A162956 -id:A162956 - cp's OEIS frontend

A162958 Equals A162956 convolved with (1, 3, 3, 3, ...).

1, 4, 10, 19, 25, 40, 67, 94, 100, 115, 142, 175, 208, 280, 388, 469, 475, 490, 517, 550, 583, 655, 763, 850, 883, 955, 1069, 1201, 1372, 1696, 2101, 2344, 2350, 2365, 2392, 2425, 2458, 2530, 2638, 2725, 2758, 2830, 2944, 3076, 3247, 3571, 3976, 4225, 4258

Offset: 1

Gary W. Adamson, Jul 18 2009

nonn

Can be considered a toothpick sequence for N=3, following rules analogous to those in A160552 (= special case of "A"), A151548 = special case "B", and the toothpick sequence A139250 (N=2) = special case "C".
To obtain the infinite set of toothpick sequences, (N = 2, 3, 4, ...), replace the multiplier "2" in A160552 with any N, getting a triangle with 2^n terms. Convolve this A sequence with (1, N, 0, 0, 0, ...) = B such that row terms of A triangles converge to B.
Then generalized toothpick sequences (C) = A convolved with (1, N, N, N, ...).
Examples: A160552 * (1, 2, 0, 0, 0,...) = a B-type sequence A151548.
A160552 * (1, 2, 2, 2, 2,...) = the toothpick sequence A139250 for N=2.
A162956 is analogous to A160552 but replaces "2" with the multiplier "3".
Row terms of A162956 tend to A162957 = (1, 3, 0, 0, 0, ...) * A162956.
Toothpick sequence for N = 3 = A162958 = A162956 * (1, 3, 3, 3, ...).
Row sums of "A"-type triangles = powers of (N+2); since row sums of A160552 = (1, 4, 16, 64, ...), while row sums of A162956 = (1, 5, 25, 125, ...).
Is there an illustration of this sequence using toothpicks? - Omar E. Pol, Dec 13 2016

Alois P. Heinz, Table of n, a(n) for n = 1..16384
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A152548, A160552, A162956, A163267.

Third diagonal of A163311.

b:= proc(n) option remember; `if`(n<2, n,
      (j-> 3*b(j)+b(j+1))(n-2^ilog2(n)))
    end:
a:= proc(n) option remember;
      `if`(n=0, 0, a(n-1)+2*b(n-1)+b(n))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 28 2017

b[n_] := b[n] = If[n<2, n, Function[j, 3*b[j]+b[j+1]][n-2^Floor[Log[2, n]] ]];
a[n_] := a[n] = If[n == 0, 0, a[n-1] + 2*b[n-1] + b[n]];
Array[a, 100] (* Jean-François Alcover, Jun 11 2018, after Alois P. Heinz *)

Clarified definition by Omar E. Pol, Feb 06 2017

A162957 A162956 convolved with (1, 3, 0, 0, 0, ...).

Original entry on oeis.org

1, 4, 7, 13, 7, 19, 34, 40, 7, 19, 34, 46, 40, 91, 142, 121, 7, 19, 34, 46, 40, 91, 142, 127, 40, 91, 148, 178, 211, 415, 547, 364, 7, 19, 34, 46, 40, 91, 142, 127, 40, 91, 148, 178, 211, 415, 547, 370, 40, 91, 148, 178, 211, 415, 553, 421, 211, 421, 622, 745, 1048, 1792, 2005, 1093

Offset: 1

Gary W. Adamson, Jul 18 2009

nonn

Rows of A162956 tend to this sequence. Analogous to rows of A160552 tending to A151548.

Cf. A162956, A162958.

Equals (1, 3, 0, 0, 0, ...) * (1, 1, 4, 1, 4, 7, 13, 1, 4, 7, 13, 7, 19, 34, 40, ...).

a(20), a(21) corrected and more terms from Georg Fischer, Jul 17 2025

A160552 a(0)=0, a(1)=1; a(2^i+j) = 2*a(j) + a(j+1) for 0 <= j < 2^i.

