A152948 -id:A152948 - cp's OEIS frontend

A245300 Triangle T(n,k) = (n+k)*(n+k+1)/2 + k, 0 <= k <= n, read by rows.

0, 1, 4, 3, 7, 12, 6, 11, 17, 24, 10, 16, 23, 31, 40, 15, 22, 30, 39, 49, 60, 21, 29, 38, 48, 59, 71, 84, 28, 37, 47, 58, 70, 83, 97, 112, 36, 46, 57, 69, 82, 96, 111, 127, 144, 45, 56, 68, 81, 95, 110, 126, 143, 161, 180, 55, 67, 80, 94, 109, 125, 142, 160, 179, 199, 220

Offset: 0

Reinhard Zumkeller, Jul 17 2014

			First rows and their row sums (A245301):
   0                                                                  0;
   1,  4                                                              5;
   3,  7,  12                                                        22;
   6, 11,  17,  24                                                   58;
  10, 16,  23,  31,  40                                             120;
  15, 22,  30,  39,  49,  60                                        215;
  21, 29,  38,  48,  59,  71,  84                                   350;
  28, 37,  47,  58,  70,  83,  97, 112                              532;
  36, 46,  57,  69,  82,  96, 111, 127, 144                         768;
  45, 56,  68,  81,  95, 110, 126, 143, 161, 180                   1065;
  55, 67,  80,  94, 109, 125, 142, 160, 179, 199, 220              1430;
  66, 79,  93, 108, 124, 141, 159, 178, 198, 219, 241, 264         1870;
  78, 92, 107, 123, 140, 158, 177, 197, 218, 240, 263, 287, 312    2392.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Cf. A000217, A046092, A062725, A245301.

a245300 n k = (n + k) * (n + k + 1) `div` 2 + k
a245300_row n = map (a245300 n) [0..n]
a245300_tabl = map a245300_row [0..]
a245300_list = concat a245300_tabl

[k + Binomial(n+k+1,2): k in [0..n], n in [0..15]]; // G. C. Greubel, Apr 01 2021

Table[k + Binomial[n+k+1,2], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 01 2021 *)

flatten([[k + binomial(n+k+1,2) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Apr 01 2021

T(n, 0) = A000217(n).
T(n, n) = A046092(n).
T(2*n, n) = A062725(n) (central terms).
Sum_{k=0..n} T(n, k) = A245301(n).
From G. C. Greubel, Apr 01 2021: (Start)
T(n, 1) = A000124(n+1) = A134869(n+1), n >= 1.
T(n, 2) = A152948(n+4), n >= 2.
T(n, 3) = A152950(n+4), n >= 3.
T(n, 4) = A145018(n+5), n >= 4.
T(n, 5) = A167499(n+4), n >= 5.
T(n, 6) = A166136(n+5), n >= 6.
T(n, 7) = A167487(n+6), n >= 7.
T(n, n-1) = A056220(n), n >= 1.
T(n, n-2) = A142463(n-1), n >= 2.
T(n, n-3) = A054000(n-1), n >= 3.
T(n, n-4) = A090288(n-3), n >= 4.
T(n, n-5) = A268581(n-4), n >= 5.
T(n, n-6) = A059993(n-4), n >= 6.
T(n, n-7) = (-1)*A147973(n), n >= 7.
T(n, n-8) = A139570(n-5), n >= 8.
T(n, n-9) = A271625(n-5), n >= 9.
T(n, n-10) = A222182(n-4), n >= 10.
T(2*n, n-1) = A081270(n-1), n >= 1.
T(2*n, n+1) = A117625(n+1), n >= 1. (End)

A115990 Riordan array (1/sqrt(1-2x-3x^2), (1-2x-3x^2)/(2(1-3x)) - sqrt(1-2x-3x^2)/2).

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 7, 5, 3, 1, 19, 13, 8, 4, 1, 51, 35, 22, 12, 5, 1, 141, 96, 61, 35, 17, 6, 1, 393, 267, 171, 101, 53, 23, 7, 1, 1107, 750, 483, 291, 160, 77, 30, 8, 1, 3139, 2123, 1373, 839, 476, 244, 108, 38, 9, 1, 8953, 6046, 3923, 2423, 1406, 752, 360, 147, 47, 10

Offset: 0

Paul Barry, Feb 10 2006

First column is central trinomial coefficients A002426. Second column is number of directed animals of size n+1, A005773(n+1). Row sums are A005717 (number of horizontal steps in all Motzkin paths of length n). First column has e.g.f. exp(x) I_0(2x). Row sums have e.g.f. dif(exp(x) I_1(2x),x).

