A247903 -id:A247903 - cp's OEIS frontend

A074909 Running sum of Pascal's triangle (A007318), or beheaded Pascal's triangle read by beheaded rows.

1, 1, 2, 1, 3, 3, 1, 4, 6, 4, 1, 5, 10, 10, 5, 1, 6, 15, 20, 15, 6, 1, 7, 21, 35, 35, 21, 7, 1, 8, 28, 56, 70, 56, 28, 8, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11

Offset: 0

Wouter Meeussen, Oct 01 2002

This sequence counts the "almost triangular" partitions of n. A partition is triangular if it is of the form 0+1+2+...+k. Examples: 3=0+1+2, 6=0+1+2+3. An "almost triangular" partition is a triangular partition with at most 1 added to each of the parts. Examples: 7 = 1+1+2+3 = 0+2+2+3 = 0+1+3+3 = 0+1+2+4. Thus a(7)=4. 8 = 1+2+2+3 = 1+1+3+3 = 1+1+2+4 = 0+2+3+3 = 0+2+2+4 = 0+1+3+4 so a(8)=6. - Moshe Shmuel Newman, Dec 19 2002
The "almost triangular" partitions are the ones cycled by the operation of "Bulgarian solitaire", as defined by Martin Gardner.
Start with A007318 - I (I = Identity matrix), then delete right border of zeros. - Gary W. Adamson, Jun 15 2007
Also the number of increasing acyclic functions from {1..n-k+1} to {1..n+2}. A function f is acyclic if for every subset B of the domain the image of B under f does not equal B. For example, T(3,1)=4 since there are exactly 4 increasing acyclic functions from {1,2,3} to {1,2,3,4,5}: f1={(1,2),(2,3),(3,4)}, f2={(1,2),(2,3),(3,5)}, f3={(1,2),(2,4),(3,5)} and f4={(1,3),(2,4),(4,5)}. - Dennis P. Walsh, Mar 14 2008
Second Bernoulli polynomials are (from A164555 instead of A027641) B2(n,x) = 1; 1/2, 1; 1/6, 1, 1; 0, 1/2, 3/2, 1; -1/30, 0, 1, 2, 1; 0, -1/6, 0, 5/3, 5/2, 1; ... . Then (B2(n,x)/A002260) = 1; 1/2, 1/2; 1/6, 1/2, 1/3; 0, 1/4, 1/2, 1/4; -1/30, 0, 1/3, 1/2, 1/5; 0, -1/12, 0, 5/12, 1/2, 1/6; ... . See (from Faulhaber 1631) Jacob Bernoulli Summae Potestatum (sum of powers) in A159688. Inverse polynomials are 1; -1, 2; 1, -3, 3; -1, 4, -6, 4; ... = A074909 with negative even diagonals. Reflected A053382/A053383 = reflected B(n,x) = RB(n,x) = 1; -1/2, 1; 1/6, -1, 1; 0, 1/2, -3/2, 1; ... . A074909 is inverse of RB(n,x)/A002260 = 1; -1/2, 1/2; 1/6, -1/2, 1/3; 0, 1/4, -1/2, 1/4; ... . - Paul Curtz, Jun 21 2010
A054143 is the fission of the polynomial sequence (p(n,x)) given by p(n,x) = x^n + x^(n-1) + ... + x + 1 by the polynomial sequence ((x+1)^n). See A193842 for the definition of fission. - Clark Kimberling, Aug 07 2011
Reversal of A135278. - Philippe Deléham, Feb 11 2012
For a closed-form formula for arbitrary left and right borders of Pascal-like triangles see A228196. - Boris Putievskiy, Aug 19 2013
For a closed-form formula for generalized Pascal's triangle see A228576. - Boris Putievskiy, Sep 09 2013
From A238363, the operator equation d/d(:xD:)f(xD)={exp[d/d(xD)]-1}f(xD) = f(xD+1)-f(xD) follows. Choosing f(x) = x^n and using :xD:^n/n! = binomial(xD,n) and (xD)^n = Bell(n,:xD:), the Bell polynomials of A008277, it follows that the lower triangular matrix [padded A074909]
  A) = [St2]*[dP]*[St1] = A048993*A132440*[padded A008275]
  B) = [St2]*[dP]*[St2]^(-1)
  C) = [St1]^(-1)*[dP]*[St1],
  where [St1]=padded A008275 just as [St2]=A048993=padded A008277 whereas [padded A074909]=A007318-I with I=identity matrix. - Tom Copeland, Apr 25 2014
T(n,k) generated by m-gon expansions in the case of odd m with "vertex to side" version or even m with "vertex to vertes" version. Refer to triangle expansions in A061777 and A101946 (and their companions for m-gons) which are "vertex to vertex" and "vertex to side" versions respectively. The label values at each iteration can be arranged as a triangle. Any m-gon can also be arranged as the same triangle with conditions: (i) m is odd and expansion is "vertex to side" version or (ii) m is even and expansion is "vertex to vertex" version. m*Sum_{i=1..k} T(n,k) gives the total label value at the n-th iteration. See also A247976. Vertex to vertex: A061777, A247618, A247619, A247620. Vertex to side: A101946, A247903, A247904, A247905. - Kival Ngaokrajang Sep 28 2014
From Tom Copeland, Nov 12 2014: (Start)
With P(n,x) = [(x+1)^(n+1)-x^(n+1)], the row polynomials of this entry, Up(n,x) = P(n,x)/(n+1) form an Appell sequence of polynomials that are the umbral compositional inverses of the Bernoulli polynomials B(n,x), i.e., B[n,Up(.,x)] = x^n = Up[n,B(.,x)] under umbral substitution, e.g., B(.,x)^n = B(n,x).
