A002597 -id:A002597 - cp's OEIS frontend

A104249 a(n) = (3*n^2 + n + 2)/2.

1, 3, 8, 16, 27, 41, 58, 78, 101, 127, 156, 188, 223, 261, 302, 346, 393, 443, 496, 552, 611, 673, 738, 806, 877, 951, 1028, 1108, 1191, 1277, 1366, 1458, 1553, 1651, 1752, 1856, 1963, 2073, 2186, 2302, 2421, 2543, 2668, 2796, 2927, 3061, 3198, 3338, 3481

Offset: 0

Thomas Wieder, Feb 26 2005

Second differences are all 3.
Related to the sequence of odd numbers A005408 since for these numbers the first differences are all 2.
Column 2 of A114202. - Paul Barry, Nov 17 2005
Equals third row of A167560 divided by 2. - Johannes W. Meijer, Nov 12 2009
A242357(a(n)) = n + 1. - Reinhard Zumkeller, May 11 2014
Also, this sequence is related to A011379, for n>0, by A011379(n) = n*a(n) - Sum_{i=0..n-1} a(i). - Bruno Berselli, Jul 08 2015
The number of Hamiltonian nonisomorphic unfoldings in an n-gonal Archimedean antiprism. See sequence A284647. - Rick Mabry, Apr 10 2021

			The sequence of first differences delta_a(n) = a(n+1) - a(n) is 2, 5, 8, 11, 14, 17, 20, 23, 26, ...
The sequence of second differences delta_delta_a(n) = a(n+2) - 2*a(n+1) + a(n) is: 3, 3, 3, 3, 3, 3, 3, ... E.g., 78 - 2*58 + 41 = 3.

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Rick Mabry, Fibonacci Numbers, Integer Compositions, and Nets of Antiprisms, The American Mathematical Monthly, Vol. 126 (2019), no. 9, pp. 786-801.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001399, A002597, A005408, A011379, A016777, A143689.

Counts special cases of A284647.

a104249 n = n*(3*n+1) `div` 2 + 1 -- Reinhard Zumkeller, May 11 2014

[(3*n^2+n+2)/2: n in [0..50]]; // Vincenzo Librandi, May 09 2011

a := proc (n) local i, u; option remember; u[0] := 1; u[1] := 3; u[2] := 8; for i from 3 to n do u[i] := -(4*u[i-3]-8*u[i-2]-2*u[i-1]+(-2*u[i-3]+2*u[i-2]-u[i-1])*i)/i end do; [seq(u[i],i = 0 .. n)] end proc;

A104249[n_] := (3*n^2 + n + 2)/2; Table[A104249[n], {n,0,100}] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2011 *)
LinearRecurrence[{3,-3,1},{1,3,8},70] (* Harvey P. Dale, Jul 21 2023 *)

a(n)=n*(3*n+1)/2+1 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (1 + 2*x^2)/(1 - x)^3.
Recurrence: (n+3)*u(n+3) + (-5-n)*u(n+2)*(-2+2*n)*u(n+1) + (-2-2*n)*u(n) = 0 for n >= 0 with u(0) = 1, u(1) = 3, and u(2) = 8.
From Paul Barry, Nov 17 2005: (Start)
a(0) = 1, a(n) = a(n-1) + 3*n - 1 for n > 0;
a(n) = Sum_{k=0..n} C(n, k)*C(2, k)*J(k+1), where J(n) = A001045(n). (End)
Binomial transform of [1, 2, 3, 0, 0, 0, ...]. - Gary W. Adamson, Apr 23 2008
E.g.f.: exp(x)*(2 + 4*x + 3*x^2)/2. - Stefano Spezia, Apr 10 2021

A064349 Generating function: 1/((1-x)(1-x^2)^2(1-x^3)^3*(1-x^4)^4).

