A006046 -id:A006046 - cp's OEIS frontend

A006048 Number of entries in first n rows of Pascal's triangle not divisible by 3.

1, 3, 6, 8, 12, 18, 21, 27, 36, 38, 42, 48, 52, 60, 72, 78, 90, 108, 111, 117, 126, 132, 144, 162, 171, 189, 216, 218, 222, 228, 232, 240, 252, 258, 270, 288, 292, 300, 312, 320, 336, 360, 372, 396, 432, 438, 450, 468, 480, 504, 540, 558, 594, 648, 651, 657, 666, 672, 684, 702, 711, 729, 756, 762, 774, 792, 804, 828, 864, 882, 918, 972, 981, 999, 1026, 1044, 1080, 1134, 1161, 1215, 1296

Offset: 0

Jeffrey Shallit

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 53.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Periodic minimum in the count of binomial coefficients not divisible by a prime, arXiv:2408.06817 [math.NT], 2024. See p. 1.
Akhlesh Lakhtakia and Russell Messier, Self-similar sequences and chaos from Gauss sums, Computers & graphics 13.1 (1989): 59-62.
Akhlesh Lakhtakia and Russell Messier, Self-similar sequences and chaos from Gauss sums, Computers & Graphics 13.1 (1989), 59-62. (Annotated scanned copy)
Akhlesh Lakhtakia and Russell Messier, Self-similar sequences and chaos from Gauss sums, Computers & Graphics 13.1 (1989), 59-60. (Annotated scanned copy)
A. Lakhtakia et al., Fractal sequences derived from the self-similar extensions of the Sierpinski gasket, J. Phys. A 21 (1988), 1925-1928.

Partial sums of A006047.

from math import prod
from gmpy2 import digits
def A006048(n): return sum(prod(int(d)+1 for d in digits(m,3)) for m in range(n+1)) # Chai Wah Wu, Aug 10 2025

from math import prod
from gmpy2 import digits
def A006048(n):
    d = list(map(lambda x:int(x)+1,digits(n+1,3)[::-1]))
    return sum((b-1)*prod(d[a:])*6**a for a, b in enumerate(d))>>1  # Chai Wah Wu, Aug 13 2025

More terms from N. J. A. Sloane, Apr 23 2025

A130667 a(1) = 1; a(n) = max{ 5*a(k) + a(n-k) | 1 <= k <= n/2 } for n > 1.

Original entry on oeis.org

1, 6, 11, 36, 41, 66, 91, 216, 221, 246, 271, 396, 421, 546, 671, 1296, 1301, 1326, 1351, 1476, 1501, 1626, 1751, 2376, 2401, 2526, 2651, 3276, 3401, 4026, 4651, 7776, 7781, 7806, 7831, 7956, 7981, 8106, 8231, 8856, 8881, 9006, 9131, 9756, 9881, 10506, 11131

Offset: 1

N. J. A. Sloane, based on a message from Don Knuth, Jun 23 2007

nonn

From Gary W. Adamson, Aug 27 2016: (Start)
The formula of Mar 26 2010 is equivalent to the following: Given the production matrix M below, lim_{k->infinity} M^k as a left-shifted vector generates the sequence.
  1, 0, 0, 0, 0, ...
  6, 0, 0, 0, 0, ...
  5, 1, 0, 0, 0, ...
  0, 6, 0, 0, 0, ...
  0, 5, 1, 0, 0, ...
  0, 0, 6, 0, 0, ...
  0, 0, 5, 1, 0, ...
  ...
The sequence divided by its aerated variant is (1, 6, 5, 0, 0, 0, ...). (End)

Alois P. Heinz, Table of n, a(n) for n = 1..16383
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 27, 32-33.
D. E. Knuth, Problem 11320, The American Mathematical Monthly, Vol. 114, No. 9 (Nov., 2007), p. 835.