The e.g.f. for the Bernoulli polynomials is [t/(e^t - 1)] e^(x*t), and for Up(n,x) it's exp[Up(.,x)t] = [(e^t - 1)/t] e^(x*t).
Another g.f. is G(t,x) = log[(1-x*t)/(1-(1+x)*t)] = log[1 + t /(1 + -(1+x)t)] = t/(1-t*Up(.,x)) = Up(0,x)*t + Up(1,x)*t^2 + Up(2,x)*t^3 + ... = t + (1+2x)/2 t^2 + (1+3x+3x^2)/3 t^3 + (1+4x+6x^2+4x^3)/4 t^4 + ... = -log(1-t*P(.,x)), expressed umbrally.
The inverse, Ginv(t,x), in t of the g.f. may be found in A008292 from Copeland's list of formulas (Sep 2014) with a=(1+x) and b=x. This relates these two sets of polynomials to algebraic geometry, e.g., elliptic curves, trigonometric expansions, Chebyshev polynomials, and the combinatorics of permutahedra and their duals.
Ginv(t,x) = [e^((1+x)t) - e^(xt)] / [(1+x) * e^((1+x)t) - x * e^(xt)] = [e^(t/2) - e^(-t/2)] / [(1+x)e^(t/2) - x*e^(-t/2)] = (e^t - 1) / [1 + (1+x) (e^t - 1)] = t - (1 + 2 x) t^2/2! + (1 + 6 x + 6 x^2) t^3/3! - (1 + 14 x + 36 x^2 + 24 x^3) t^4/4! + ... = -exp[-Perm(.,x)t], where Perm(n,x) are the reverse face polynomials, or reverse f-vectors, for the permutahedra, i.e., the face polynomials for the duals of the permutahedra. Cf. A090582, A019538, A049019, A133314, A135278.
With L(t,x) = t/(1+t*x) with inverse L(t,-x) in t, and Cinv(t) = e^t - 1 with inverse C(t) = log(1 + t). Then Ginv(t,x) = L[Cinv(t),(1+x)] and G(t,x) = C[L[t,-(1+x)]]. Note L is the special linear fractional (Mobius) transformation.
Connections among the combinatorics of the permutahedra, simplices (cf. A135278), and the associahedra can be made through the Lagrange inversion formula (LIF) of A133437 applied to G(t,x) (cf. A111785 and the Schroeder paths A126216 also), and similarly for the LIF A134685 applied to Ginv(t,x) involving the simplicial Whitehouse complex, phylogenetic trees, and other structures. (See also the LIFs A145271 and A133932). (End)
R = x - exp[-[B(n+1)/(n+1)]D] = x - exp[zeta(-n)D] is the raising operator for this normalized sequence UP(n,x) = P(n,x) / (n+1), that is, R UP(n,x) = UP(n+1,x), where D = d/dx, zeta(-n) is the value of the Riemann zeta function evaluated at -n, and B(n) is the n-th Bernoulli number, or constant B(n,0) of the Bernoulli polynomials. The raising operator for the Bernoulli polynomials is then x + exp[-[B(n+1)/(n+1)]D]. [Note added Nov 25 2014: exp[zeta(-n)D] is abbreviation of exp(a.D) with (a.)^n = a_n = zeta(-n)]. - Tom Copeland, Nov 17 2014
The diagonals T(n, n-m), for n >= m, give the m-th iterated partial sum of the positive integers; that is A000027(n+1), A000217(n), A000292(n-1), A000332(n+1), A000389(n+1), A000579(n+1), A000580(n+1), A000581(n+1), A000582(n+1), ... . - Wolfdieter Lang, May 21 2015
The transpose gives the numerical coefficients of the Maurer-Cartan form matrix for the general linear group GL(n,1) (cf. Olver, but note that the formula at the bottom of p. 6 has an error--the 12 should be a 15). - Tom Copeland, Nov 05 2015
The left invariant Maurer-Cartan form polynomial on p. 7 of the Olver paper for the group GL^n(1) is essentially a binomial convolution of the row polynomials of this entry with those of A133314, or equivalently the row polynomials generated by the product of the e.g.f. of this entry with that of A133314, with some reindexing. - Tom Copeland, Jul 03 2018
From Tom Copeland, Jul 10 2018: (Start)
The first column of the inverse matrix is the sequence of Bernoulli numbers, which follows from the umbral definition of the Bernoulli polynomials (B.(0) + x)^n = B_n(x) evaluated at x = 1 and the relation B_n(0) = B_n(1) for n > 1 and -B_1(0) = 1/2 = B_1(1), so the Bernoulli numbers can be calculated using Cramer's rule acting on this entry's matrix and, therefore, from the ratios of volumes of parallelepipeds determined by the columns of this entry's square submatrices. - Tom Copeland, Jul 10 2018
Umbrally composing the row polynomials with B_n(x), the Bernoulli polynomials, gives (B.(x)+1)^(n+1) - (B.(x))^(n+1) = d[x^(n+1)]/dx = (n+1)*x^n, so multiplying this entry as a lower triangular matrix (LTM) by the LTM of the coefficients of the Bernoulli polynomials gives the diagonal matrix of the natural numbers. Then the inverse matrix of this entry has the elements B_(n,k)/(k+1), where B_(n,k) is the coefficient of x^k for B_n(x), and the e.g.f. (1/x) (e^(xt)-1)/(e^t-1). (End)

