A096940 -id:A096940 - cp's OEIS frontend

A096941 Fourth column of (1,5)-Pascal triangle A096940.

5, 16, 34, 60, 95, 140, 196, 264, 345, 440, 550, 676, 819, 980, 1160, 1360, 1581, 1824, 2090, 2380, 2695, 3036, 3404, 3800, 4225, 4680, 5166, 5684, 6235, 6820, 7440, 8096, 8789, 9520, 10290, 11100, 11951, 12844, 13780, 14760, 15785, 16856, 17974, 19140

Offset: 0

Wolfdieter Lang, Jul 16 2004

If Y is a 5-subset of an n-set X then, for n>=7, a(n-7) is the number of 3-subsets of X having at most one element in common with Y. - Milan Janjic, Dec 08 2007

Third column: A056000; fifth column: A096942.

Table[(n^3 + 15 n^2 + 14 n)/6, {n, 100}] (* Vladimir Joseph Stephan Orlovsky, Jul 06 2011 *)

a(n)= (n+15)*(n+2)*(n+1)/6 = 5*b(n)-4*b(n-1), with b(n):=A000292(n)=binomial(n+3, 3).

G.f.: (5-4*x)/(1-x)^4.

A096942 Fifth column of (1,5)-Pascal triangle A096940.

Original entry on oeis.org

5, 21, 55, 115, 210, 350, 546, 810, 1155, 1595, 2145, 2821, 3640, 4620, 5780, 7140, 8721, 10545, 12635, 15015, 17710, 20746, 24150, 27950, 32175, 36855, 42021, 47705, 53940, 60760, 68200, 76296, 85085, 94605, 104895, 115995, 127946, 140790, 154570, 169330, 185115

Offset: 0

Wolfdieter Lang, Jul 16 2004

If Y is a 5-subset of an n-set X then, for n>=8, a(n-8) is the number of 4-subsets of X having at most one element in common with Y. - Milan Janjic, Dec 08 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Fourth column: A096941; sixth column: A096943.

[(n + 20)*Binomial(n + 3, 3) div 4: n in [0..50]]; // Vincenzo Librandi, Oct 01 2013

Table[(n + 20) Binomial[n + 3, 3]/4, {n, 0, 100}]
CoefficientList[Series[(5 - 4 x)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Oct 01 2013 *)

a(n) = (n+20)*binomial(n+3, 3)/4 = 5*b(n) - 4*b(n-1), with b(n) = A000332(n+4) = binomial(n+4, 4).
G.f.: (5-4*x)/(1-x)^5.
a(n) = Sum_{k=1..n} (Sum_{i=1..k} i*(n-k+5)). - Wesley Ivan Hurt, Sep 26 2013

A096943 Sixth column of (1,5)-Pascal triangle A096940.

Original entry on oeis.org

5, 26, 81, 196, 406, 756, 1302, 2112, 3267, 4862, 7007, 9828, 13468, 18088, 23868, 31008, 39729, 50274, 62909, 77924, 95634, 116380, 140530, 168480, 200655, 237510, 279531, 327236, 381176, 441936, 510136, 586432, 671517, 766122, 871017, 987012

Offset: 0

Wolfdieter Lang, Jul 16 2004

If Y is a 5-subset of an n-set X then, for n>=9, a(n-9) is the number of 5-subsets of X having at most one element in common with Y. - Milan Janjic, Dec 08 2007

Index entries for linear recurrences with constant coefficients, signature (6, -15, 20, -15, 6, -1).

Fifth column: A096942; seventh column: A096944.

