: y^3*2*x^8/((1-2*x)^3*(1-2*x^2)*(1-2*x^3)).
G.f. of column 4: y^4*x^12*(1 + 4*x - 10*x^3)/((1-2*x)^4*(1-2*x^2)^2*(1-2*x^3)*(1-2*x^4)).
G.f. of column 5: y^5*x^17*(4 + 2*x + 8*x^2 - 28*x^4)/((1-2*x)^5*(1-2*x^2)^2*(1-2*x^3)*(1-2*x^4)*(1-2*x^5)).
G.f. of column 6: y^6*x^22*(6 + 8*x - 20*x^3 - 24*x^4 - 36*x^5 - 56*x^6 + 16*x^7 + 176*x^8 + 224*x^9 - 336*x^11)/((1-2*x)^6*(1-2*x^2)^3*(1-2*x^3)^2*(1-2*x^4)*(1-2*x^5)*(1-2*x^6)).
G.f. of column 7: y^7*x^27*(4 + 24*x + 4*x^2 - 12*x^3 - 72*x^5 - 112*x^6 - 96*x^7 + 112*x^8 - 64*x^9 + 64*x^10 + 496*x^11 + 576*x^12 - 1056*x^14) / ((1-2*x)^7*(1-2*x^2)^3*(1-2*x^3)^2*(1-2*x^4)*(1-2*x^5)*(1-2*x^6)*(1-2*x^7)).
G.f. of column 8: y^8*x^32*(1 + 24*x + 36*x^2 - 4*x^3 - 88*x^4 - 202*x^5 - 14*x^6 - 82*x^7 - 168*x^8 + 400*x^9 + 440*x^10 + 892*x^11 + 1292*x^12 - 660*x^13 - 800*x^14 - 688*x^15 - 1776*x^16 - 1136*x^17 - 4504*x^18 - 2672*x^19 + 4672*x^20 + 5664*x^21 + 12672*x^22 - 13728*x^24) / ((1-2*x)^8*(1-2*x^2)^4*(1-2*x^3)^2*(1-2*x^4)^2*(1-2*x^5)*(1-2*x^6)*(1-2*x^7)*(1-2*x^8)).
G.f. of column 9: y^9*x^38*(8 + 60*x + 72*x^2 + 16*x^3 - 238*x^4 - 584*x^5 - 232*x^6 + 172*x^7 + 328*x^8 + 52*x^9 + 1012*x^10 + 2636*x^11 + 1464*x^12 + 520*x^13 - 2040*x^14 - 664*x^15 - 2360*x^16 - 8712*x^17 - 13008*x^18 - 3696*x^19 + 12080*x^20 + 15392*x^21 + 1456*x^22 - 11040*x^23 + 18112*x^24 + 37728*x^25 + 47040*x^26 - 34304*x^27 - 78144*x^28 - 73216*x^29 + 91520*x^31) / ((1-2*x)^9*(1-2*x^2)^4*(1-2*x^3)^3*(1-2*x^4)^2*(1-2*x^5)*(1-2*x^6)*(1-2*x^7)*(1-2*x^8)*(1-2*x^9)).
G.f. of column 10: y^10*x^44*(28 + 96*x + 198*x^2 - 160*x^3 - 864*x^4 - 596*x^5 - 856*x^6 - 384*x^7 + 3652*x^8 + 4752*x^9 + 696*x^10 - 2972*x^11 + 3928*x^12 + 4848*x^13 - 8360*x^14 - 18768*x^15 - 11000*x^16 - 14184*x^17 - 9896*x^18 + 17184*x^19 + 23664*x^20 + 7904*x^21 + 34480*x^22 + 53472*x^23 + 54160*x^24 + 68160*x^25 + 10560*x^26 - 166208*x^27 - 203488*x^28 - 86720*x^29 - 23552*x^30 + 13632*x^31 + 67584*x^32 - 95232*x^33 - 232256*x^34 + 129536*x^35 + 677632*x^36 + 624000*x^37 + 355840*x^38 - 67584*x^39 - 988416*x^40 - 1464320*x^41 + 1244672*x^43) / ((1-2*x)^10*(1-2*x^2)^5*(1-2*x^3)^3*(1-2*x^4)^2*(1-2*x^5)^2*(1-2*x^6)*(1-2*x^7)*(1-2*x^8)*(1-2*x^9)*(1-2*x^10)).
...
The g.f. of column n, y^n * x^A033156(n) * P(n,x)/Q(n,x), appears to have the following denominator:
Q(n,x) = Product_{k=1..n} (1 - 2*x^k)^floor(n/k), with
P(n,x) being a polynomial of degree A024916(n) - A033156(n),
where A024916(n) = Sum_{k=1..n} k*floor(n/k).
...

Links

Crossrefs

Cf. A274965 (row sums), A275691 (antidiagonal sums), A033156.
Cf. variant: A138157.

Programs

  • PARI
    /* Print first N rows of this triangle: */ N=32;
{a(n) = my(A=1 +x*O(x^n)); for(k=0, n, A = A^2 + y*x^(n+1-k)); polcoeff(A, n)}
{for(n=0, N, for(k=0,n, if(k==0,print1(polcoeff(a(n)+y*O(y^n),k,y)", "), if(polcoeff(a(n)+y*O(y^n),k,y)==0,break,print1(polcoeff(a(n)+y*O(y^n),k,y),", "))));print(""))}

Formula

G.f. A(x,y) satisfies: 1 = ...(((((A(x,y) - x*y)^(1/2) - x^2*y)^(1/2) - x^3*y)^(1/2) - x^4*y)^(1/2) - x^5*y)^(1/2) -...- x^n*y)^(1/2) -..., an infinite series of nested square roots.
Showing 1-2 of 2 results.

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