Original entry on oeis.org

1, 1, 3, 6, 13, 19, 37, 58, 97, 143, 227, 328, 492, 688, 992, 1364, 1903, 2551, 3473, 4586, 6097, 7911, 10333, 13226, 16988, 21454, 27172, 33938, 42437, 52423, 64833, 79354, 97130, 117824, 142930, 172018, 206925, 247179, 295105, 350154, 415124

Offset: 0

Henry Bottomley, Sep 17 2001

nonn

Number of partitions of n into parts 1 (of one kind), 2 (of two kinds), 3 (of three kinds), and 4 (of 4 kinds). [Joerg Arndt, Jul 11 2013]

Ray Chandler, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1, 2, 1, 0, -9, -5, 2, 13, 21, -4, -17, -30, -13, 25, 28, 25, -13, -30, -17, -4, 21, 13, 2, -5, -9, 0, 1, 2, 1, -1).

The sequence of sequences A000007, A000012, A008805, A002597, A064349, etc. approaches A000219.
Essentially the same as A002598.
Cf. A002598.

a(n) = floor( ([13, 28, -44][n%3+1]+(9/2)*(n\3+2)*((n+1)%3-1)) * (n\3+1)/729 - (n\2+1)*(-1)^(n\2) * (3*[-8, 11]+(n\2+2)*(2*[-1, 3]+(n\2+3)*(1/3)*[0, 1]))[n%2+1]/512 + (2*n^9 +270*n^8 +15600*n^7 +504000*n^6 +9977730*n^5 +124629750*n^4 +973069200*n^3 +4521339000*n^2 +11137512613*n +16461579435 +5103*(n+15)*(2*n^4 +120*n^3 +2440*n^2 +19200*n +48213)*(-1)^n) / 20065812480 ) \\ Tani Akinari, Jul 12 2013

Vec(1/((1-x)*(1-x^2)^2*(1-x^3)^3*(1-x^4)^4)+O(x^66)) \\ Joerg Arndt, Jul 11 2013

A002598 A generalized partition function.

Original entry on oeis.org

1, 6, 9, 13, 19, 37, 58, 97, 143, 227, 328, 492, 688, 992, 1364, 1903, 2551, 3473, 4586, 6097, 7911, 10333, 13226, 16988, 21454, 27172, 33938, 42437, 52423, 64833, 79354, 97130, 117824, 142930, 172018, 206925, 247179, 295105, 350154, 415124, 489414, 576540

Offset: 1

N. J. A. Sloane

Hansraj Gupta, A generalization of the partition function. Proc. Nat. Inst. Sci. India 17, (1951), 231-238.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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 = 1..1000
Hansraj Gupta, A generalization of the partition function, Proc. Nat. Inst. Sci. India 17, (1951). 231-238. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (1, 2, 1, 0, -9, -5, 2, 13, 21, -4, -17, -30, -13, 25, 28, 25, -13, -30, -17, -4, 21, 13, 2, -5, -9, 0, 1, 2, 1, -1).

Essentially the same as A064349.

CoefficientList[Series[(3 x^32 - 9 x^30 - 10 x^29 - 2 x^28 + 29 x^27 + 43 x^26 + 9 x^25 - 54 x^24 - 107 x^23 - 49 x^22 + 76 x^21 + 162 x^20 + 125 x^19 - 53 x^18 - 189 x^17 - 172 x^16 - 11 x^15 + 157 x^14 + 166 x^13 + 50 x^12 - 81 x^11 - 119 x^10 - 49 x^9 + 30 x^8 + 55 x^7 + 29 x^6 - 8 x^5 - 18 x^4 - 9 x^3 + x^2 + 5 x + 1)/((x - 1)^10 (x + 1)^6 (x^2 + 1)^4 (x^2 + x + 1)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 13 2013 *)

G.f.: x*(3*x^32 -9*x^30 -10*x^29 -2*x^28 +29*x^27 +43*x^26 +9*x^25 -54*x^24 -107*x^23 -49*x^22 +76*x^21 +162*x^20 +125*x^19 -53*x^18 -189*x^17 -172*x^16 -11*x^15 +157*x^14 +166*x^13 +50*x^12 -81*x^11 -119*x^10 -49*x^9 +30*x^8 +55*x^7 +29*x^6 -8*x^5 -18*x^4 -9*x^3 +x^2 +5*x +1)/((x -1)^10*(x +1)^6*(x^2 +1)^4*(x^2 +x +1)^3). [Colin Barker, Oct 02 2012]

More terms from Vincenzo Librandi, Oct 13 2013

A002600 A generalized partition function.