Cf. A000120, A006046, A116520, A130665, A360189.

import Data.List (transpose)
a130667 n = a130667_list !! (n-1)
a130667_list = 1 : (concat $ transpose
   [zipWith (+) vs a130667_list, zipWith (+) vs $ tail a130667_list])
   where vs = map (* 5) a130667_list
-- Reinhard Zumkeller, Apr 18 2012

[&+[5^(2*k - Valuation(Factorial(2*k), 2)): k in [0..n]]: n in [0..50]]; // Vincenzo Librandi, Mar 15 2019

a:= proc(n) option remember;
      `if`(n=1, 1, `if`(irem(n, 2, 'm')=0, 6*a(m), 5*a(m)+a(n-m)))
    end:
seq(a(n), n=1..70); # Alois P. Heinz, Apr 09 2012

a[1]=1; a[n_] := a[n] = If[EvenQ[n], 6a[n/2], 5a[(n-1)/2]+a[(n+1)/2]]; Array[a, 50] (* Jean-François Alcover, Feb 13 2015 *)

first(n)=my(v=vector(n),r,t); v[1]=1; for(i=2,n, r=0; for(k=1,i\2, t=5*v[k]+v[i-k]; if(t>r, r=t)); v[i]=r); v \\ Charles R Greathouse IV, Aug 29 2016

a(2*n) = 6*a(n) and a(2*n+1) = 5*a(n) + a(n+1).
Let r(x) = (1 + 6*x + 5*x^2). Then (1 + 6*x + 11*x^2 + 36*x^3 + ...) = r(x) * r(x^2) * r(x^4) * r(x^8) * ... - Gary W. Adamson, Mar 26 2010
a(n) = Sum_{k=0..n} 5^wt(k), where wt = A000120. - Mike Warburton, Mar 14 2019
a(n) = Sum_{k=0..floor(log_2(n))} 5^k*A360189(n-1,k). - Alois P. Heinz, Mar 06 2023

A073121 a(n) = ra(ceiling(n/2)) + sa(floor(n/2)) with a(1)=1 and (r,s)=(2,2).

Original entry on oeis.org

1, 4, 10, 16, 28, 40, 52, 64, 88, 112, 136, 160, 184, 208, 232, 256, 304, 352, 400, 448, 496, 544, 592, 640, 688, 736, 784, 832, 880, 928, 976, 1024, 1120, 1216, 1312, 1408, 1504, 1600, 1696, 1792, 1888, 1984, 2080, 2176, 2272, 2368, 2464, 2560, 2656, 2752

Offset: 1

Jeffrey Shallit, Aug 25 2002

nonn

A recurrence occurring in the analysis of a regular expression algorithm.

			a(1)=1, a(2) = 2*(a(1)+a(1)) = 4, a(3) = 2*(a(2)+a(1)) = 10.

K. Ellul, J. Shallit and M.-w. Wang, Regular expressions: new results and open problems, in Descriptional Complexity of Formal Systems (DCFS), Proceedings of workshop, London, Ontario, Canada, 21-24 August 2002, pp. 17-34.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Jean-Paul Allouche and Jeffrey Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
Jean-Paul Allouche and Jeffrey Shallit, The Ring of k-regular Sequences, II
F. Barbero, G. Gutin, M. Jones, and B. Sheng, Parameterized and approximation algorithms for the load coloring problem, Algorithmica 79, No. 1, 211-229 (2017). Prop 3.
Keh-Ning Chang and Shi-Chun Tsai, Exact solution of a minimal recurrence, Inform. Process. Lett. 75 (2000), 61-64.
Erik D. Demaine, Martin L. Demaine, Yair N. Minsky, Joseph S. B. Mitchell, Ronald L. Rivest, and Mihai Patrascu, Picture-Hanging Puzzles, arXiv:1203.3602 [cs.DS], 2012-2014.
Keith Ellul, Bryan Krawetz, Jeffrey Shallit, and Ming-wei Wang, Regular expressions: new results and open problems, Journal of Automata, Languages and Combinatorics, preprint.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 27, 37.
Jean-Marc Luck, Revisiting log-periodic oscillations, arXiv:2403.00432 [cond-mat.stat-mech], 2024. See p. 14.
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions

Cf. A053644, A053645, A254575.