			T(4,2) = 0+0+1+3+6 = 10 = binomial(5, 2).
Triangle T(n,k) begins:
n\k 0  1  2   3   4   5   6   7   8   9 10 11
0:  1
1:  1  2
2:  1  3  3
3:  1  4  6   4
4:  1  5 10  10   5
5:  1  6 15  20  15   6
6:  1  7 21  35  35  21   7
7:  1  8 28  56  70  56  28   8
8:  1  9 36  84 126 126  84  36  9
9:  1 10 45 120 210 252 210 120 45   10
10: 1 11 55 165 330 462 462 330 165  55 11
11: 1 12 66 220 495 792 924 792 495 220 66 12
... Reformatted. - _Wolfdieter Lang_, Nov 04 2014
.
Can be seen as the square array A(n, k) = binomial(n + k + 1, n) read by descending antidiagonals. A(n, k) is the number of monotone nondecreasing functions f: {1,2,..,k} -> {1,2,..,n}. - _Peter Luschny_, Aug 25 2019
[0]  1,  1,   1,   1,    1,    1,     1,     1,     1, ... A000012
[1]  2,  3,   4,   5,    6,    7,     8,     9,    10, ... A000027
[2]  3,  6,  10,  15,   21,   28,    36,    45,    55, ... A000217
[3]  4, 10,  20,  35,   56,   84,   120,   165,   220, ... A000292
[4]  5, 15,  35,  70,  126,  210,   330,   495,   715, ... A000332
[5]  6, 21,  56, 126,  252,  462,   792,  1287,  2002, ... A000389
[6]  7, 28,  84, 210,  462,  924,  1716,  3003,  5005, ... A000579
[7]  8, 36, 120, 330,  792, 1716,  3432,  6435, 11440, ... A000580
[8]  9, 45, 165, 495, 1287, 3003,  6435, 12870, 24310, ... A000581
[9] 10, 55, 220, 715, 2002, 5005, 11440, 24310, 48620, ... A000582

Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened
Feryal Alayont and Evan Henning, Edge Covers of Caterpillars, Cycles with Pendants, and Spider Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.9.4.
Paul Barry, On the f-Matrices of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1805.02274 [math.CO], 2018.
Tom Copeland, Appell polynomials, cumulants, noncrossing partitions, Dyck lattice paths, and inversion, 2014.
Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, 2015.
Sela Fried, The expected degree of noninvertibility of compositions of functions and a related combinatorial identity, arXiv:2202.13061 [math.CO], 2022.
J. R. Griggs, The Cycling of Partitions and Compositions under Repeated Shifts, Advances in Applied Mathematics, Volume 21, Issue 2, August 1998, Pages 205-227.
P. Olver, The canonical contact form p. 7.
D. P. Walsh, A short note on increasing acyclic functions

Cf. A007318, A181971, A228196, A228576.
Row sums are A000225, diagonal sums are A052952.
The number of acyclic functions is A058127.
Cf. A008292, A090582, A019538, A049019, A133314, A135278, A133437, A111785, A126216, A134685, A133932, A248727, A033282, A134264.
Cf. A000027, A000217, A000292, A000332, A000389, A000579, A000580, A000581, A000582.
Cf. A325191.

Flat(List([0..10],n->List([0..n],k->Binomial(n+1,k)))); # Muniru A Asiru, Jul 10 2018

a074909 n k = a074909_tabl !! n !! k
a074909_row n = a074909_tabl !! n
a074909_tabl = iterate
   (\row -> zipWith (+) ([0] ++ row) (row ++ [1])) [1]
-- Reinhard Zumkeller, Feb 25 2012

/* As triangle */ [[Binomial(n+1,k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jul 22 2018

A074909 := proc(n,k)
    if k > n or k < 0 then
        0;
    else
        binomial(n+1,k) ;
    end if;
end proc: # Zerinvary Lajos, Nov 09 2006

Flatten[Join[{1}, Table[Sum[Binomial[k, m], {k, 0, n}], {n, 0, 12}, {m, 0, n}] ]] (* or *) Flatten[Join[{1}, Table[Binomial[n, m], {n, 12}, {m, n}]]]

print1(1);for(n=1,10,for(k=1,n,print1(", "binomial(n,k)))) \\ Charles R Greathouse IV, Mar 26 2013

from math import comb, isqrt
def A074909(n): return comb(r:=(m:=isqrt(k:=n+1<<1))+(k>m*(m+1)),n-comb(r,2)) # Chai Wah Wu, Nov 12 2024

T(n, k) = Sum_{i=0..n} C(i, n-k) = C(n+1, k).
Row n has g.f. (1+x)^(n+1)-x^(n+1).
E.g.f.: ((1+x)*e^t - x) e^(x*t). The row polynomials p_n(x) satisfy dp_n(x)/dx = (n+1)*p_(n-1)(x). - Tom Copeland, Jul 10 2018
T(n, k) = T(n-1, k-1) + T(n-1, k) for k: 0Reinhard Zumkeller, Apr 18 2005
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) - T(n-2,k-2), T(0,0)=1, T(1,0)=1, T(1,1)=2, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Dec 27 2013
G.f. for column k (with leading zeros): x^(k-1)*(1/(1-x)^(k+1)-1), k >= 0. - Wolfdieter Lang, Nov 04 2014
Up(n, x+y) = (Up(.,x)+ y)^n = Sum_{k=0..n} binomial(n,k) Up(k,x)*y^(n-k), where Up(n,x) = ((x+1)^(n+1)-x^(n+1)) / (n+1) = P(n,x)/(n+1) with P(n,x) the n-th row polynomial of this entry. dUp(n,x)/dx = n * Up(n-1,x) and dP(n,x)/dx = (n+1)*P(n-1,x). - Tom Copeland, Nov 14 2014
The o.g.f. GF(x,t) = x / ((1-t*x)*(1-(1+t)x)) = x + (1+2t)*x^2 + (1+3t+3t^2)*x^3 + ... has the inverse GFinv(x,t) = (1+(1+2t)x-sqrt(1+(1+2t)*2x+x^2))/(2t(1+t)x) in x about 0, which generates the row polynomials (mod row signs) of A033282. The reciprocal of the o.g.f., i.e., x/GF(x,t), gives the free cumulants (1, -(1+2t) , t(1+t) , 0, 0, ...) associated with the moments defined by GFinv, and, in fact, these free cumulants generate these moments through the noncrossing partitions of A134264. The associated e.g.f. and relations to Grassmannians are described in A248727, whose polynomials are the basis for an Appell sequence of polynomials that are umbral compositional inverses of the Appell sequence formed from this entry's polynomials (distinct from the one described in the comments above, without the normalizing reciprocal). - Tom Copeland, Jan 07 2015
T(n, k) = (1/k!) * Sum_{i=0..k} Stirling1(k,i)*(n+1)^i, for 0<=k<=n. - Ridouane Oudra, Oct 23 2022

