A117142 -id:A117142 - cp's OEIS frontend

A139601 Square array of polygonal numbers read by ascending antidiagonals: T(n, k) = (n + 1)(k - 1)k/2 + k.

0, 0, 1, 0, 1, 3, 0, 1, 4, 6, 0, 1, 5, 9, 10, 0, 1, 6, 12, 16, 15, 0, 1, 7, 15, 22, 25, 21, 0, 1, 8, 18, 28, 35, 36, 28, 0, 1, 9, 21, 34, 45, 51, 49, 36, 0, 1, 10, 24, 40, 55, 66, 70, 64, 45, 0, 1, 11, 27, 46, 65, 81, 91, 92, 81, 55, 0, 1, 12, 30, 52, 75, 96, 112, 120, 117, 100, 66

Offset: 0

Omar E. Pol, Apr 27 2008

A general formula for polygonal numbers is P(n,k) = (n-2)(k-1)k/2 + k, where P(n,k) is the k-th n-gonal number. - Omar E. Pol, Dec 21 2008

			The square array of polygonal numbers begins:
========================================================
Triangulars .. A000217: 0, 1,  3,  6, 10,  15,  21,  28,
Squares ...... A000290: 0, 1,  4,  9, 16,  25,  36,  49,
Pentagonals .. A000326: 0, 1,  5, 12, 22,  35,  51,  70,
Hexagonals ... A000384: 0, 1,  6, 15, 28,  45,  66,  91,
Heptagonals .. A000566: 0, 1,  7, 18, 34,  55,  81, 112,
Octagonals ... A000567: 0, 1,  8, 21, 40,  65,  96, 133,
9-gonals ..... A001106: 0, 1,  9, 24, 46,  75, 111, 154,
10-gonals .... A001107: 0, 1, 10, 27, 52,  85, 126, 175,
11-gonals .... A051682: 0, 1, 11, 30, 58,  95, 141, 196,
12-gonals .... A051624: 0, 1, 12, 33, 64, 105, 156, 217,
And so on ..............................................
========================================================

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Peter Luschny, Figurate number — a very short introduction. With plots from Stefan Friedrich Birkner.
Omar E. Pol, Polygonal numbers, An alternative illustration of initial terms.

Sequences of m-gonal numbers: A000217 (m=3), A000290 (m=4), A000326 (m=5), A000384 (m=6), A000566 (m=7), A000567 (m=8), A001106 (m=9), A001107 (m=10), A051682 (m=11), A051624 (m=12), A051865 (m=13), A051866 (m=14), A051867 (m=15), A051868 (m=16), A051869 (m=17), A051870 (m=18), A051871 (m=19), A051872 (m=20), A051873 (m=21), A051874 (m=22), A051875 (m=23), A051876 (m=24), A255184 (m=25), A255185 (m=26), A255186 (m=27), A161935 (m=28), A255187 (m=29), A254474 (m=30).
Cf. A000007, A000012, A000027, A002411, A006000, A006003, A006522.
Cf. A008585, A016957, A017329, A086270, A117142, A139606, A139607.
Cf. A139608, A139609, A139610, A139611, A139612, A139613, A139614.
Cf. A139615, A139616, A057145, A086271, A139600, A139617, A139618.
Cf. A139619, A139620.

T:= func< n,k | k*((n+1)*(k-1) +2)/2 >;
A139601:= func< n,k | T(n-k, k) >;
[A139601(n,k): k in  [0..n], n in [0..12]]; // G. C. Greubel, Jul 12 2024

T[n_, k_] := (n + 1)*(k - 1)*k/2 + k; Table[ T[n - k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Jul 12 2009 *)

def T(n,k): return k*((n+1)*(k-1)+2)/2
def A139601(n,k): return T(n-k, k)
flatten([[A139601(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 12 2024

T(n,k) = A086270(n,k), k>0. - R. J. Mathar, Aug 06 2008
T(n,k) = (n+1)*(k-1)*k/2 +k, n>=0, k>=0. - Omar E. Pol, Jan 07 2009
From G. C. Greubel, Jul 12 2024: (Start)
t(n, k) = (k/2)*( (k-1)*(n-k+1) + 2), where t(n,k) is this array read by rising antidiagonals.
t(2*n, n) = A006003(n).
t(2*n+1, n) = A002411(n).
t(2*n-1, n) = A006000(n-1).
Sum_{k=0..n} t(n, k) = A006522(n+2).
Sum_{k=0..n} (-1)^k*t(n, k) = (-1)^n * A117142(n).
Sum_{k=0..n} t(n-k, k) = (2*n^4 + 34*n^2 + 48*n - 15 + 3*(-1)^n*(2*n^2 + 16*n + 5))/384. (End)