CoefficientList[Series[(5-4x)/(1-x)^6,{x,0,40}],x] (* or *) LinearRecurrence[ {6,-15,20,-15,6,-1},{5,26,81,196,406,756},40] (* Harvey P. Dale, Jan 11 2014 *)

G.f.: (5-4*x)/(1-x)^6.
a(n)= (n+25)*binomial(n+4, 4)/5 = 5*b(n)-4*b(n-1), with b(n):= A000389(n+5)=binomial(n+5, 5).
a(0)=5, a(1)=26, a(2)=81, a(3)=196, a(4)=406, a(5)=756, a(n)=6*a(n-1)- 15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6). - Harvey P. Dale, Jan 11 2014

A096944 Seventh column of (1,5)-Pascal triangle A096940.

Original entry on oeis.org

5, 31, 112, 308, 714, 1470, 2772, 4884, 8151, 13013, 20020, 29848, 43316, 61404, 85272, 116280, 156009, 206283, 269192, 347116, 442750, 559130, 699660, 868140, 1068795, 1306305, 1585836, 1913072, 2294248, 2736184, 3246320, 3832752

Offset: 0

Wolfdieter Lang, Jul 16 2004

If Y is a 5-subset of an n-set X then, for n>=10, a(n-10) is the number of 6-subsets of X having at most one element in common with Y. > - Milan Janjic, Dec 08 2007

Sixth column: A096943; eighth column: A096945.

G.f.: (5-4*x)/(1-x)^7.

a(n)= (n+30)*binomial(n+5, 5)/6 = 5*b(n)-4*b(n-1), with b(n):=A000579(n+6)=binomial(n+6, 6).

A096945 Eighth column of (1,5)-Pascal triangle A096940.

Original entry on oeis.org

5, 36, 148, 456, 1170, 2640, 5412, 10296, 18447, 31460, 51480, 81328, 124644, 186048, 271320, 387600, 543609, 749892, 1019084, 1366200, 1808950, 2368080, 3067740, 3935880, 5004675, 6310980, 7896816, 9809888, 12104136, 14840320

Offset: 0

Wolfdieter Lang, Jul 16 2004

If Y is a 5-subset of an n-set X then, for n>=11, a(n-11) is the number of 7-subsets of X having at most one element in common with Y. > - Milan Janjic, Dec 08 2007

Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

Seventh column: A096944; ninth column: A096946.

CoefficientList[Series[(5-4*x)/(1-x)^8,{x,0,30}],x] (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{5,36,148,456,1170,2640,5412,10296},30] (* Harvey P. Dale, Aug 16 2014 *)

G.f.: (5-4*x)/(1-x)^8.
a(n)= (n+35)*binomial(n+6, 6)/7 = 5*b(n)-4*b(n-1), with b(n):=A000580(n+7)=binomial(n+7, 7).
a(0)=5, a(1)=36, a(2)=148, a(3)=456, a(4)=1170, a(5)=2640, a(6)=5412, a(7)=10296, a(n)=8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)- 28*a(n-6)+ 8*a(n-7)-a(n-8). - Harvey P. Dale, Aug 16 2014

A096946 Ninth column of (1,5)-Pascal triangle A096940.

Original entry on oeis.org

5, 41, 189, 645, 1815, 4455, 9867, 20163, 38610, 70070, 121550, 202878, 327522, 513570, 784890, 1172490, 1716099, 2465991, 3485075, 4851275, 6660225, 9028305, 12096045, 16031925, 21036600, 27347580, 35244396, 45054284, 57158420

Offset: 0

Wolfdieter Lang, Jul 16 2004

If Y is a 5-subset of an n-set X then, for n>=12, a(n-12) is the number of 8-subsets of X having at most one element in common with Y. > - Milan Janjic, Dec 08 2007

Eighth column: A096945; tenth column: A096947.

a(n)= (n+40)*binomial(n+7, 7)/8 = 5*b(n)-4*b(n-1), with b(n):=A000581(n+8)=binomial(n+8, 8).

G.f.: (5-4*x)/(1-x)^9.

A096947 Tenth column of (1,5)-Pascal triangle A096940.