I added an initial 1 at the suggestion of Paul Barry, which makes the triangle a little nicer but may mean that some of the formulas will now need adjusting. - N. J. A. Sloane, Feb 11 2003

Formula section edited, checked and corrected by Wolfdieter Lang, Nov 04 2014

A248969 Start with a single equilateral triangle; at odd n-th generation add a similar triangle at each expandable vertex (this is the "vertex to vertex" version); alternate with the "side to vertex" version for even n-th generation; a(n) is the number of triangle for each generation.

Original entry on oeis.org

1, 3, 6, 15, 18, 42, 24, 57, 30, 72, 36, 87, 48, 114, 54, 129, 60, 144, 66, 159, 78, 186, 84, 201, 90, 216, 96, 231, 108, 258, 114, 273, 120, 288, 126, 303, 138, 330, 144, 345, 150, 360, 156, 375, 168, 402, 174, 417, 180, 432, 186, 447, 198, 474, 204, 489, 210, 504, 216, 519

Offset: 0

Kival Ngaokrajang, Oct 18 2014

nonn

The construction rules alternate between "vertex to vertex" (A061777 & companions) and "side to vertex" (A101946 & companions). See the link for an illustration.

Kival Ngaokrajang, Illustration of initial terms

Vertex to vertex: A061777, A247618, A247619, A247620.

Side to side: A101946, A247903, A247904, A247905.

{
c2=0;c3=0;c6=3;c7=1;c8=0;
for(n=0,100,
   if (Mod(n,2)==0,
   \\even
       if (n<1,a(n)=1,c3=c3+c2;a=6*c3);
       c1=n/8+3/4;
       if (c1==floor(c1),c2=2,c2=1)
   ,
   \\odd
       c4=(n^2-1)/16;
       if (c4==floor(c4),c5=-1,c5=1);
       if (n>4, c6=c6+c5);
       if (n>=2, c7=c7+c6);
       if (c6<>4, c9=0,c9=2);
       a=3*(c7+c8+c9);
       c8=c7
   );
   print1(a", ")
)
}

Empirical g.f.: (3*x^11 +x^10 +12*x^9 +5*x^8 +15*x^7 +6*x^6 +27*x^5 +12*x^4 +12*x^3 +5*x^2 +3*x +1) / ((x -1)^2*(x +1)^2*(x^2 +1)*(x^4 +1)). - Colin Barker, Oct 18 2014

Edited. Small changes in the text. - Wolfdieter Lang, Nov 10 2014

A249246 Start with a single equilateral triangle for n=0; for the odd n-th generation add a triangle at each expandable side of the triangles of the (n-1)-th generation (this is the "vertex to side" version); for the even n-th generation use the "vertex to vertex" version; a(n) is the number of triangles in the n-th generation.