A194621 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) is the number of partitions of n in which any two parts differ by at most k.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 5, 5, 5, 2, 5, 6, 7, 7, 7, 4, 6, 9, 10, 11, 11, 11, 2, 7, 10, 13, 14, 15, 15, 15, 4, 8, 14, 17, 20, 21, 22, 22, 22, 3, 9, 15, 22, 25, 28, 29, 30, 30, 30, 4, 10, 20, 27, 34, 37, 40, 41, 42, 42, 42, 2, 11, 21, 33, 41, 48, 51, 54, 55, 56, 56, 56

Offset: 0

Alois P. Heinz, Aug 30 2011

T(n,k) = A000041(n) for n >= 0 and k >= n.

			T(6,0) = 4: [6], [3,3], [2,2,2], [1,1,1,1,1,1].
T(6,1) = 6: [6], [3,3], [2,1,1,1,1], [2,2,1,1], [2,2,2], [1,1,1,1,1,1].
T(6,2) = 9: [6], [4,2], [3,1,1,1], [3,2,1], [3,3], [2,1,1,1,1], [2,2,1,1], [2,2,2], [1,1,1,1,1,1].
Triangle begins:
  1;
  1, 1;
  2, 2,  2;
  2, 3,  3,  3;
  3, 4,  5,  5,  5;
  2, 5,  6,  7,  7,  7;
  4, 6,  9, 10, 11, 11, 11;
  2, 7, 10, 13, 14, 15, 15, 15;

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-10 give (for n>0): A000005, A000027, A117142, A117143, A218506, A218507, A218508, A218509, A218510, A218511, A218512.

Main diagonal gives: A000041.

b:= proc(n, i, k) option remember;
      if n<0 or k<0 then 0
    elif n=0 then 1
    elif i<1 then 0
    else b(n, i-1, k-1) +b(n-i, i, k)
      fi
    end:
T:= (n, k)-> `if`(n=0, 1, 0) +add(b(n-i, i, k), i=1..n):
seq(seq(T(n, k), k=0..n), n=0..20);

b[n_, i_, k_] := b[n, i, k] = If[n < 0 || k < 0, 0, If[n == 0, 1, If[i < 1, 0, b[n, i-1, k-1] + b[n-i, i, k]]]]; t[n_, k_] := If[n == 0, 1, 0] + Sum[b[n-i, i, k], {i, 1, n}]; Table[Table[t[n, k], {k, 0, n}], {n, 0, 20}] // Flatten (* Jean-François Alcover, Dec 09 2013, translated from Maple *)

G.f. of column k: 1 + Sum_{j>0} x^j / Product_{i=0..k} (1-x^(i+j)).

A004197 Triangle read by rows. T(n, k) = n - k if n - k < k, otherwise k.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 3, 2, 1, 0, 0, 1, 2, 3, 3, 2, 1, 0, 0, 1, 2, 3, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 5, 6, 6, 5, 4, 3, 2, 1, 0, 0, 1, 2

Offset: 0

David W. Wilson

Table of min(x,y), where (x,y) = (0,0),(0,1),(1,0),(0,2),(1,1),(2,0),...
Highest power of 6 that divides A036561. - Fred Daniel Kline, May 29 2012
Triangle T(n,k) read by rows: T(n,k) = min(k,n-k). - Philippe Deléham, Feb 25 2014