Sequences of form a(n) = r*a(ceiling(n/2)) + s*a(floor(n/2)), a(1)=1, for (r,s) = (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), (1,4), (2,3), (3,2), (4,1): A000027, A006046, A064194, A130665, A073121, A268524, A116520, A268525, A268526, A268527.

a073121 n = a053644 n * (fromIntegral n + 2 * a053645 n)
-- Reinhard Zumkeller, Mar 23 2012

a:= proc(n) option remember; `if`(n=1, 1,
      2*((m-> a(m)+a(n-m))(iquo(n, 2))))
    end:
seq(a(n), n=1..70);  # Alois P. Heinz, Feb 01 2015

a[n_] := a[n] = If[n == 1, 1, 2*(a[Quotient[n, 2]] + a[n - Quotient[n, 2]])]; Table[a[n], {n, 1, 70}] (* Jean-François Alcover, Feb 24 2016, after Alois P. Heinz *)
a[ n_] := If[ n < 1, 0, Module[{m = 1, A = 1}, While[m < n, m *= 2; A = (Normal[A] /. x -> x^2) 2 (1 + x)^2 - 1 + O[x]^m]; Coefficient[A, x, n - 1]]]; (* Michael Somos, Jul 04 2017 *)

{a(n) = n--; if( n<0, 0, my(m=1, A = 1 + O(x)); while(m<=n, m*=2; A = subst(A, x, x^2) * 2 * (1 + x)^2 - 1); polcoeff(A, n))}; /* Michael Somos, Jul 04 2017 */

a(n) = 2*(a(floor(n/2)) + a(ceiling(n/2))) for n >= 2; alternatively, a(n) = 2^c(n+2b) where n = 2^c + b, 0 <= b < 2^c.
a(n) == 1 (mod 3), a(n+1)-a(n) = 3*A053644(n). If k >= 1: a(2^k)=4^k, a(3*2^k)=(10/9)*4^k. More generally a(m*2^k) = a(m)*4^k. Hence for any n, n^2 <= a(n) <= C*n^2 where C is a constant 1.125 < C < 1.14 and it seems that C = lim_{k->infinity} a(A001045(k))/A001045(k)^2 where A001045(k) =(2^n - (-1)^n)/3 is the Jacobsthal sequence. In other words, in the range 2^k <= n <= 2^(k+1) the maximum of a(n)/n^2 is reached for the only possible n in the Jacobsthal sequence. - Benoit Cloitre, Aug 26 2002
For any n, n^2 <= a(n) <= 9/8 * n^2. - Arnoud van der Leer, Sep 01 2019
a(n) = 2*(a(floor(n/2)) + a(ceiling(n/2))) for n >= 2; alternatively, a(n) = 2^c(n+2b) where n = 2^c + b, 0 <= b < 2^c
G.f.: 3*x/(1-x)^2 * ((2*x+1)/3 + Sum_{k>=1} 2^(k-1)*x^2^k). - Ralf Stephan, Apr 18 2003
G.f.: A(x) = 2 * (1/x + 2 + x) * A(x^2) - x. - Michael Somos, Jul 04 2017

Edited by N. J. A. Sloane, Feb 16 2016

A194458 Total number of entries in rows 0,1,...,n of Pascal's triangle not divisible by 5.

Original entry on oeis.org

1, 3, 6, 10, 15, 17, 21, 27, 35, 45, 48, 54, 63, 75, 90, 94, 102, 114, 130, 150, 155, 165, 180, 200, 225, 227, 231, 237, 245, 255, 259, 267, 279, 295, 315, 321, 333, 351, 375, 405, 413, 429, 453, 485, 525, 535, 555, 585, 625, 675, 678, 684, 693, 705, 720, 726

Offset: 0

Paul Weisenhorn, Aug 24 2011

nonn

The number of zeros in the first n rows is binomial(n+1,2) - a(n).

			n = 38: n+1 = 39 = 124_5, thus a(38) = (C(5,2)*15^0*3 + C(3,2)*15^1)*2 + C(2,2)*15^2 = (10*1*3 + 3*15)*2 + 1*225 = 375.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 53.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Periodic minimum in the count of binomial coefficients not divisible by a prime, arXiv:2408.06817 [math.NT], 2024. See p. 1.