Original entry on oeis.org

1, 10, 25, 37, 42, 48, 79, 145, 244, 415, 672, 1100, 1722, 2727, 4193, 6428, 9658, 14478, 21313, 31304, 45329, 65311, 93074, 132026, 185413, 259242, 359395, 495839, 679175, 926064, 1254360, 1691753, 2268267, 3028345, 4021954, 5320139, 7003154

Offset: 1

N. J. A. Sloane

nonn

Hansraj Gupta, A generalization of the partition function. Proc. Nat. Inst. Sci. India 17 (1951), 231-238.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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 = 1..1000
Hansraj Gupta, A generalization of the partition function, Proc. Nat. Inst. Sci. India 17 (1951), 231-238. [Annotated scanned copy]

J:= m-> product((1-x^j)^(-j), j=1..m): a:= t-> coeff(series(J(min(6, t)), x, 1+max(6, t)), x, max(6, t)): seq(a(n), n=1..40); # Alois P. Heinz, Jul 20 2009

J[m_] := Product[(1-x^j)^-j, {j, 1, m}]; a[t_] := SeriesCoefficient[J[Min[6, t]], {x, 0, Max[6, t]}]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Mar 13 2014, after Alois P. Heinz *)

More terms from Alois P. Heinz, Jul 20 2009

A002599 A generalized partition function.

Original entry on oeis.org

1, 6, 15, 19, 24, 42, 73, 127, 208, 337, 528, 827, 1263, 1902, 2819, 4133, 5986, 8578, 12146, 17057, 23711, 32708, 44726, 60713, 81800, 109468, 145526, 192288, 252521, 329792, 428316, 553478, 711596, 910563, 1159790, 1470798, 1857286, 2335838

Offset: 1

N. J. A. Sloane

nonn

Hansraj Gupta, A generalization of the partition function. Proc. Nat. Inst. Sci. India 17 (1951), 231-238.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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 = 1..1000
Hansraj Gupta, A generalization of the partition function, Proc. Nat. Inst. Sci. India 17 (1951), 231-238. [Annotated scanned copy]

J:= m-> product((1-x^j)^(-j), j=1..m): a:= t-> coeff(series(J(min(5, t)), x, 1+max(5, t)), x, max(5, t)): seq(a(n), n=1..40); # Alois P. Heinz, Jul 20 2009

J[m_] := Product[(1-x^j)^-j, {j, 1, m}]; a[t_] := SeriesCoefficient[J[Min[5, t]], {x, 0, Max[5, t]}]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Mar 13 2014, after Alois P. Heinz *)

More terms from Alois P. Heinz, Jul 20 2009

A002601 A generalized partition function.

Original entry on oeis.org

1, 10, 34, 58, 73, 79, 86, 152, 265, 457, 763, 1268, 2058, 3308, 5236, 8220, 12731, 19546, 29685, 44702, 66714, 98806, 145154, 211756, 306667, 441249, 630771, 896344, 1266146, 1778692, 2485086, 3454206, 4777165, 6575350, 9008159

Offset: 1

N. J. A. Sloane

nonn

Hansraj Gupta, A generalization of the partition function. Proc. Nat. Inst. Sci. India 17 (1951), 231-238.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Hansraj Gupta, A generalization of the partition function, Proc. Nat. Inst. Sci. India 17 (1951), 231-238. [Annotated scanned copy]

J:= m-> product((1-x^j)^(-j), j=1..m): a:= t-> coeff(series(J(min(7, t)), x, 1+max(7, t)), x, max(7, t)): seq(a(n), n=1..40); # Alois P. Heinz, Jul 20 2009

J[m_] := Product [(1-x^j)^-j, {j, 1, m}]; a[t_] := SeriesCoefficient[J[Min[7, t]], {x, 0, Max[7, t]}]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *)

More terms from Alois P. Heinz, Jul 20 2009

A002602 A generalized partition function.