			From _Philippe Deléham_, Feb 25 2014: (Start)
Top left corner of table:
  0 0 0 0
  0 1 1 1
  0 1 2 2
  0 1 2 3
Triangle T(n,k) begins:
  0;
  0, 0;
  0, 1, 0;
  0, 1, 1, 0;
  0, 1, 2, 1, 0;
  0, 1, 2, 2, 1, 0;
  0, 1, 2, 3, 2, 1, 0;
  0, 1, 2, 3, 3, 2, 1, 0;
  0, 1, 2, 3, 4, 3, 2, 1, 0;
  0, 1, 2, 3, 4, 4, 3, 2, 1, 0;
  0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0;
  0, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 0;
  0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0;
  0, 1, 2, 3, 4, 5, 6, 6, 5, 4, 3, 2, 1, 0;
  0, 1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1, 0;
  0, 1, 2, 3, 4, 5, 6, 7, 7, 6, 5, 4, 3, 2, 1, 0;
  ... (End)

Reinhard Zumkeller, Rows n=0..100 of triangle, flattened

Similar to but strictly different from A087062 and A261684.
Row sums give A002620. - Reinhard Zumkeller, Jul 27 2005
Positions of zero are given in A117142. - Ridouane Oudra, Apr 30 2019
Cf. A144464, A152714, A152716, A152717.

a004197 n k = a004197_tabl !! n !! k
a004197_tabl = map a004197_row [0..]
a004197_row n = hs ++ drop (1 - n `mod` 2) (reverse hs)
   where hs = [0..n `div` 2]
-- Reinhard Zumkeller, Aug 14 2011

T := (n, k) -> if n - k < k then n - k else k fi:
for n from 0 to 9 do seq(T(n, k), k=0..n) od; # Peter Luschny, May 07 2023

Flatten[Table[IntegerExponent[2^(n - k) 3^k, 6], {n, 0, 20}, {k, 0, n}]] (* Fred Daniel Kline, May 29 2012 *)

T(x,y)=min(x,y) \\ Charles R Greathouse IV, Feb 07 2017

a(n) = A003983(n) - 1.
G.f.: x*y/((1-x)*(1-y)*(1-x*y)). - Franklin T. Adams-Watters, Feb 06 2006
2^T(n,k) = A144464(n,k), 3^T(n,k) = A152714(n,k), 4^T(n,k) = A152716(n,k), 5^T(n,k) = A152717(n,k). - Philippe Deléham, Feb 25 2014
a(n) = (1/2)*(t - 1 - abs(t^2 - 2*n - 1)), where t = floor(sqrt(2*n+1)+1/2). - Ridouane Oudra, May 03 2019

Mathematica program fixed by Harvey P. Dale, Nov 26 2020

Name edited by Peter Luschny, May 07 2023

A117143 Number of partitions of n in which any two parts differ by at most 3.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 13, 17, 22, 27, 33, 41, 48, 57, 68, 78, 90, 105, 118, 134, 153, 170, 190, 214, 235, 260, 289, 315, 345, 380, 411, 447, 488, 525, 567, 615, 658, 707, 762, 812, 868, 931, 988, 1052, 1123, 1188, 1260, 1340, 1413, 1494, 1583, 1665, 1755, 1854

Offset: 1

Emeric Deutsch, Feb 27 2006

			a(6) = 10 because we have [6], [4,2], [4,1,1], [3,3], [3,2,1], [3,1,1,1], [2,2,2], [2,2,1,1], [2,1,1,1,1] and [1,1,1,1,1,1] ([5,1] does not qualify).

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Jonathan Bloom, Nathan McNew, Counting pattern-avoiding integer partitions, arXiv:1908.03953 [math.CO], 2019.
Index entries for linear recurrences with constant coefficients, signature (1,1,1,-2,-2,1,1,1,-1).

Cf. A117142.
Column k=3 of A194621.
Cf. A002717, A045947, A248851.

[(2*Floor((n+2)/3)*(14*Floor((n+2)/3)^2-(10*n+21)*Floor((n+2)/3)+2*(n^2+5*n+7))-(1-(-1)^Floor((n+2)/3))*(-1)^(n+2-Floor((n+2)/3)))/16: n in [1..60]]; // Vincenzo Librandi, May 12 2015

g:=sum(x^k/(1-x^k)/(1-x^(k+1))/(1-x^(k+2))/(1-x^(k+3)),k=1..85): gser:=series(g,x=0,65): seq(coeff(gser,x^n),n=1..59); with(combinat): for n from 1 to 7 do P:=partition(n): A:={}: for j from 1 to nops(P) do if P[j][nops(P[j])]-P[j][1]<4 then A:=A union {P[j]} else A:=A fi od: print(A); od: # this program yields the partitions