A006046(n+1) = A006046(n) + A001316(n) for p=2.
A006048(n+1) = A006048(n) + A006047(n+1) for p=3.
a(n+1) = a(n) + A194459(n+1) for p=5.

a:= proc(n) local l, m, h, j;
      m:= n+1;
      l:= [];
      while m>0 do l:= [l[], irem (m, 5, 'm')+1] od;
      h:= 0;
      for j to nops(l) do h:= h*l[j] +binomial (l[j], 2) *15^(j-1) od:
      h
    end:
seq(a(n), n=0..100);

a[n_] := Module[{l, m, r, h, j}, m = n+1; l = {}; While[m>0, l = Append[l, {m, r} = QuotientRemainder[m, 5]; r+1]]; h = 0; For[j = 1, j <= Length[l], j++, h = h*l[[j]] + Binomial [l[[j]], 2] *15^(j-1)]; h]; Table [a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 26 2017, translated from Maple *)

from math import prod
from gmpy2 import digits
def A194458(n): return sum(prod(int(d)+1 for d in digits(m,5)) for m in range(n+1)) # Chai Wah Wu, Aug 10 2025

from math import prod
from gmpy2 import digits
def A194458(n):
    d = list(map(lambda x:int(x)+1,digits(n+1,5)[::-1]))
    return sum((b-1)*prod(d[a:])*15**a for a, b in enumerate(d))>>1 # Chai Wah Wu, Aug 13 2025

a(n) = ((C(d0+1,2)*15^0*(d1+1) + C(d1+1,2)*15^1)*(d1+1) + C(d1+1,2)*15^1)*(d2+1) + C(d2+1,2)*15^2 ..., where d_k...d_1d_0 is the base 5 expansion of n+1 and 15 = binomial(5+1,2). The formula generalizes to other prime bases p.

Edited by Alois P. Heinz, Sep 06 2011

A161342 Number of "ON" cubic cells at n-th stage in simple 3-dimensional cellular automaton: a(n) = A160428(n)/8.

Original entry on oeis.org

0, 1, 8, 15, 64, 71, 120, 169, 512, 519, 568, 617, 960, 1009, 1352, 1695, 4096, 4103, 4152, 4201, 4544, 4593, 4936, 5279, 7680, 7729, 8072, 8415, 10816, 11159, 13560, 15961, 32768, 32775, 32824, 32873, 33216, 33265, 33608, 33951, 36352, 36401, 36744, 37087, 39488

Offset: 0

Omar E. Pol, Jun 14 2009

nonn

First differences are in A161343. - Omar E. Pol, May 03 2015
From Gary W. Adamson, Aug 30 2016: (Start)
Let M =
  1, 0, 0, 0, 0, ...
  8, 0, 0, 0, 0, ...
  7, 1, 0, 0, 0, ...
  0, 8, 0, 0, 0, ...
  0, 7, 1, 0, 0, ...
  0, 0, 8, 0, 0, ...
  0, 0, 7, 1, 0, ...
  ...
Then M^k converges to a single nonzero column giving the sequence.
The sequence with offset 1 divided by its aerated variant is (1, 8, 7, 0, 0, 0, ...). (End)

Alois P. Heinz, Table of n, a(n) for n = 0..16383
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A160410, A160428, A161343, A006046, A130665, A116520, A130667, A116522, A116526, A116525, A360189.

b:= proc(n) option remember; `if`(n<0, 0,
      b(n-1)+x^add(i, i=Bits[Split](n)))
    end:
a:= n-> subs(x=7, b(n-1)):
seq(a(n), n=0..44);  # Alois P. Heinz, Mar 06 2023