Original entry on oeis.org

5, 46, 235, 880, 2695, 7150, 17017, 37180, 75790, 145860, 267410, 470288, 797810, 1311380, 2096270, 3268760, 4984859, 7450850, 10935925, 15787200, 22447425, 31475730, 43571775, 59603700, 80640300, 107987880, 143232276, 188286560

Offset: 0

Wolfdieter Lang, Jul 16 2004

If Y is a 5-subset of an n-set X then, for n>=13, a(n-13) is the number of 9-subsets of X having at most one element in common with Y. - Milan Janjic, Dec 08 2007

Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Ninth column: A096946.

CoefficientList[Series[(5-4x)/(1-x)^10,{x,0,40}],x] (* Harvey P. Dale, Jan 06 2020 *)

a(n) = (n+45)*binomial(n+8, 8)/9.
a(n) = 5*b(n)-4*b(n-1), with b(n) = A000582(n+9) = binomial(n+9, 9).
G.f.: (5-4*x)/(1-x)^10.

A007283 a(n) = 3*2^n.

Original entry on oeis.org

3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472, 6442450944, 12884901888

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(3, 6), L(3, 6), P(3, 6), T(3, 6). See A008776 for definitions of Pisot sequences.
Numbers k such that A006530(A000010(k)) = A000010(A006530(k)) = 2. - Labos Elemer, May 07 2002
Also least number m such that 2^n is the smallest proper divisor of m which is also a suffix of m in binary representation, see A080940. - Reinhard Zumkeller, Feb 25 2003
Length of the period of the sequence Fibonacci(k) (mod 2^(n+1)). - Benoit Cloitre, Mar 12 2003
The sequence of first differences is this sequence itself. - Alexandre Wajnberg and Eric Angelini, Sep 07 2005
Subsequence of A122132. - Reinhard Zumkeller, Aug 21 2006
Apart from the first term, a subsequence of A124509. - Reinhard Zumkeller, Nov 04 2006
Total number of Latin n-dimensional hypercubes (Latin polyhedra) of order 3. - Kenji Ohkuma (k-ookuma(AT)ipa.go.jp), Jan 10 2007
Number of different ternary hypercubes of dimension n. - Edwin Soedarmadji (edwin(AT)systems.caltech.edu), Dec 10 2005
For n >= 1, a(n) is equal to the number of functions f:{1, 2, ..., n + 1} -> {1, 2, 3} such that for fixed, different x_1, x_2,...,x_n in {1, 2, ..., n + 1} and fixed y_1, y_2,...,y_n in {1, 2, 3} we have f(x_i) <> y_i, (i = 1,2,...,n). - Milan Janjic, May 10 2007
a(n) written in base 2: 11, 110, 11000, 110000, ..., i.e.: 2 times 1, n times 0 (see A003953). - Jaroslav Krizek, Aug 17 2009
Subsequence of A051916. - Reinhard Zumkeller, Mar 20 2010
Numbers containing the number 3 in their Collatz trajectories. - Reinhard Zumkeller, Feb 20 2012
a(n-1) gives the number of ternary numbers with n digits with no two adjacent digits in common; e.g., for n=3 we have 010, 012, 020, 021, 101, 102, 120, 121, 201, 202, 210 and 212. - Jon Perry, Oct 10 2012
If n > 1, then a(n) is a solution for the equation sigma(x) + phi(x) = 3x-4. This equation also has solutions 84, 3348, 1450092, ...  which are not of the form 3*2^n. - Farideh Firoozbakht, Nov 30 2013
a(n) is the upper bound for the "X-ray number" of any convex body in E^(n + 2), conjectured by Bezdek and Zamfirescu, and proved in the plane E^2 (see the paper by Bezdek and Zamfirescu). - L. Edson Jeffery, Jan 11 2014
If T is a topology on a set V of size n and T is not the discrete topology, then T has at most 3 * 2^(n-2) many open sets. See Brown and Stephen references. - Ross La Haye, Jan 19 2014
Comment from Charles Fefferman, courtesy of Doron Zeilberger, Dec 02 2014: (Start)
Fix a dimension n. For a real-valued function f defined on a finite set E in R^n, let Norm(f, E) denote the inf of the C^2 norms of all functions F on R^n that agree with f on E. Then there exist constants k and C depending only on the dimension n such that Norm(f, E) <= C*max{ Norm(f, S) }, where the max is taken over all k-point subsets S in E.  Moreover, the best possible k is 3 * 2^(n-1).
The analogous result, with the same k, holds when the C^2 norm is replaced, e.g., by the C^1, alpha norm (0 < alpha <= 1). However, the optimal analogous k, e.g., for the C^3 norm is unknown.
For the above results, see Y. Brudnyi and P. Shvartsman (1994). (End)
Also, coordination sequence for (infinity, infinity, infinity) tiling of hyperbolic plane. - N. J. A. Sloane, Dec 29 2015
The average of consecutive powers of 2 beginning with 2^1. - Melvin Peralta and Miriam Ong Ante, May 14 2016
For n > 1, a(n) is the smallest Zumkeller number with n divisors that are also Zumkeller numbers (A083207). - Ivan N. Ianakiev, Dec 09 2016
Also, for n >= 2, the number of length-n strings over the alphabet {0,1,2,3} having only the single letters as nonempty palindromic subwords. (Corollary 21 in Fleischer and Shallit) - Jeffrey Shallit, Dec 02 2019
Also, a(n) is the minimum link-length of any covering trail, circuit, path, and cycle for the set of the 2^(n+2) vertices of an (n+2)-dimensional hypercube. - Marco Ripà, Aug 22 2022
The known fixed points of maps n -> A163511(n) and n -> A243071(n). [See comments in A163511]. - Antti Karttunen, Sep 06 2023
The finite subsequence a(3), a(4), a(5), a(6) = 24, 48, 96, 192 is one of only two geometric sequences that can be formed with all interior angles (all integer, in degrees) of a simple polygon. The other sequence is a subsequence of A000244 (see comment there). - Felix Huber, Feb 15 2024
A level 1 Sierpiński triangle is a triangle. Level n+1 is formed from three copies of level n by identifying pairs of corner vertices of each pair of triangles.  For n>2, a(n-3) is the radius of the level n Sierpiński triangle graph. - Allan Bickle, Aug 03 2024