Table[Count[IntegerPartitions[n], ?(Max[#] - Min[#] <= 3 &)], {n, 30}] (* _Birkas Gyorgy, Feb 20 2011 *)

Vec(x*(x^5-x^4-x^3+x+1)/((x-1)^4*(x+1)*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, Mar 05 2015

G.f.: sum(x^k/[(1-x^k)(1-x^(k+1))(1-x^(k+2))(1-x^(k+3))], k=1..infinity). More generally, the g.f. of the number of partitions in which any two parts differ by at most b is sum(x^k/product(1-x^j, j=k..k+b), k=1..infinity).
G.f.: x*(x^5-x^4-x^3+x+1) / ((x-1)^4*(x+1)*(x^2+x+1)^2). - Colin Barker, Mar 05 2015
a(n)=(2*floor((n+2)/3)*(14*floor((n+2)/3)^2-(10*n+21)*floor((n+2)/3)+2*(n^2+5*n+7))-(1-(-1)^floor((n+2)/3))*(-1)^(n+2-floor((n+2)/3)))/16. - Luce ETIENNE, May 12 2015

A128508 Number of partitions p of n such that max(p) - min(p) = 3.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 3, 3, 7, 7, 12, 14, 20, 22, 32, 34, 45, 51, 63, 69, 87, 93, 112, 124, 144, 156, 184, 196, 225, 245, 275, 295, 335, 355, 396, 426, 468, 498, 552, 582, 637, 679, 735, 777, 847, 889, 960, 1016, 1088, 1144, 1232, 1288, 1377, 1449, 1539, 1611, 1719

Offset: 0

John W. Layman, May 07 2007

nonn

See A008805 and A049820 for the numbers of partitions p of n such that max(p)-min(p)=1 or 2, respectively.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
G. E. Andrews, M. Beck and N. Robbins, Partitions with fixed differences between largest and smallest parts, arXiv:1406.3374 [math.NT], 2014

Cf. A008805, A049820, A097364, A116685, A117142, A117143.

np[n_]:=Length[Select[IntegerPartitions[n],Max[#]-Min[#]==3&]]; Array[np,60] (* Harvey P. Dale, Jul 02 2012 *)

Conjecture. a(1)=0 and, for n>1, a(n+1)=a(n)+d(n), where d(n) is defined as follows: d=0,0,0,1,0 for n=1,...,5 and, for n>5, d(n)=d(n-2)+1 if n=6k or n=6k+4, d(n)=d(n-2) if n=6k+1 or n=6k+3, d(n)=d(n-2)+2Floor[n/6] if n=6k+2 and d(n)=d(n-5) if n=6k+5.
G.f. for number of partitions p of n such that max(p)-min(p) = m is Sum_{k>0} x^(2*k+m)/Product_{i=0..m} (1-x^(k+i)). - Vladeta Jovovic, Jul 04 2007
a(n) = A097364(n,3) = A116685(n,3) = A117143(n) - A117142(n). - Alois P. Heinz, Nov 02 2012

More terms from Vladeta Jovovic, Jul 04 2007

A123158 Square array related to Bell numbers read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 5, 3, 1, 15, 15, 10, 5, 1, 52, 52, 37, 22, 6, 1, 203, 203, 151, 99, 31, 9, 1, 877, 877, 674, 471, 160, 61, 10, 1, 4140, 4140, 3263, 2386, 856, 385, 75, 14, 1, 21147, 21147, 17007, 12867, 4802, 2416, 520, 135, 15, 1

Offset: 0

Philippe Deléham, Oct 01 2006

			Square array, A(n, k), begins:
   1,   1,   1,    1,    1, ... (Row n=0: A000012);
   1,   2,   3,    5,    6, ... (Row n=1: A117142);
   2,   5,  10,   22,   31, ...;
   5,  15,  37,   99,  160, ...;
  15,  52, 151,  471,  856, ...;
  52, 203, 674, 2386, 4802, ...;
Antidiagonals, T(n, k), begin as:
    1;
    1,   1;
    2,   2,   1;
    5,   5,   3,   1;
   15,  15,  10,   5,   1;
   52,  52,  37,  22,   6,  1;
  203, 203, 151,  99,  31,  9,   1;
  877, 877, 674, 471, 160, 61,  10,  1;

G. C. Greubel, Antidiagonals n = 0..50, flattened

Columns include: A000110 (Bell numbers), A003128, A005493, A033452.