A161342list[nmax_]:=Join[{0},Accumulate[7^DigitCount[Range[0,nmax-1],2,1]]];A161342list[100] (* Paolo Xausa, Aug 05 2023 *)

From Nathaniel Johnston, Nov 13 2010: (Start)
a(n) = Sum_{k=0..n-1} 7^A000120(k).
a(n) = 1 + 7 * Sum_{k=1..n-1} A151785(k), for n >= 1.
a(2^n) = 2^(3n).
(End)
a(n) = Sum_{k=0..floor(log_2(n))} 7^k*A360189(n-1,k). - Alois P. Heinz, Mar 06 2023

More terms from Nathaniel Johnston, Nov 13 2010

A268524 a(n) = ra(ceiling(n/2))+sa(floor(n/2)) with a(1)=1 and (r,s)=(3,1).

Original entry on oeis.org

1, 4, 13, 16, 43, 52, 61, 64, 145, 172, 199, 208, 235, 244, 253, 256, 499, 580, 661, 688, 769, 796, 823, 832, 913, 940, 967, 976, 1003, 1012, 1021, 1024, 1753, 1996, 2239, 2320, 2563, 2644, 2725, 2752, 2995, 3076, 3157, 3184, 3265, 3292, 3319, 3328, 3571, 3652, 3733, 3760, 3841, 3868, 3895

Offset: 1

N. J. A. Sloane, Feb 16 2016

nonn

Number of triples 0 <= i, j, k < n such that bitwise AND of all pairs (i, j), (j, k), (k, i) is 0. - Peter Karpov, Mar 01 2016

Start with A = [[[1]]], iteratively replace every element Aijk with Aijk * [[[1, 1], [1, 0]], [[1, 0], [0, 0]]]. a(n) is the sum of the resulting array inside the cubic region i, j, k < n. - Peter Karpov, Mar 01 2016

K.-N. Chang and S.-C. Tsai, Exact solution of a minimal recurrence, Inform. Process. Lett. 75 (2000), 61-64.

Sequences of form a(n) = r*a(ceiling(n/2))+s*a(floor(n/2)) with a(1)=1 and (r,s) = (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), (1,4), (2,3), (3,2), (4,1): A000027, A006046, A064194, A130665, A073121, A268524, A116520, A268525, A268526, A268527.

a(n) = if (n==1, 1, 3*a(ceil(n/2)) + a(floor(n/2))); \\ Michel Marcus, Mar 24 2016

A160722 Number of "ON" cells at n-th stage in a certain 2-dimensional cellular automaton based on Sierpinski triangles (see Comments for precise definition).

Original entry on oeis.org

0, 1, 5, 9, 19, 23, 33, 43, 65, 69, 79, 89, 111, 121, 143, 165, 211, 215, 225, 235, 257, 267, 289, 311, 357, 367, 389, 411, 457, 479, 525, 571, 665, 669, 679, 689, 711, 721, 743, 765, 811, 821, 843, 865, 911, 933, 979, 1025, 1119, 1129, 1151, 1173, 1219, 1241

Offset: 0

Omar E. Pol, May 25 2009, Jan 03 2010

nonn

This cellular automata is formed by the concatenation of three Sierpinski triangles, starting from a central vertex. Adjacent polygons are fused. The ON cells are triangles, but we only count after fusion. The sequence gives the number of polygons at the n-th round.

If instead we start from four Sierpinski triangles we get A160720.

			We start at round 0 with no polygons, a(0) = 0.
At round 1 we turn ON the first triangle in each of the three Sierpinski triangles. After fusion we have a concave pentagon, so a(1) = 1.
At round 2 we turn ON two triangles in each the three Sierpinski triangles. After fusions we have the concave pentagon and four triangles. So a(2) = 1 + 4 = 5.

Michael De Vlieger, Table of n, a(n) for n = 0..9999
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.]
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 30.
Omar E. Pol, Illustration if initial terms
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

A160723 gives the first differences.

