cp's OEIS Frontend

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.

Showing 1-4 of 4 results.

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.

A275751 Self-convolution square root of the even bisection of A274965.

Original entry on oeis.org

1, 1, 4, 19, 92, 446, 2150, 10280, 48761, 229558, 1073278, 4986624, 23037102, 105877968, 484337300, 2206188412, 10010589904, 45264063504, 204016241794, 916898737038, 4109984712933, 18379240912034, 82012499946246, 365245641944278, 1623757696702586, 7207073607368924, 31941896126213722, 141377838141158888, 624983649220555836, 2759711619634526196, 12173102200970091434
Offset: 0

Views

Author

Paul D. Hanna, Aug 14 2016

Keywords

Comments

The g.f. of A274965 equals G(x,1), where G(x,y) = x*y + G(x,x*y)^2 is the g.f. of A275670.
First negative term is at a(646).

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 19*x^3 + 92*x^4 + 446*x^5 + 2150*x^6 + 10280*x^7 + 48761*x^8 + 229558*x^9 + 1073278*x^10 + 4986624*x^11 + 23037102*x^12 +...
where
A(x)^2 = 1 + 2*x + 9*x^2 + 46*x^3 + 238*x^4 + 1228*x^5 + 6289*x^6 + 31924*x^7 + 160694*x^8 + 802642*x^9 + 3980916*x^10 +...+ A274965(2*n)*x^n +...

Links

Crossrefs

Cf. A274965, A275752.

Programs

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

A275752 Self-convolution square root of the odd bisection of A274965.

Original entry on oeis.org

1, 2, 8, 36, 166, 770, 3574, 16560, 76516, 352498, 1619014, 7414134, 33855996, 154181234, 700333366, 3173299648, 14345094004, 64704125888, 291235313046, 1308229210186, 5865335253474, 26248821086374, 117265700856282, 523010482541564, 2328947839518852, 10354971182171076, 45973304229373220, 203824525466826232, 902455230607927616, 3990584636812405052, 17624255201680536016
Offset: 0

Views

Author

Paul D. Hanna, Aug 14 2016

Keywords

Comments

The g.f. of A274965 equals G(x,1), where G(x,y) = x*y + G(x,x*y)^2 is the g.f. of A275670.

Examples

			G.f.: A(x) = 1 + 2*x + 8*x^2 + 36*x^3 + 166*x^4 + 770*x^5 + 3574*x^6 + 16560*x^7 + 76516*x^8 + 352498*x^9 + 1619014*x^10 + 7414134*x^11 + 33855996*x^12 +...
where
A(x)^2 = 1 + 4*x + 20*x^2 + 104*x^3 + 540*x^4 + 2780*x^5 + 14180*x^6 + 71688*x^7 + 359452*x^8 + 1788988*x^9 + 8844064*x^10 +...+ A274965(2*n+1)*x^n +...

Links

Crossrefs

Cf. A274965, A275751.

Programs

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

A275691 G.f. A(x) satisfies: 1 = ...(((((A(x) - x^2)^(1/2) - x^3)^(1/2) - x^4)^(1/2) - x^5)^(1/2) - x^6)^(1/2) -...- x^n)^(1/2) -..., an infinite series of nested square roots.

Original entry on oeis.org

1, 0, 1, 2, 4, 8, 17, 36, 78, 168, 364, 786, 1700, 3668, 7916, 17056, 36729, 78996, 169772, 364472, 781814, 1675464, 3587660, 7675722, 16409240, 35052552, 74822496, 159599700, 340199178, 724675528, 1542673868, 3281957116, 6977971852, 14827596904, 31489490296, 66837617960, 141789447876, 300636048724, 637116434912, 1349532001896, 2857195771769, 6046370298448
Offset: 0

Views

Author

Paul D. Hanna, Aug 05 2016

Keywords

Comments

Compare definition with that of A274965.

Examples

			G.f.: A(x) = 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 + 3668*x^13 + 7916*x^14 +...
The g.f. of related sequence A274965 begins:
A(x)^2 + 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 + 2780*x^11 + 6289*x^12 +...

Crossrefs

Cf. A274965.
Antidiagonal sums of triangle A275670.

Programs

  • PARI
    {a(n) = my(A=1 +x*O(x^n)); for(k=0, n, A = A^2 + x^(n+2-k)); polcoeff(A, n)}
for(n=0, 60, print1(a(n), ", "))

Formula

G.f.: A(x) = G(x,x), where G(x,y) = x*y + G(x,x*y)^2 is the g.f. of A275670.
G.f.: A(x) = sqrt(F(x) - x), where F(x) is the g.f. of A274965.
Showing 1-4 of 4 results.

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