Rows include: A000012, A117142.

function A(n,k)
  if k lt 0 or n lt 0 then return 0;
  elif n eq 0 then return 1;
  elif (k mod 2) eq 1 then return A(n,k-1) + (1/2)*(k+1)*A(n-1,k+1);
  else return A(n,k-1) + A(n-1,k+1);
  end if;
end function;
T:= func< n,k | A(n-k,k) >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 18 2023

A[0, _?NonNegative] = 1;
A[n_, k_]:= A[n, k]= If[n<0 || k<0, 0, If[OddQ[k], A[n, k-1] + (1/2)(k+1) A[n-1, k+1], A[n, k-1] + A[n-1, k+1]]];
Table[A[n-k, k], {n,0,10}, {k,0,n}]//Flatten (* Jean-François Alcover, Feb 21 2020 *)

@CachedFunction
def A(n,k):
    if (k<0 or n<0): return 0
    elif (n==0): return 1
    elif (k%2==1): return A(n,k-1) +(1/2)*(k+1)*A(n-1,k+1)
    else: return A(n,k-1) +A(n-1,k+1)
def T(n,k): return A(n-k,k)
flatten([[T(n,k) for k in range(n+1)] for n in range(12)]) # G. C. Greubel, Jul 18 2023

A(n, k) = 0 if n < 0, A(0, k) = 1 for k >= 0, A(n, k) = A(n, k-1) + (1/2)*(k+1)*A(n-1, k+1) if k is an odd number, A(n, k) = A(n, k-1) + A(n-1, k+1) if k is an even number (array).
A(n, 0) = A000110(n).
A(n, 1) = A000110(n+1).
A(n, 2) = A005493(n).
A(n, 3) = A033452(n).
A(n, 4) = A003128(n+2).
T(n, k) = A(n-k, k) (antidiagonals).

A165157 Zero followed by partial sums of A133622.

Original entry on oeis.org

0, 1, 3, 4, 7, 8, 12, 13, 18, 19, 25, 26, 33, 34, 42, 43, 52, 53, 63, 64, 75, 76, 88, 89, 102, 103, 117, 118, 133, 134, 150, 151, 168, 169, 187, 188, 207, 208, 228, 229, 250, 251, 273, 274, 297, 298, 322, 323, 348, 349, 375, 376, 403, 404, 432, 433, 462, 463, 493, 494, 525

Offset: 0

Jaroslav Krizek, Sep 05 2009

A133622 is a toothed sequence.

Interleaving of A055998 and A034856.

			From _Stefano Spezia_, Jul 10 2020: (Start)
Illustration of the initial terms for n > 0:
o    o      o      o         o        o
     o o    o o    o o       o o      o o
            o      o         o        o
                   o o o     o o o    o o o
                             o        o
                                      o o o o
(1)  (3)   (4)    (7)       (8)      (12)
(End)

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

Equals -1+A101881.
a(n) = A117142(n+2)-2 = A055802(n+6)-3 = A114220(n+5)-3 = A134519(n+3)-3.
Cf. A133622, A055998, A034856.

a165157 n = a165157_list !! n
a165157_list = scanl (+) 0 a133622_list
-- Reinhard Zumkeller, Feb 20 2015

m:=60; T:=[ 1+(1+(-1)^n)*n/4: n in [1..m] ]; [0] cat [ n eq 1 select T[1] else Self(n-1)+T[n]: n in [1..m] ]; // Klaus Brockhaus, Sep 06 2009

[ n le 2 select n-1 else n le 4 select n else 2*Self(n-2)-Self(n-4)+1: n in [1..61] ]; // Klaus Brockhaus, Sep 06 2009