Original entry on oeis.org

1, 15, 51, 97, 127, 145, 152, 160, 273, 481, 811, 1372, 2250, 3692, 5924, 9472, 14887, 23310, 36005, 55314, 84042, 126998, 190138, 283108, 418175, 614429, 896439, 1301168, 1876826, 2693988, 3845134, 5462744, 7720947, 10864828, 15216527

Offset: 1

N. J. A. Sloane

nonn

Hansraj Gupta, A generalization of the partition function. Proc. Nat. Inst. Sci. India 17 (1951), 231-238.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Hansraj Gupta, A generalization of the partition function, Proc. Nat. Inst. Sci. India 17 (1951), 231-238. [Annotated scanned copy]

J:= m-> product((1-x^j)^(-j), j=1..m): a:= t-> coeff(series(J(min(8, t)), x, 1+max(8, t)), x, max(8, t)): seq(a(n), n=1..40); # Alois P. Heinz, Jul 20 2009

J[m_] := Product[(1-x^j)^-j, {j, 1, m}]; a[t_] := SeriesCoefficient[J[Min[8, t]], {x, 0, Max[8, t]}]; Table[ a[n], {n, 1, 40}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *)

More terms from Alois P. Heinz, Jul 20 2009

A002603 A generalized partition function.

Original entry on oeis.org

1, 15, 73, 143, 208, 244, 265, 273, 282, 490, 838, 1426, 2367, 3908, 6356, 10246, 16327, 25812, 40379, 62748, 96660, 147833, 224446, 338584, 507293, 755612, 1118679, 1647023, 2411642, 3513096, 5091511, 7344086, 10543419, 15068833, 21442703, 30385111, 42880601

Offset: 1

N. J. A. Sloane

nonn

Hansraj Gupta, A generalization of the partition function. Proc. Nat. Inst. Sci. India 17 (1951), 231-238.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Hansraj Gupta, A generalization of the partition function, Proc. Nat. Inst. Sci. India 17 (1951), 231-238. [Annotated scanned copy]

J:= m-> product((1-x^j)^(-j), j=1..m): a:= t-> coeff(series(J(min(9, t)), x, 1+max(9, t)), x, max(9, t)): seq(a(n), n=1..40); # Alois P. Heinz, Jul 20 2009

J[m_] := Product[(1-x^j)^-j, {j, 1, m}]; a[t_] := SeriesCoefficient[J[Min[9, t]], {x, 0, Max[9, t]}]; Table[ a[n], {n, 1, 40}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *)

More terms from Alois P. Heinz, Jul 20 2009

cp's OEIS Frontend

A104249 a(n) = (3*n^2 + n + 2)/2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Formula

A064349 Generating function: 1/((1-x)*(1-x^2)^2*(1-x^3)^3*(1-x^4)^4).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

A002598 A generalized partition function.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A002600 A generalized partition function.

Views

Author

Keywords

References

Links

Programs

Maple

Mathematica

Extensions

A002599 A generalized partition function.

Views

Author

Keywords

References

Links

Programs

Maple

Mathematica

Extensions

A002601 A generalized partition function.

Views

Author

Keywords

References

Links

Programs

Maple

Mathematica

Extensions

A002602 A generalized partition function.

Views

Author

Keywords

References

Links

Programs

Maple

Mathematica

Extensions

A002603 A generalized partition function.

Views

Author

A064349 Generating function: 1/((1-x)(1-x^2)^2(1-x^3)^3*(1-x^4)^4).