Riordan array (1/sqrt(1-2*x-3*x^2), (1+x-sqrt(1-2*x-3*x^2))/2).

			Triangle begins
    1;
    1,  1;
    3,  2,  1;
    7,  5,  3,  1;
   19, 13,  8,  4,  1;
   51, 35, 22, 12,  5,  1;
  141, 96, 61, 35, 17,  6,  1;

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. A115991, A005773 (k=1), A025566 (k=2), A035045 (k=3), A152948 (diag. n=k+2), .

Flat(List([0..10], n-> List([0..n], k-> Sum([0..n], j-> Binomial(n-k, j-k)*Binomial(j, n-j)) ))); # G. C. Greubel, May 09 2019

[[(&+[Binomial(n-k, j-k)*Binomial(j, n-j): j in [0..n]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 09 2019

A115990 := proc(n,k)
    add(binomial(n-k,j-k)*binomial(j,n-j),j=0..n) ;
end proc:
seq(seq(A115990(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Jun 25 2023

Table[Sum[ Binomial[n-k, j-k]*Binomial[j, n-j], {j, 0, n}], {n, 0, 10}, {k, 0, n} ] // Flatten (* G. C. Greubel, Mar 07 2017 *)

{T(n, k) = sum(j=0, n, binomial(n-k, j-k)*binomial(j, n-j))}; \\ G. C. Greubel, May 09 2019

[[sum(binomial(n-k, j-k)*binomial(j, n-j) for j in (0..n)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 09 2019

Number triangle T(n,k) = Sum_{j=0..n} C(n-k,j-k)*C(j,n-j).

A152949 a(n) = 3 + binomial(n-1,2).

Original entry on oeis.org

3, 3, 4, 6, 9, 13, 18, 24, 31, 39, 48, 58, 69, 81, 94, 108, 123, 139, 156, 174, 193, 213, 234, 256, 279, 303, 328, 354, 381, 409, 438, 468, 499, 531, 564, 598, 633, 669, 706, 744, 783, 823, 864, 906, 949, 993, 1038, 1084, 1131, 1179, 1228, 1278, 1329, 1381

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 15 2008

a(1)=3; then add 0 to the first number, then 1,2,3,4,... and so on.

Muniru A Asiru, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000124, A000217, A016028, A152947, A152948, A152950.

List([1..55],n->3+Binomial(n-1,2)); # Muniru A Asiru, Oct 28 2018

seq(coeff(series(x*(4*x^2-6*x+3)/(1-x)^3,x,n+1), x, n), n = 1 .. 55); # Muniru A Asiru, Oct 28 2018

s=3;lst={3};Do[s+=n;AppendTo[lst,s],{n,0,5!}];lst
Table[Binomial[n-1,2],{n,60}]+3 (* Harvey P. Dale, Feb 27 2013 *)

Vec( x*(3-6*x+4*x^2)/(1-x)^3 + O(x^66) ) \\ Joerg Arndt, Jul 24 2013

[3+binomial(n,2) for n in range(0, 54)] # Zerinvary Lajos, Mar 12 2009

a(n) = a(n-1) + n - 2 (with a(1)=3). - Vincenzo Librandi, Nov 27 2010
G.f.: x*(3-6*x+4*x^2)/(1-x)^3. - Nikita Gogin, Jul 24 2013
a(n) = A016028(n+1) for n >= 2. - Georg Fischer, Oct 28 2018
Sum_{n>=1} 1/a(n) = 1/3 + 2*Pi*tanh(sqrt(23)*Pi/2)/sqrt(23). - Amiram Eldar, Dec 13 2022
From Elmo R. Oliveira, Sep 05 2025: (Start)
a(n) = (n^2 - 3*n + 8)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3.
E.g.f.: exp(x)*(8 - 2*x + x^2)/2 - 4. (End)

A236257 a(n) = 2n^2 - 7n + 9.