Jason I. Brown, Discrete Structures and Their Interactions, CRC Press, 2013, p. 71.
T. Ito, Method, equipment, program and storage media for producing tables, Publication number JP2004-272104A, Japan Patent Office (written in Japanese, a(2)=12, a(3)=24, a(4)=48, a(5)=96, a(6)=192, a(7)=384 (a(7)=284 was corrected)).
Kenji Ohkuma, Atsuhiro Yamagishi and Toru Ito, Cryptography Research Group Technical report, IT Security Center, Information-Technology Promotion Agency, JAPAN.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
K. Bezdek and Tudor Zamfirescu, A Characterization of 3-dimensional Convex Sets with an Infinite X-ray Number, in: Coll. Math. Soc. J. Bolyai 63., Intuitive Geometry, Szeged (Hungary), North-Holland, Amsterdam, 1991, pp. 33-38.
Allan Bickle, Properties of Sierpinski Triangle Graphs, Springer PROMS 448 (2021) 295-303.
Yuri Brudnyi and Pavel Shvartsman, Generalizations of Whitney's extension theorem, International Mathematics Research Notices 1994.3 (1994): 129-139.
J. W. Cannon and P. Wagreich, Growth functions of surface groups, Mathematische Annalen, 1992, Volume 293, pp. 239-257. See Prop. 3.1.
Tomislav Došlić, Kepler-Bouwkamp Radius of Combinatorial Sequences, Journal of Integer Sequences, Vol. 17, 2014, #14.11.3.
John Elias, Illustration: 2^n+1 hexagram perimeters
Lukas Fleischer and Jeffrey Shallit, Words With Few Palindromes, Revisited, arxiv preprint arXiv:1911.12464 [cs.FL], November 27 2019.
A. Hinz, S. Klavzar, and S. Zemljic, A survey and classification of Sierpinski-type graphs, Discrete Applied Mathematics 217 3 (2017), 565-600.
Tanya Khovanova, Recursive Sequences
Roberto Rinaldi and Marco Ripà, Optimal cycles enclosing all the nodes of a k-dimensional hypercube, arXiv:2212.11216 [math.CO], 2022.
Edwin Soedarmadji, Latin Hypercubes and MDS Codes, Discrete Mathematics, Volume 306, Issue 12, Jun 28 2006, Pages 1232-1239
D. Stephen, Topology on Finite Sets, American Mathematical Monthly, 75: 739 - 741, 1968.
Index entries for linear recurrences with constant coefficients, signature (2).

Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.
Subsequence of the following sequences: A029744, A029747, A029748, A029750, A362804 (after 3), A364494, A364496, A364289, A364291, A364292, A364295, A364497, A364964, A365422.
Essentially same as A003945 and A042950.
Row sums of (5, 1)-Pascal triangle A093562 and of (1, 5) Pascal triangle A096940.
Cf. A000079, A100540, A124508, A221718.
Cf. Latin squares: A000315, A002860, A003090, A040082, A003191; Latin cubes: A098843, A098846, A098679, A099321.
Cf. A133466, A225546, A163511, A243071.
Cf. A007283, A029858, A067771, A193277, A233774, A233775, A246959.

a007283 = (* 3) . (2 ^)
a007283_list = iterate (* 2) 3
-- Reinhard Zumkeller, Mar 18 2012, Feb 20 2012

[3*2^n: n in [0..30]]; // Vincenzo Librandi, May 18 2011

A007283:=n->3*2^n; seq(A007283(n), n=0..50); # Wesley Ivan Hurt, Dec 03 2013

Table[3(2^n), {n, 0, 32}] (* Alonso del Arte, Mar 24 2011 *)

A007283(n):=3*2^n$
makelist(A007283(n),n,0,30); /* Martin Ettl, Nov 11 2012 */

PARI
```
a(n)=3*2^n
```

a(n)=3<Charles R Greathouse IV, Oct 10 2012

def A007283(n): return 3<Chai Wah Wu, Feb 14 2023

(List.fill(40)(2: BigInt)).scanLeft(1: BigInt)( * ).map(3 * ) // _Alonso del Arte, Nov 28 2019

G.f.: 3/(1-2*x).
a(n) = 2*a(n - 1), n > 0; a(0) = 3.
a(n) = Sum_{k = 0..n} (-1)^(k reduced (mod 3))*binomial(n, k). - Benoit Cloitre, Aug 20 2002
a(n) = A118416(n + 1, 2) for n > 1. - Reinhard Zumkeller, Apr 27 2006
a(n) = A000079(n) + A000079(n + 1). - Zerinvary Lajos, May 12 2007
a(n) = A000079(n)*3. - Omar E. Pol, Dec 16 2008
From Paul Curtz, Feb 05 2009: (Start)
a(n) = b(n) + b(n+3) for b = A001045, A078008, A154879.
a(n) = abs(b(n) - b(n+3)) with b(n) = (-1)^n*A084247(n). (End)
a(n) = 2^n + 2^(n + 1). - Jaroslav Krizek, Aug 17 2009
a(n) = A173786(n + 1, n) = A173787(n + 2, n). - Reinhard Zumkeller, Feb 28 2010
A216022(a(n)) = 6 and A216059(a(n)) = 7, for n > 0. - Reinhard Zumkeller, Sep 01 2012
a(n) = (A000225(n) + 1)*3. - Martin Ettl, Nov 11 2012
E.g.f.: 3*exp(2*x). - Ilya Gutkovskiy, May 15 2016
A020651(a(n)) = 2. - Yosu Yurramendi, Jun 01 2016
a(n) = sqrt(A014551(n + 1)*A014551(n + 2) + A014551(n)^2). - Ezhilarasu Velayutham, Sep 01 2019
a(A048672(n)) = A225546(A133466(n)). - Michel Marcus and Peter Munn, Nov 29 2019
Sum_{n>=1} 1/a(n) = 2/3. - Amiram Eldar, Oct 28 2020