a(0) = 0, a(2*n) = a(2*n-1) + n + 1, a(2*n+1) = a(2*n) + 1.
a(n) = (n^2+10*n)/8 if n is even, a(n) = (n^2+8*n-1)/8 if n is odd.
a(2*k) = A055998(k) = k*(k+5)/2; a(2*k+1) = A034856(k+1) = k*(k+5)/2+1.
a(n) = 2*a(n-2)-a(n-4)+1 for n > 3; a(0)=0, a(1)=1, a(2)=3, a(3)=4. - Klaus Brockhaus, Sep 06 2009
a(n) = (2*n*(n+9)-1+(2*n+1)*(-1)^n)/16. - Klaus Brockhaus, Sep 06 2009
a(n) = n+binomial(1+floor(n/2),2). - Mircea Merca, Feb 18 2012
G.f.: x*(1+2*x-x^2-x^3)/((1-x)^3*(1+x)^2). - Klaus Brockhaus, Sep 06 2009
From Stefano Spezia, Jul 10 2020: (Start)
E.g.f.: (x*(9 + x)*cosh(x) + (-1 + 11*x + x^2)*sinh(x))/8.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 4. (End)

Edited and extended by Klaus Brockhaus, Sep 06 2009

A134519 Numbers remaining when the natural numbers (A000027) are arranged into a triangle and only the beginning and end terms of each row are retained.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 10, 11, 15, 16, 21, 22, 28, 29, 36, 37, 45, 46, 55, 56, 66, 67, 78, 79, 91, 92, 105, 106, 120, 121, 136, 137, 153, 154, 171, 172, 190, 191, 210, 211, 231, 232, 253, 254, 276, 277, 300, 301, 325, 326, 351, 352, 378, 379, 406, 407, 435, 436, 465, 466

Offset: 1

Rick L. Shepherd, Oct 29 2007

Equivalently, this is TriRet(A000027,{1}) = TriRem(A000027,{2,3,4,...}), using the operations defined in A134509. Bisections are A000217-{0} and A000124-{1}. A055802 and A114220 appear to be this sequence with two and three additional leading terms, respectively.

Muniru A Asiru, Table of n, a(n) for n = 1..10000
Bruno Berselli, An interpretation of initial terms.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000027, A000124, A000217, A055802, A057979, A114220, A134509.
Cf. A084263: A000217(m) + (1 + (-1)^m)/2.
Cf. A117142: A000217(floor(m/2)+1) - (1 + (-1)^m)/2.

a:=[];; for n in [1..60] do if n mod 2=0 then Add(a,(16+4*n+2*n^2)/16); else Add(a,(3+4*n+n^2)/8); fi; od; a; # Muniru A Asiru, Dec 21 2018

T:=func; [T(Floor((n+1)/2))+(1+(-1)^n)/2: n in [1..60]]; // Bruno Berselli, Aug 20 2019

Maple

seq(coeff(series(-x*(x^4-x^3-x^2+x+1)/((x-1)^3*(x+1)^2),x,n+1), x, n), n = 1 .. 60); # Muniru A Asiru, Dec 21 2018

Mathematica

Table[Sum[If[EvenQ[k], 1, (k - 1)/2], {k, 0, n}], {n, 60}] (* Jon Maiga, Dec 21 2018 *) LinearRecurrence[{1,2,-2,-1,1},{1,2,3,4,6},60] (* Harvey P. Dale, Oct 13 2024 *)

Formula

From Colin Barker, Jul 17 2013: (Start)

a(n) = (16 + 4*n + 2*n^2)/16 for n even, a(n) = (3 + 4*n + n^2)/8 for n odd.

G.f.: -x*(x^4 - x^3 - x^2 + x + 1) / ((x - 1)^3*(x + 1)^2). (End)

a(n) = Sum_{k=0..n-1} A057979(k). - Jon Maiga, Dec 21 2018

a(n) = A000217(floor(n+1)/2) + (1 + (-1)^n)/2. - Bruno Berselli, Aug 20 2019

cp's OEIS Frontend

A139601 Square array of polygonal numbers read by ascending antidiagonals: T(n, k) = (n + 1)*(k - 1)*k/2 + k.

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A194621 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) is the number of partitions of n in which any two parts differ by at most k.

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A004197 Triangle read by rows. T(n, k) = n - k if n - k < k, otherwise k.

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A117143 Number of partitions of n in which any two parts differ by at most 3.

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A128508 Number of partitions p of n such that max(p) - min(p) = 3.

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A123158 Square array related to Bell numbers read by antidiagonals.

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A165157 Zero followed by partial sums of A133622.

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A139601 Square array of polygonal numbers read by ascending antidiagonals: T(n, k) = (n + 1)(k - 1)k/2 + k.