Original entry on oeis.org

1, 3, 6, 15, 18, 30, 24, 45, 30, 60, 36, 75, 48, 90, 54, 105, 60, 120, 66, 135, 78, 150, 84, 165, 90, 180, 96, 195, 108, 210, 114, 225, 120, 240, 126, 255, 138, 270, 144, 285, 150, 300, 156, 315, 168, 330, 174, 345, 180, 360, 186, 375, 198, 390, 204, 405, 210, 420, 216, 435

Offset: 0

Kival Ngaokrajang, Oct 23 2014

nonn

The construction rules alternate between "vertex to side" (A101946 & companions) and "vertex to vertex" (A061777 & companions). 'Vertex to side' means vertex of n-th generation triangle touches the middle of a side of the (n-1)-th generation triangle. See the link with an illustration. The even terms are the same as in A248969. Note that the triangles overlap.

Kival Ngaokrajang, Illustration of initial terms

Vertex to vertex: A061777, A247618, A247619, A247620.
Vertex to side: A101946, A247903, A247904, A247905.
Cf. A248969.

{
c2=0;c3=0;c5=3;
for(n=0,100,
   if (Mod(n,2)==0,
   \\even
       if (n<1,a(n)=1,c3=c3+c2;a=6*c3);
       c1=n/8+3/4;
       if (c1==floor(c1),c2=2,c2=1)
   ,
   \\odd
       a=c5;
       if (n<=1,c4=12,c4=15);
       c5=c5+c4
   );
   print1(a", ")
)
}

Empirical g.f.: (3*x^11 + x^10 + 12*x^9 + 5*x^8 + 15*x^7 + 6*x^6 + 15*x^5 + 12*x^4 + 12*x^3 + 5*x^2 + 3*x + 1) / ((x-1)^2*(x+1)^2*(x^2+1)*(x^4+1)). - Colin Barker, Oct 24 2014

Edited. Name and comment reformulated. - Wolfdieter Lang, Nov 04 2014

A247904 Start with a single pentagon; at n-th generation add a pentagon at each expandable vertex (this is the "vertex to side" version); a(n) is the sum of all label values at n-th generation. (See comment for construction rules.)

Original entry on oeis.org

1, 6, 21, 56, 131, 286, 601, 1236, 2511, 5066, 10181, 20416, 40891, 81846, 163761, 327596, 655271, 1310626, 2621341, 5242776, 10485651, 20971406, 41942921, 83885956, 167772031, 335544186, 671088501, 1342177136, 2684354411, 5368708966, 10737418081

Offset: 0

Kival Ngaokrajang, Sep 26 2014

Refer to A247619, which is the "vertex to vertex" expansion version. For this case, the expandable vertices of the existing generation will contact the sides of the new ones i.e."vertex to side" expansion version. Let us assign the label "1" to the pentagon at the origin; at n-th generation add a pentagon at each expandable vertex, i.e. each vertex where the added generations will not overlap the existing ones, although overlaps among new generations are allowed. The non-overlapping pentagons will have the same label value as a predecessor; for the overlapping ones, the label value will be sum of label values of predecessors. a(n) is the sum of all label values at n-th generation. The pentagons count is A005891. See illustration. For n >= 1, (a(n) - a(n-1))/5 is A000225.

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

Cf. Vertex to vertex version: A061777, A247618, A247619, A247620.
Cf. Vertex to side version: A101946, A247903, A247905.
Cf. A000225, A005891.

[10*2^n -(5*n+9): n in [0..50]]; // G. C. Greubel, Feb 18 2022

LinearRecurrence[{4,-5,2}, {1,6,21}, 51] (* G. C. Greubel, Feb 18 2022 *)

a(n) = if (n<1, 1, 5*(2^n-1)+a(n-1))
for (n=0, 50, print1(a(n), ", "))

Vec(-(2*x^2+2*x+1)/((x-1)^2*(2*x-1)) + O(x^100)) \\ Colin Barker, Sep 26 2014

[5*2^(n+1) -(5*n+9) for n in (0..50)] # G. C. Greubel, Feb 18 2022

a(0) = 1, for n >= 1, a(n) = 5*A000225(n) + a(n-1).
a(n) = 4*a(n-1)-5*a(n-2)+2*a(n-3). - Colin Barker, Sep 26 2014
G.f.: (1+2*x+2*x^2)/((1-x)^2*(1-2*x)). - Colin Barker, Sep 26 2014
From G. C. Greubel, Feb 18 2022: (Start)
a(n) = 10*2^n - (5*n + 9).
E.g.f.: 10*exp(2*x) - (9 + 5*x)*exp(x). (End)

More terms from Colin Barker, Sep 26 2014

A247905 Start with a single hexagon; at n-th generation add a hexagon at each expandable vertex (this is the "vertex to side" version); a(n) is the sum of all label values at n-th generation. (See comment for construction rules.)