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 5, 7, 1, 3, 5, 7, 5, 11, 17, 15, 1, 3, 5, 7, 5, 11, 17, 15, 5, 11, 17, 19, 21, 39, 49, 31, 1, 3, 5, 7, 5, 11, 17, 15, 5, 11, 17, 19, 21, 39, 49, 31, 5, 11, 17, 19, 21, 39, 49, 35, 21, 39, 53, 59, 81, 127, 129, 63, 1, 3, 5, 7, 5, 11, 17, 15, 5, 11, 17, 19, 21, 39, 49, 31

Offset: 0

David Applegate, May 18 2009

nonn

This recurrence is patterned after the one for A152980, but without the special cases.
Sequence viewed as triangle:
  0,
  1,
  1, 3,
  1, 3, 5, 7,
  1, 3, 5, 7, 5, 11, 17, 15,
  1, 3, 5, 7, 5, 11, 17, 15, 5, 11, 17, 19, 21, 39, 49, 31.
The rows converge to A151548.
Also the sum of the terms in the k-th row (k >= 1) is 4^(k-1). Proof by induction. - N. J. A. Sloane, Jan 21 2010
If this sequence [1, 1, 3, 1, 3, 5, 7, 1, 3, 5, 7, 5, 11, 17, 15, ...] is convolved with [1, 2, 2, 2, 2, 2, 2, ...] we obtain A139250, the toothpick sequence. Example: A139250(5) = 15 = (1, 2, 2, 2, 2) * (3, 1, 3, 1, 1). - Gary W. Adamson, May 19 2009
Starting with 1 and convolved with [1, 2, 0, 0, 0, ...] = A151548. - Gary W. Adamson, Jun 04 2009
Refer to A162956 for the analogous triangle using N=3. - Gary W. Adamson, Jul 20 2009
It appears that the sums of two successive terms give the positive terms of A139251. - Omar E. Pol, Feb 18 2015

			a(2) = a(2^1+0) = 2*a(0) + a(1) = 1, a(3) = a(2^1+1) = 2*a(1) + a(2) = 3*a(2^i) = 2*a(0) + a(1) = 1.

N. J. A. Sloane, Table of n, a(n) for n = 0..16384
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

For the recurrence a(2^i+j) = C*a(j) + D*a(j+1), a(0) = A, a(1) = B for following values of (A B C D) see: (0 1 1 1) A118977, (1 0 1 1) A151702, (1 1 1 1) A151570, (1 2 1 1) A151571, (0 1 1 2) A151572, (1 0 1 2) A151703, (1 1 1 2) A151573, (1 2 1 2) A151574, (0 1 2 1) A160552, (1 0 2 1) A151704, (1 1 2 1) A151568, (1 2 2 1) A151569, (0 1 2 2) A151705, (1 0 2 2) A151706, (1 1 2 2) A151707, (1 2 2 2) A151708.
Cf. A152980, A139250, A139251, A151548, A160570, A151568.
Cf. A162956, A170903.

S:=proc(n) option remember; local i,j; if n <= 1 then RETURN(n); fi; i:=floor(log(n)/log(2)); j:=n-2^i; 2*S(j)+S(j+1); end; # N. J. A. Sloane, May 18 2009
H := x*(1+2*x)/(1+x) + (4*x^2/(1+2*x))*(mul(1+x^(2^k-1)+2*x^(2^k),k=1..20)-1); series(H,x,120); # N. J. A. Sloane, May 23 2009

Nest[Join[#, 2 # + Append[Rest@#, 1]] &, {0}, 7] (* Ivan Neretin, Feb 09 2017 *)

G.f.: x*(1+2*x)/(1+x) + (4*x^2/(1+2*x))*(-1 + Product_{k>=1} (1 + x^(2^k-1) + 2*x^(2^k))). - N. J. A. Sloane, May 23 2009, based on Gary W. Adamson's comment above and the known g.f. for A139250.
It appears that a(n) = A169708(n)/4, n >= 1. - Omar E. Pol, Feb 15 2015
It appears that a(n) = A139251(n) - a(n-1), n >= 1. - Omar E. Pol, Feb 18 2015

A163267 Partial sums of A118977.

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 10, 13, 14, 16, 19, 22, 25, 30, 36, 40, 41, 43, 46, 49, 52, 57, 63, 67, 70, 75, 81, 87, 95, 106, 116, 121, 122, 124, 127, 130, 133, 138, 144, 148, 151, 156, 162, 168, 176, 187, 197, 202, 205, 210, 216, 222, 230, 241, 251, 258, 266, 277, 289, 303, 322, 343, 358