Cf. A139250, A160720.

a[0] = 0; a[1] = 1; a[n_] := a[n] = 2 a[Floor[#]] + a[Ceiling[#]] &[n/2]; Array[3 a[#] - 2 # &, 54, 0] (* Michael De Vlieger, Nov 01 2022 *)

a(n) = 3*A006046(n) - 2*n. - Max Alekseyev, Jan 21 2010

Extended by Max Alekseyev, Jan 21 2010

A233774 Total number of vertices in the first n rows of Sierpinski gasket, with a(0) = 1.

Original entry on oeis.org

1, 3, 6, 10, 15, 19, 25, 33, 42, 46, 52, 60, 70, 78, 90, 106, 123, 127, 133, 141, 151, 159, 171, 187, 205, 213, 225, 241, 261, 277, 301, 333, 366, 370, 376, 384, 394, 402, 414, 430, 448, 456, 468, 484, 504, 520, 544, 576, 610, 618, 630, 646, 666, 682

Offset: 0

Omar E. Pol, Dec 16 2013

nonn

			Illustration of initial terms:
-----------------------------------------------------
           Diagram            n     A233775(n)   a(n)
-----------------------------------------------------
              *               0         1         1
             /T\
            *---*             1         2         3
           /T\ /T\
          *---*---*           2         3         6
         /T\     /T\
        *---*   *---*         3         4        10
       /T\ /T\ /T\ /T\
      *---*---*---*---*       4         5        15
     /T\             /T\
    *---*           *---*     5         4        19
-----------------------------------------------------
After five stages the number of "black" triangles in the structure is A006046(5) = 11. The total number of vertices is 19, so a(5) = 19.

Paolo Xausa, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Eric Weisstein's World of Mathematics, Sierpiński Gasket Graph.
Wikipedia, Sierpinski triangle.

Partial sums of A233775.

Cf. A001316, A006046, A047999, A067771.

A233775[n_] := If[n == 0, 1, (2^IntegerExponent[n, 2]+1)*2^(DigitSum[n, 2]-1)];
Accumulate[Array[A233775, 100, 0]] (* Paolo Xausa, Aug 07 2024 *)

a(2^k) = A067771(k), k >= 0.

A233775 Number of vertices in the n-th row of the Sierpinski gasket (cf. A047999).

Original entry on oeis.org

1, 2, 3, 4, 5, 4, 6, 8, 9, 4, 6, 8, 10, 8, 12, 16, 17, 4, 6, 8, 10, 8, 12, 16, 18, 8, 12, 16, 20, 16, 24, 32, 33, 4, 6, 8, 10, 8, 12, 16, 18, 8, 12, 16, 20, 16, 24, 32, 34, 8, 12, 16, 20, 16, 24, 32, 36, 16, 24, 32, 40, 32, 48, 64, 65, 4, 6, 8, 10, 8, 12

Offset: 0

Omar E. Pol, Dec 16 2013

Partial sums give A233774.
The subsequence of odd terms is A083318. - Gary W. Adamson, Jan 13 2014
Equivalently, this is the coordination sequence for the Sierpinski gasket with respect to the apex. - N. J. A. Sloane, Sep 19 2020

			Illustration of initial terms:
--------------------------------------------------------
           Diagram            n        a(n)   A233774(n)
--------------------------------------------------------
              *               0         1         1
             /T\
            *---*             1         2         3
           /T\ /T\
          *---*---*           2         3         6
         /T\     /T\
        *---*   *---*         3         4        10
       /T\ /T\ /T\ /T\
      *---*---*---*---*       4         5        15
     /T\             /T\
    *---*           *---*     5         4        19
--------------------------------------------------------
After five stages the number of "black" triangles in the structure is A006046(5) = 11 and the number of "black" triangles in row 5 is A001316(5-1) = 2. The number of vertices in row 5 is equal to 4, so a(5) = 4.
Written as an irregular triangle the sequence begins:
  1;
  2;
  3;
  4,5;
  4,6,8,9;
  4,6,8,10,8,12,16,17;
  4,6,8,10,8,12,16,18,8,12,16,20,16,24,32,33;
  ...