Original entry on oeis.org

9, 4, 3, 6, 13, 24, 39, 58, 81, 108, 139, 174, 213, 256, 303, 354, 409, 468, 531, 598, 669, 744, 823, 906, 993, 1084, 1179, 1278, 1381, 1488, 1599, 1714, 1833, 1956, 2083, 2214, 2349, 2488, 2631, 2778, 2929, 3084, 3243, 3406, 3573, 3744, 3919, 4098, 4281, 4468

Offset: 0

Vladimir Shevelev, Jan 21 2014

If zero polygonal numbers are ignored, then for n >= 3, the a(n)-th n-gonal number is a sum of the (a(n)-1)-th n-gonal number and the (2*n-3)-th n-gonal number.

			a(7)=58. This means that the 58th heptagonal number 8323 (cf. A000566) is a sum of two heptagonal numbers. We have 8323 = 8037 + 286 with indices in A000566 58,57,11.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
W. Burrows, C. Tuffley, Maximising common fixtures in a round robin tournament with two divisions, arXiv preprint arXiv:1502.06664 [math.CO], 2015.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A000290, A000326, A000384, A000566, A000567, A001106, A001107, A051624, A051682, A051865-A051876, A152948.

Table[2 n^2 - 7 n + 9, {n, 0, 48}] (* Michael De Vlieger, Apr 19 2015 *)
LinearRecurrence[{3,-3,1},{9,4,3},50] (* Harvey P. Dale, Nov 24 2017 *)

Vec(-(18*x^2-23*x+9)/(x-1)^3 + O(x^100)) \\ Colin Barker, Jan 21 2014

From Colin Barker, Jan 21 2014: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: -(18*x^2 - 23*x + 9)/(x-1)^3. (End)
E.g.f.: exp(x)*(9 - 5*x + 2*x^2). - Elmo R. Oliveira, Nov 13 2024

A278436 Numbers such that A279966(n) = 0.

Original entry on oeis.org

17, 23, 47, 57, 93, 107, 173, 233, 353, 437, 467, 563, 677, 743, 817, 829, 851, 863, 955, 1037, 1187, 1213, 1277, 1387, 1433, 1487, 1549, 2089, 2147, 2213, 2287, 2293, 2417, 2473, 2689, 2777, 2911, 3083, 3323, 3391, 6691, 9337, 22969, 38557, 47347, 75391, 104999, 130927, 146719

Offset: 1

Alec Jones, Dec 24 2016

nonn

Not all numbers in this list are prime; nonprime elements include 57, 93, 437, 817, 851, 955, 1037, 1387, 2147.

This sequence is neither a subset nor a superset of sequence A281533 since 155 and 817 are the first numbers in one, but not the other, respectively. - Hartmut F. W. Hoft, Jan 23 2017

			Number 817 = 19*43, equivalent to array position (4, 37), is in the sequence since none of the numbers in the prior column, diagonal, row and antidiagonal contain the counts of 1, 19, 43 and 817. - _Hartmut F. W. Hoft_, Jan 23 2017

Cf. A279966, A000124, A145018, A152948, A152950, A167499, A279967.

(* support functions are in A279967 *)
a278436[k_] := Module[{ut=upperTriangle[k], ms=Table[" ", {i, 1, k}, {j, 1, k}], h, pos, val, seqL={}}, ms[[1, 1]]=1; For[h=2, h<=Length[ut], h++, pos=ut[[h]]; val=Length[Select[Map[ms[[Apply[Sequence, #]]]&, priorPos[pos]], #!=0 && Mod[seqPos[pos], #]==0&]]; If[val==0, AppendTo[seqL, h]]; ms[[Apply[Sequence, pos]]]=val]; seqL]
a278436[100] (* data through 3391. - Hartmut F. W. Hoft, Jan 23 2017 *)

A159881 Triangle read by rows : T(n,0) = n+1, T(n,k)=0 if k<0 or if k>n, T(n,k) = k*T(n-1,k) - T(n-1,k-1).