A228196 A triangle formed like Pascal's triangle, but with n^2 on the left border and 2^n on the right border instead of 1.

Original entry on oeis.org

0, 1, 2, 4, 3, 4, 9, 7, 7, 8, 16, 16, 14, 15, 16, 25, 32, 30, 29, 31, 32, 36, 57, 62, 59, 60, 63, 64, 49, 93, 119, 121, 119, 123, 127, 128, 64, 142, 212, 240, 240, 242, 250, 255, 256, 81, 206, 354, 452, 480, 482, 492, 505, 511, 512, 100, 287, 560, 806, 932, 962, 974, 997, 1016, 1023, 1024

Offset: 1

Boris Putievskiy, Aug 15 2013

The third row is (n^4 - n^2 + 24*n + 24)/12.

For a closed-form formula for generalized Pascal's triangle see A228576. - Boris Putievskiy, Sep 04 2013

			The start of the sequence as a triangular array read by rows:
   0;
   1,  2;
   4,  3,  4;
   9,  7,  7,  8;
  16, 16, 14, 15, 16;
  25, 32, 30, 29, 31, 32;
  36, 57, 62, 59, 60, 63, 64;

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Rely Pellicer and David Alvo, Modified Pascal Triangle and Pascal Surfaces p.4
Index entries for triangles and arrays related to Pascal's triangle

Cf. We denote Pascal-like triangle with L(n) on the left border and R(n) on the right border by (L(n),R(n)). A007318 (1,1), A008949 (1,2^n), A029600 (2,3), A029618 (3,2), A029635 (1,2), A029653 (2,1), A037027 (Fibonacci(n),1), A051601 (n,n) n>=0, A051597 (n,n) n>0, A051666 (n^2,n^2), A071919 (1,0), A074829 (Fibonacci(n), Fibonacci(n)), A074909 (1,n), A093560 (3,1), A093561 (4,1), A093562 (5,1), A093563 (6,1), A093564 (7,1), A093565 (8,1), A093644 (9,1), A093645 (10,1), A095660 (1,3), A095666 (1,4), A096940 (1,5), A096956 (1,6), A106516 (3^n,1), A108561(1,(-1)^n), A132200 (4,4), A134636 (2n+1,2n+1), A137688 (2^n,2^n), A160760 (3^(n-1),1), A164844(1,10^n), A164847 (100^n,1), A164855 (101*100^n,1), A164866 (101^n,1), A172171 (1,9), A172185 (9,11), A172283 (-9,11), A177954 (int(n/2),1), A193820 (1,2^n), A214292 (n,-n), A227074 (4^n,4^n), A227075 (3^n,3^n), A227076 (5^n,5^n), A227550 (n!,n!), A228053 ((-1)^n,(-1)^n), A228074 (Fibonacci(n), n).

Cf. A000290 (row 1), A153056 (row 2), A000079 (column 1), A000225 (column 2), A132753 (column 3), A118885 (row sums of triangle array + 1), A228576 (generalized Pascal's triangle).