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Eric Weisstein's World of Mathematics, Sierpiński Gasket Graph.
Wikipedia, Sierpinski triangle.
Index entries for coordination sequences

Right border gives A094373.

Cf. A001316, A006046, A047999, A233774, A083318.

A000120 := n -> add(i, i=convert(n, base, 2)):
A007814 := n -> padic[ordp](n, 2):
A233775 := n->(2^A007814(n)+1)*(2^(A000120(n)-1); # N. J. A. Sloane, Sep 19 2020

A233775[n_] := If[n == 0, 1, (2^IntegerExponent[n, 2]+1)*2^(DigitSum[n, 2]-1)];
Array[A233775, 100, 0] (* Paolo Xausa, Aug 05 2024 *)

print1("1, "); for(k=1,70, print1((2^valuation(k,2)+1) *2^(hammingweight(k)-1),", ")) \\ Hugo Pfoertner, Jul 27 2020

a(0)=1, a(n) = (2^t(n) + 1) * 2^(c(n) - 1) where t(n) = A007814(n) is the number of trailing zeros in the binary representation of n and c(n) = A000120(n) is the total number of ones in the binary representation of n. - Johan Falk, Jun 24 2020

A268525 a(n) = ra(ceiling(n/2))+sa(floor(n/2)) with a(1)=1 and (r,s)=(2,3).

Original entry on oeis.org

1, 5, 13, 25, 41, 65, 89, 125, 157, 205, 253, 325, 373, 445, 517, 625, 689, 785, 881, 1025, 1121, 1265, 1409, 1625, 1721, 1865, 2009, 2225, 2369, 2585, 2801, 3125, 3253, 3445, 3637, 3925, 4117, 4405, 4693, 5125, 5317, 5605, 5893, 6325, 6613, 7045, 7477, 8125, 8317, 8605, 8893, 9325, 9613, 10045

Offset: 1

N. J. A. Sloane, Feb 16 2016

nonn

K.-N. Chang and S.-C. Tsai, Exact solution of a minimal recurrence, Inform. Process. Lett. 75 (2000), 61-64.

Sequences of form a(n) = r*a(ceiling(n/2))+s*a(floor(n/2)) with a(1)=1 and (r,s) = (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), (1,4), (2,3), (3,2), (4,1): A000027, A006046, A064194, A130665, A073121, A268524, A116520, A268525, A268526, A268527.

[n le 1 select 1 else 2*Self(Ceiling(n/2))+3*Self(Floor(n/2)): n in [1..60]]; // Vincenzo Librandi, Aug 30 2016

a(n) = if (n==1, 1, 2*a(ceil(n/2))+3*a(floor(n/2))); \\ Michel Marcus, Aug 30 2016

cp's OEIS Frontend

A006048 Number of entries in first n rows of Pascal's triangle not divisible by 3.

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A130667 a(1) = 1; a(n) = max{ 5*a(k) + a(n-k) | 1 <= k <= n/2 } for n > 1.

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A073121 a(n) = r*a(ceiling(n/2)) + s*a(floor(n/2)) with a(1)=1 and (r,s)=(2,2).

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A194458 Total number of entries in rows 0,1,...,n of Pascal's triangle not divisible by 5.

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A161342 Number of "ON" cubic cells at n-th stage in simple 3-dimensional cellular automaton: a(n) = A160428(n)/8.

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A268524 a(n) = r*a(ceiling(n/2))+s*a(floor(n/2)) with a(1)=1 and (r,s)=(3,1).

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A160722 Number of "ON" cells at n-th stage in a certain 2-dimensional cellular automaton based on Sierpinski triangles (see Comments for precise definition).

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A073121 a(n) = ra(ceiling(n/2)) + sa(floor(n/2)) with a(1)=1 and (r,s)=(2,2).

A268524 a(n) = ra(ceiling(n/2))+sa(floor(n/2)) with a(1)=1 and (r,s)=(3,1).

A268525 a(n) = ra(ceiling(n/2))+sa(floor(n/2)) with a(1)=1 and (r,s)=(2,3).