Original entry on oeis.org

1, 2, -1, 3, -3, 1, 4, -6, 5, -1, 5, -10, 16, -8, 1, 6, -15, 42, -40, 12, -1, 7, -21, 99, -162, 88, -17, 1, 8, -28, 219, -585, 514, -173, 23, -1, 9, -36, 466, -1974, 2641, -1379, 311, -30, 1, 10, -45, 968, -6388, 12538, -9536, 3245, -521, 38, -1

Offset: 0

Philippe Deléham, Apr 25 2009

			Triangle begins :
1;
2, -1;
3, -3, 1;
4, -6, 5, -1;
5, -10, 16, -8, 1;
6, -15, 42, -40, 12, -1;
7, -21, 99, -162, 88, -17, 1;
8, -28, 219, -585, 514, -173, 23, -1;
9, -36, 466, -1974, 2641, -1379, 311, -30, 1;
10, -45, 968, -6388, 12538, -9536, 3245, -521, 38, -1;
11, -55, 1981, -20132, 56540, -60218, 29006, -6892, 825, -47, 1;

G. C. Greubel, Rows n=0..100 of triangle, flattened

Cf. A000027, A000217, A002662, A152948

A159881 := proc(n,k) option remember; if k = 0 then n+1; elif k < 0 or k > n then 0 ; else k*procname(n-1,k)-procname(n-1,k-1) ; fi; end: for n from 0 to 10 do for k from 0 to n do printf("%d,",A159881(n,k)) ; od: od: # R. J. Mathar, May 29 2009

T[n_,0]:= n+1; T[n_,k_]:= T[n,k] = If[k < 0 || k > n, 0, k*T[n-1, k] - T[n-1, k-1]]; Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Jul 27 2018 *)

{T(n,k) = if(k==0, n+1, if(k<0 || k>n, 0, k*T(n-1,k) - T(n-1,k-1)))};
for(n=0,15, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Jul 27 2018

Conjecture row sums: Sum_{k=0..n} |T(n,k)| = A029761(n). - R. J. Mathar, May 29 2009

A214928 A209293 as table read layer by layer clockwise.

Original entry on oeis.org

1, 2, 4, 3, 5, 9, 14, 7, 6, 8, 12, 17, 23, 20, 11, 10, 13, 19, 26, 34, 43, 30, 27, 16, 15, 18, 24, 31, 39, 48, 58, 53, 38, 35, 22, 21, 25, 33, 42, 52, 63, 75, 88, 69, 64, 47, 44, 29, 28, 32, 40, 49, 59, 70, 82, 95, 109, 102, 81, 76, 57, 54, 37, 36, 41, 51, 62

Offset: 1

Boris Putievskiy, Mar 11 2013

Permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1). The order of the list:
T(1,1)=1;
T(1,2), T(2,2), T(2,1);
. . .
T(1,n), T(2,n), ... T(n-1,n), T(n,n), T(n,n-1), ... T(n,1);
. . .

			The start of the sequence as table:
  1....2...5...8..13..18...
  3....4...9..12..19..24...
  6....7..14..17..26..31...
  10..11..20..23..34..39...
  15..16..27..30..43..48...
  21..22..35..38..53..58...
  . . .
The start of the sequence as triangle array read by rows:
  1;
  2,4,3;
  5,9,14,7,6;
  8,12,17,23,20,11,10;
  13,19,26,34,43,30,27,16,15;
  18,24,31,39,48,58,53,38,35,22,21;
  . . .
Row number r contains 2*r-1 numbers.

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A209293, A209279, A209278, A185180, A060734, A060736; table T(n,k) contains: in rows A000982, A097063; in columns A000217, A000124, A000096, A152948, A034856, A152950, A055998, A000982, A097063.

t=int((math.sqrt(n-1)))+1
i=min(t,n-(t-1)**2)
j=min(t,t**2-n+1)
m1=int((i+j)/2)+int(i/2)*(-1)**(2*i+j-1)
m2=int((i+j+1)/2)+int(i/2)*(-1)**(2*i+j-2)
result=(m1+m2-1)*(m1+m2-2)/2+m1

As table
T(n,k) = n*n/2+4*(floor((k-1)/2)+1)*n+ceiling((k-1)^2/2), n,k > 0.
As linear sequence
a(n)= (m1+m2-1)*(m1+m2-2)/2+m1, where m1=floor((i+j)/2) + floor(i/2)*(-1)^(2*i+j-1), m2=int((i+j+1)/2)+int(i/2)*(-1)^(2*i+j-2), where i=min(t; n-(t-1)^2), j=min(t; t^2-n+1), t=floor(sqrt(n-1))+1.