T:= function(n,k)
    if k=0 then return n^2;
    elif k=n then return 2^n;
    else return T(n-1,k-1) + T(n-1,k);
    fi;
  end;
Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Nov 12 2019

T:= proc(n, k) option remember;
      if k=0 then n^2
    elif k=n then 2^k
    else T(n-1, k-1) + T(n-1, k)
      fi
    end:
seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Nov 12 2019

T[n_, k_]:= T[n, k] = If[k==0, n^2, If[k==n, 2^k, T[n-1, k-1] + T[n-1, k]]]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 12 2019 *)
Flatten[Table[Sum[i^2 Binomial[n-1-i, n-k-i], {i,1,n-k}] + Sum[2^i Binomial[n-1-i, k-i], {i,1,k}], {n,0,10}, {k,0,n}]] (* Greg Dresden, Aug 06 2022 *)

T(n,k) = if(k==0, n^2, if(k==n, 2^k, T(n-1, k-1) + T(n-1, k) )); \\ G. C. Greubel, Nov 12 2019

def funcL(n):
   q = n**2
   return q
def funcR(n):
   q = 2**n
   return q
for n in range (1,9871):
   t=int((math.sqrt(8*n-7) - 1)/ 2)
   i=n-t*(t+1)/2-1
   j=(t*t+3*t+4)/2-n-1
   sum1=0
   sum2=0
   for m1 in range (1,i+1):
      sum1=sum1+funcR(m1)*binomial(i+j-m1-1,i-m1)
   for m2 in range (1,j+1):
      sum2=sum2+funcL(m2)*binomial(i+j-m2-1,j-m2)
   sum=sum1+sum2

@CachedFunction
def T(n, k):
    if (k==0): return n^2
    elif (k==n): return 2^n
    else: return T(n-1, k-1) + T(n-1, k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 12 2019

T(n,0) = n^2, n>0; T(0,k) = 2^k; T(n, k) = T(n-1, k-1) + T(n-1, k) for n,k > 0. [corrected by G. C. Greubel, Nov 12 2019]
Closed-form formula for general case. Let L(m) and R(m) be the left border and the right border of Pascal like triangle, respectively. We denote binomial(n,k) by C(n,k).
As table read by antidiagonals T(n,k) = Sum_{m1=1..n} R(m1)*C(n+k-m1-1, n-m1) + Sum_{m2=1..k} L(m2)*C(n+k-m2-1, k-m2); n,k >=0.
As linear sequence a(n) = Sum_{m1=1..i} R(m1)*C(i+j-m1-1, i-m1) + Sum_{m2=1..j} L(m2)*C(i+j-m2-1, j-m2), where i=n-t*(t+1)/2-1, j=(t*t+3*t+4)/2-n-1, t=floor((-1+sqrt(8*n-7))/2); n>0.
Some special cases. If L(m)={b,b,b...} b*A000012, then the second sum takes form b*C(n+k-1,j). If L(m) is {0,b,2b,...} b*A001477, then the second sum takes form b*C(n+k,n-1). Similarly for R(m) and the first sum.
For this sequence L(m)=m^2 and R(m)=2^m.
As table read by antidiagonals T(n,k) = Sum_{m1=1..n} (2^m1)*C(n+k-m1-1, n-m1) + Sum_{m2=1..k} (m2^2)*C(n+k-m2-1, k-m2); n,k >=0.
As linear sequence a(n) = Sum_{m1=1..i} (2^m1)*C(i+j-m1-1, i-m1) + Sum_{m2=1..j} (m2^2)*C(i+j-m2-1, j-m2), where i=n-t*(t+1)/2-1, j=(t*t+3*t+4)/2-n-1, t=floor((-1+sqrt(8*n-7))/2).
As a triangular array read by rows, T(n,k) = Sum_{i=1..n-k} i^2*C(n-1-i, n-k-i) + Sum_{i=1..k} 2^i*C(n-1-i, k-i); n,k >=0. - Greg Dresden, Aug 06 2022

Cross-references corrected and extended by Philippe Deléham, Dec 27 2013

A056000 a(n) = n*(n+9)/2.