Original entry on oeis.org

1, 7, 19, 43, 79, 139, 223, 355, 535, 811, 1183, 1747, 2503, 3643, 5167, 7459, 10519, 15115, 21247, 30451, 42727, 61147, 85711, 122563, 171703, 245419, 343711, 491155, 687751, 982651, 1375855, 1965667, 2752087, 3931723, 5504575, 7863859, 11009575, 15728155, 22019599, 31456771, 44039671, 62914027, 88079839, 125828563, 176160199, 251657659

Offset: 0

Kival Ngaokrajang, Sep 26 2014

Refer to A247620, which is the "vertex to vertex" expansion version. For this case, the expandable vertices of the existing generation will contact the sides of the new ones i.e. "vertex to side" expansion version. Let us assign the label "1" the hexagon at the origin; at n-th generation add a hexagon at each expandable vertex, i.e. each vertex where the added generations will not overlap the existing ones, although overlaps among new generations are allowed. The non-overlapping hexagons will have the same label value as a predecessor; for the overlapping ones, the label value will be sum of label values of predecessors. a(n) is the sum of all label values at n-th generation. The hexagons count is A003215. See illustration. For n >= 1, (a(n) - a(n-1))/6 is A027383.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Kival Ngaokrajang, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,2).

Cf. Vertex to vertex version: A061777, A247618, A247619, A247620.
Cf. Vertex to side version: A101946, A247903, A247904.
Cf. A003215, A027383.

[3*2^(n/2)*((7+5*Sqrt(2)) + (-1)^n*(7-5*Sqrt(2))) -(12*n+41): n in [0..50]]; // G. C. Greubel, Feb 17 2022

LinearRecurrence[{2,1,-4,2}, {1,7,19,43}, 50] (* G. C. Greubel, Feb 17 2022 *)

{
b=0; a=1; print1(1, ", ");
for (n=0, 50,
     b=b+2^floor(n/2);
     a=a+6*b;
     print1(a, ", ")
    )
}

Vec(-(2*x^3+4*x^2+5*x+1)/((x-1)^2*(2*x^2-1)) + O(x^100)) \\ Colin Barker, Sep 26 2014

[3*2^(n/2)*((7+5*sqrt(2)) + (-1)^n*(7-5*sqrt(2))) -(12*n+41) for n in (0..50)] # G. C. Greubel, Feb 17 2022

a(0) = 1, for n >= 1, a(n) = 6*A027383(n) + a(n-1).
a(n) = 2*a(n-1) +a(n-2) -4*a(n-3) +2*a(n-4). - Colin Barker, Sep 26 2014
G.f.: (1+5*x+4*x^2+2*x^3)/((1-x)^2*(1-2*x^2)). - Colin Barker, Sep 26 2014
a(n) = 3*2^(n/2)*((1+sqrt(2))^3 + (-1)^n*(1-sqrt(2))^3) -12*n - 41. - G. C. Greubel, Feb 18 2022

cp's OEIS Frontend

A074909 Running sum of Pascal's triangle (A007318), or beheaded Pascal's triangle read by beheaded rows.

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A248969 Start with a single equilateral triangle; at odd n-th generation add a similar triangle at each expandable vertex (this is the "vertex to vertex" version); alternate with the "side to vertex" version for even n-th generation; a(n) is the number of triangle for each generation.

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A247904 Start with a single pentagon; at n-th generation add a pentagon at each expandable vertex (this is the "vertex to side" version); a(n) is the sum of all label values at n-th generation. (See comment for construction rules.)

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A247905 Start with a single hexagon; at n-th generation add a hexagon at each expandable vertex (this is the "vertex to side" version); a(n) is the sum of all label values at n-th generation. (See comment for construction rules.)

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A247976 Triangle read by rows: T(n,k) generated by m-gon expansions in the case of odd m with "vertex to vertex" version or even m with "vertex to side" version. (See comment for details.)

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