A214929 A209293 as table read layer by layer - layer clockwise, layer counterclockwise and so on.

Original entry on oeis.org

1, 3, 4, 2, 5, 9, 14, 7, 6, 10, 11, 20, 23, 17, 12, 8, 13, 19, 26, 34, 43, 30, 27, 16, 15, 21, 22, 35, 38, 53, 58, 48, 39, 31, 24, 18, 25, 33, 42, 52, 63, 75, 88, 69, 64, 47, 44, 29, 28, 36, 37, 54, 57, 76, 81, 102, 109, 95, 82, 70, 59, 49, 40, 32, 41, 51, 62

Offset: 1

Boris Putievskiy, Mar 11 2013

nonn

Permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1). Table read by boustrophedonic ("ox-plowing") method. Let m be natural number. The order of the list:
T(1,1)=1;
T(2,1), T(2,2), T(1,2);
. . .
T(1,2*m+1), T(2,2*m+1), ... T(2*m,2*m+1), T(2*m+1,2*m+1), T(2*m+1,2*m), ... T(2*m+1,1);
T(2*m,1),   T(2*m,2),   ... T(2*m,2*m-1), T(2*m,2*m),     T(2*m-1,2*m), ... T(1,2*m);
. . .
The first row is layer read clockwise, the second row is layer counterclockwise.

			The start of the sequence as table:
  1....2...5...8..13..18...
  3....4...9..12..19..24...
  6....7..14..17..26..31...
  10..11..20..23..34..39...
  15..16..27..30..43..48...
  21..22..35..38..53..58...
  . . .
The start of the sequence as triangle array read by rows:
  1;
  3,4,2;
  5,9,14,7,6;
  10,11,20,23,17,12,8;
  13,19,26,34,43,30,27,16,15;
  21,22,35,38,53,58,48,39,31,24,18;
  . . .
Row number r contains 2*r-1 numbers.

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A081344, A209293, A209279, A209278, A185180; table T(n,k) contains: in rows A000982, A097063; in columns A000217, A000124, A000096, A152948, A034856, A152950, A055998, A000982, A097063.

t=int((math.sqrt(n-1)))+1
i=(t % 2)*min(t,n-(t-1)**2) + ((t+1) % 2)*min(t,t**2-n+1)
j=(t % 2)*min(t,t**2-n+1) + ((t+1) % 2)*min(t,n-(t-1)**2)
m1=int((i+j)/2)+int(i/2)*(-1)**(2*i+j-1)
m2=int((i+j+1)/2)+int(i/2)*(-1)**(2*i+j-2)
result=(m1+m2-1)*(m1+m2-2)/2+m1

As table
T(n,k) = n*n/2+4*(floor((k-1)/2)+1)*n+ceiling((k-1)^2/2), n,k > 0.
As linear sequence
a(n)= (m1+m2-1)*(m1+m2-2)/2+m1, where
m1=floor((i+j)/2) + floor(i/2)*(-1)^(2*i+j-1), m2=int((i+j+1)/2)+int(i/2)*(-1)^(2*i+j-2),
where i=(t mod 2)*min(t; n-(t-1)^2) + (t+1 mod 2)*min(t; t^2-n+1), j=(t mod 2)*min(t; t^2-n+1) + (t+1 mod 2)*min(t; n-(t-1)^2), t=floor(sqrt(n-1))+1.

A238531 Expansion of (1 - x + x^2)^2 / (1 - x)^3 in powers of x.

Original entry on oeis.org

1, 1, 3, 5, 8, 12, 17, 23, 30, 38, 47, 57, 68, 80, 93, 107, 122, 138, 155, 173, 192, 212, 233, 255, 278, 302, 327, 353, 380, 408, 437, 467, 498, 530, 563, 597, 632, 668, 705, 743, 782, 822, 863, 905, 948, 992, 1037, 1083, 1130, 1178, 1227, 1277, 1328, 1380

Offset: 0

Michael Somos, Feb 28 2014

Essentially the same as A152948, A133263 and A089071. - R. J. Mathar, Mar 30 2014