Original entry on oeis.org

0, 5, 11, 18, 26, 35, 45, 56, 68, 81, 95, 110, 126, 143, 161, 180, 200, 221, 243, 266, 290, 315, 341, 368, 396, 425, 455, 486, 518, 551, 585, 620, 656, 693, 731, 770, 810, 851, 893, 936, 980, 1025, 1071, 1118, 1166, 1215, 1265, 1316, 1368, 1421, 1475

Offset: 0

Barry E. Williams, Jun 16 2000

Numbers m >= 0 such that 8m+81 is a square. - Bruce J. Nicholson, Jul 29 2017

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 193.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Leo Tavares, Illustration: Triangular Pairs.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000096, A055998, A055999, A001477.
Column m=2 of (1, 5)-Pascal triangle A096940.
Cf. numbers of the form n*(d*n+10-d)/2 indexed in A140090.
Cf. A000217, A008585.

Table[n (n + 9)/2, {n, 0, 50}] (* or *)
FoldList[#1 + #2 + 4 &, Range[0, 50]] (* or *)
Table[PolygonalNumber[n + 4] - 10, {n, 0, 50}] (* or *)
CoefficientList[Series[x (5 - 4 x)/(1 - x)^3, {x, 0, 50}], x] (* Michael De Vlieger, Jul 30 2017 *)

a(n)=n*(n+9)/2 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = A000217(n+4) - 10.
G.f.: x(5-4x)/(1-x)^3.
From Zerinvary Lajos, Oct 01 2006: (Start)
a(n) = A000096(n) + 3*n.
a(n) = A055999(n) + n.
a(n) = A056115(n) - n.
(End)
a(n) = binomial(n,2) - 4*n, n >= 9. - Zerinvary Lajos, Nov 25 2006
a(n) = A126890(n,4) for n > 3. - Reinhard Zumkeller, Dec 30 2006
a(n) = A028569(n)/2. - Zerinvary Lajos, Feb 12 2007
If we define f(n,i,a) = Sum_{k=0..(n-i)} binomial(n,k)*stirling1(n-k,i)*Product_{j=0..k-1} (-a-j), then a(n) = -f(n,n-1,5), for n >= 1. - Milan Janjic, Dec 20 2008
a(n) = n + a(n-1) + 4. - Vincenzo Librandi, Aug 07 2010
a(n) = Sum_{k=1..n} (k+4). - Gary Detlefs, Aug 10 2010
Sum_{n>=1} 1/a(n) = 7129/11340. - R. J. Mathar, Jul 14 2012
a(n) = 5n - floor(n/2) + floor(n^2/2). - Wesley Ivan Hurt, Jun 15 2013
E.g.f.: (1/2)*(x^2 + 10*x)*exp(x). - G. C. Greubel, Jul 17 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/9 - 1879/11340. - Amiram Eldar, Jul 03 2020
a(n) = A000217(n+1) + A008585(n) - 1. - Leo Tavares, Sep 22 2022
From Amiram Eldar, Feb 12 2024: (Start)
Product_{n>=1} (1 - 1/a(n)) = -567*cos(sqrt(89)*Pi/2)/(220*Pi).
Product_{n>=1} (1 + 1/a(n)) = 35*cos(sqrt(73)*Pi/2)/(4*Pi). (End)

More terms from James Sellers, Jul 04 2000

cp's OEIS Frontend

A096941 Fourth column of (1,5)-Pascal triangle A096940.

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A096942 Fifth column of (1,5)-Pascal triangle A096940.

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A096943 Sixth column of (1,5)-Pascal triangle A096940.

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A096944 Seventh column of (1,5)-Pascal triangle A096940.

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A096945 Eighth column of (1,5)-Pascal triangle A096940.

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A096946 Ninth column of (1,5)-Pascal triangle A096940.

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A096947 Tenth column of (1,5)-Pascal triangle A096940.

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A007283 a(n) = 3*2^n.

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