Offset: 0

Gary W. Adamson, Jul 24 2009

nonn

Can be considered toothpick sequence for N=1.
Based on a consistent set of rules for generating toothpick sequences.
Cf. A139250 where (1, 1, 3, 1, 3, 5, 7, ...) is convolved with (1, 2, 2, 2, ...) and A162958 where A162956 is convolved with (1, 3, 3, 3, ...); the present sequence can be considered toothpick N=1 since A118977 is convolved with (1, 1, 1, ...).
Arranged in array fashion, the first three toothpick sequences would be:
N=1: A163267: (1, 2,  4,  5,  7, 10, 13, 14, ...)
N=2: A139250: (1, 3,  7, 11, 15, 23, 35, 43, ...)
N=3: A162958: (1, 4, 10, 19, 25, 40, 67, 94, ...)
...
Is there an illustration of this sequence using toothpicks? - Omar E. Pol, Dec 13 2016

G. C. Greubel, Table of n, a(n) for n = 0..1000
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A118977, A139250, A162958.

a[0] = 0; a[1] = 1; a[n_] := a[n] = (j = n - 2^Floor[Log[2, n]]; a[j] + a[j + 1]); Table[Sum[a[n], {n, 0, k}], {k, 0, 20}] (* G. C. Greubel, Dec 12 2016 *)

Edited and extended by N. J. A. Sloane, Jan 07 2010

A170838 G.f.: Product_{k>=0} (1 + x^(2^k-1) + 3x^(2^k)).

Original entry on oeis.org

2, 5, 9, 11, 11, 24, 36, 29, 11, 24, 38, 44, 57, 108, 135, 83, 11, 24, 38, 44, 57, 108, 137, 98, 57, 110, 158, 189, 279, 459, 486, 245, 11, 24, 38, 44, 57, 108, 137, 98, 57, 110, 158, 189, 279, 459, 488, 260, 57, 110, 158, 189, 279, 461, 509, 351, 281, 488, 663, 846, 1296

Offset: 0

N. J. A. Sloane, Jan 02 2010

nonn

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

A170838-A170852, A170854-A170872 were added to supplement Gary W. Adamson's A162956.

Maple program for A170838-A170852, A162956, A170854-A170872.
read format;
G := proc(a,b,c); mul( 1 + a*x^(2^n-1) + b*x^(2^n), n=c..20); end;
f := proc(a,b,c) seriestolist(series(G(a,b,c),x,120)); end;
at:=170838:
for a from 1 to 2 do for c from 0 to 2 do
b:=3;
t1:=f(a,b,c);
lprint( format(t1,at) );
lprint("G.f.: Prod_{k >= ", c, "} (1 +",a,"* x^(2^k-1) +",b,"* x^(2^k)).");
at:=at+1; od: od:
for b from 1 to 3 do for c from 0 to 2 do
a:=3;
t1:=f(a,b,c);
lprint( format(t1,at) );
lprint("G.f.: Prod_{k >= ", c, "} (1 +",a,"* x^(2^k-1) +",b,"* x^(2^k)).");
at:=at+1; od: od:
h:=proc(r,s,a,b) local s1,n,i,j;
s1:=array(0..120);
s1[0]:=r; s1[1]:=s;
for n from 2 to 120 do i:=floor(log(n)/log(2));
j:=n-2^i; s1[n]:=a*s1[j]+b*s1[j+1]; od:
[seq(s1[n],n=0..120)];
end;
l1:=[[0,1],[1,0],[1,1],[1,2]];
l2:=[[3,1],[3,2],[1,3],[2,3],[3,3]];
for i from 1 to 4 do for j from 1 to 5 do
r:=l1[i][1];
s:=l1[i][2];
a:=l2[j][1];
b:=l2[j][2];
t1:=h(r,s,a,b);
lprint(format(t1,at)); at:=at+1;
lprint("a(0)=",r,", a(1)=", s, "; a(2^i+j)=",a,"*a(j)+",b,"a(j+1) for 0 <= j < 2^i.");
od: od:

With[{nn=60},CoefficientList[Series[Product[1+x^(2^k-1)+3x^2^k,{k,0,nn}],{x,0,nn}],x]] (* Harvey P. Dale, Dec 29 2021 *)

cp's OEIS Frontend

A162958 Equals A162956 convolved with (1, 3, 3, 3, ...).

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A162957 A162956 convolved with (1, 3, 0, 0, 0, ...).

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A160552 a(0)=0, a(1)=1; a(2^i+j) = 2*a(j) + a(j+1) for 0 <= j < 2^i.

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A163267 Partial sums of A118977.

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A170838 G.f.: Product_{k>=0} (1 + x^(2^k-1) + 3x^(2^k)).

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