			G.f. = 1 + x + 3*x^2 + 5*x^3 + 8*x^4 + 12*x^5 + 17*x^6 + 23*x^7 + 30*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A133263.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x+x^2)^2/(1-x)^3)); // G. C. Greubel, Aug 07 2018

a[ n_] := (n^2 - n) / 2 + If[ n == 0 || n == 1, 1, 2];
CoefficientList[Series[(1-x+x^2)^2/(1-x)^3, {x, 0, 50}], x] (* G. C. Greubel, Aug 07 2018 *)

{a(n) = (n^2 - n) / 2 + 2 - (n==0) - (n==1)};

{a(n) = if( n<0, n = 1-n); polcoeff( (1 - x + x^2)^2 / (1 - x)^3 + x * O(x^n), n)};

Euler transform of length 6 sequence [1, 2, 2, 0, 0, -2].
Binomial transform of [1, 0, 2, -2, 3, -4, 5, -6, ...].
a(n) = p(-1) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0, 1, ..., n.
G.f.: (1 - x + x^2)^2 / (1 - x)^3.
a(n) = a(1 - n) for all n in Z.
a(n + 1) = A133263(n) if n>=0. a(n) = (n^2 - n) / 2 + 2 unless n=0 or n=1.
(1 + x^2 + x^3 + x^4 + ...)*(1 + x + 2x^2 + 3x^3 + 4x^4 + ...) = (1 + x + 3x^2 + 5x^3 + 8x^4 + 12x^5 + ...). - Gary W. Adamson, Jul 27 2010

A173154 a(n) = n^3/6 + 3n^2/4 + 7n/3 + 7/8 + (-1)^n/8.

Original entry on oeis.org

1, 4, 10, 19, 33, 52, 78, 111, 153, 204, 266, 339, 425, 524, 638, 767, 913, 1076, 1258, 1459, 1681, 1924, 2190, 2479, 2793, 3132, 3498, 3891, 4313, 4764, 5246, 5759, 6305, 6884, 7498, 8147, 8833, 9556, 10318, 11119, 11961, 12844, 13770, 14739, 15753, 16812, 17918, 19071, 20273, 21524

Offset: 0

Paul Curtz, Feb 11 2010

Generated by reading the table shown in A172002 down the diagonal starting at 1.

The inverse binomial transform yields 1, 3, 3, 0, 2, -4, 8, -16, 32, -64, 128, -256, 512, -1024, ... with a pattern of powers of 2.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

[n^3/6 + 3*n^2/4 + 7*n/3 + 7/8 + (-1)^n/8: n in [0..50]]; // Vincenzo Librandi, Aug 05 2011

Table[n^3/6+(3n^2)/4+(7n)/3+7/8+(-1)^n/8,{n,0,50}] (* or *) LinearRecurrence[{3,-2,-2,3,-1},{1,4,10,19,33},50] (* Harvey P. Dale, Jan 04 2012 *)

G.f.: ( 1 + x - x^3 + x^4 ) / ( (1+x)*(x-1)^4 ).
a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5).
a(n+4) - a(n) = 4*A152948(n+5) = 4*A089071(n+5).
First differences: a(n+1) - a(n) = A061925(n+2).
Second differences: a(n+2) - 2*a(n+1) + a(n) = n + 5/2 + (-1)^n/2 = 3, 3, 5, 5, 7, 7, 9, 9, ... , duplicated A144396.

cp's OEIS Frontend

A245300 Triangle T(n,k) = (n+k)*(n+k+1)/2 + k, 0 <= k <= n, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

Sage

Formula

A115990 Riordan array (1/sqrt(1-2*x-3*x^2), (1-2*x-3*x^2)/(2*(1-3*x)) - sqrt(1-2*x-3*x^2)/2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A152949 a(n) = 3 + binomial(n-1,2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

Sage

Formula

A236257 a(n) = 2*n^2 - 7*n + 9.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A278436 Numbers such that A279966(n) = 0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A159881 Triangle read by rows : T(n,0) = n+1, T(n,k)=0 if k<0 or if k>n, T(n,k) = k*T(n-1,k) - T(n-1,k-1).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A214928 A209293 as table read layer by layer clockwise.

Views

Author

A115990 Riordan array (1/sqrt(1-2x-3x^2), (1-2x-3x^2)/(2(1-3x)) - sqrt(1-2x-3x^2)/2).

A236257 a(n) = 2n^2 - 7n + 9.

A173154 a(n) = n^3/6 + 3n^2/4 + 7n/3 + 7/8 + (-1)^n/8.